Introduction
A coisotropic submanifold of a Poisson manifold admits two complementary descriptions. Globally, its conormal Lie algebroid can, under suitable integrability assumptions, be integrated to a Lagrangian subgroupoid of a symplectic groupoid integrating the ambient manifold [10 , 5 , 6 ] . Infinitesimally, its Chevalley–Eilenberg complex carries the homotopy Poisson structure of Cattaneo and Felder [7 ] . These descriptions lead to different questions: whether the integrating subgroupoid is embedded, how it intersects other Lagrangian subgroupoids, and which features of the original geometry are retained by the homotopy Poisson algebra.
Shifted geometry provides a common setting for these questions [25 , 4 , 29 , 22 , 23 ] . A symplectic groupoid presents a 1-shifted symplectic differentiable stack, and a Lagrangian subgroupoid presents a 1-shifted Lagrangian morphism [35 , 2 ] . In the differential-geometric theory developed by the second author, a clean homotopy fibre product of two such Lagrangian carries a 0-shifted Poisson structure whenever the induced Dirac structure is smooth [21 ] . In previous work, we extended the integration and quantization picture to 1-shifted coisotropics under suitable hypotheses, constructing a flat P ∞ P_\infty P ∞ A ∞ A_\infty A ∞ [1 ] . Explicit examples require an understanding of the integrating subgroupoids, the cleanness of their homotopy fibre products, and the dependence of the higher brackets on choices.
In this paper, we study these questions for curves in Radko surfaces. A Radko surface is a compact oriented Poisson surface whose bivector vanishes transversely along a nonempty union of circles D D D [26 ] . We give an explicit description of its adjoint symplectic groupoid, relating the affine local model and the global gluing to the constructions of Gualtieri and Li [14 ] . Every curve in the surface is coisotropic, and its Lagrangian integrations are 1-shifted Lagrangians, but its contacts with D D D D D D D D D
Our results describe how these contacts govern the integrating Lagrangian subgroupoids, their shifted intersections, and the information retained by their homotopy Poisson algebras. In particular, we distinguish isomorphisms of the underlying L ∞ L_\infty L ∞ L ∞ L_\infty L ∞
Let N ⊂ M N \subset M N ⊂ M a = C ∞ ( ∧ ∙ ν N ) \mathfrak{a}=C^\infty(\wedge^\bullet\nu_N) a = C ∞ ( ∧ ∙ ν N ) ψ \psi ψ N N N M M M ν N \nu_N ν N N N N ν N \nu_N ν N P ∞ P_\infty P ∞ λ ψ \lambda^\psi λ ψ a \mathfrak{a} a ( a , λ ψ ) (\mathfrak{a},\lambda^\psi) ( a , λ ψ ) homotopy Poisson algebra of ( N , ψ ) (N,\psi) ( N , ψ ) λ 1 \lambda_1 λ 1 ψ \psi ψ T N ∘ TN^\circ T N ∘ σ ψ \sigma_\psi σ ψ σ \sigma σ N N N ψ \psi ψ [7 , 32 , 8 ] .
For the statement of the criterion, recall that an L ∞ L_\infty L ∞ F F F S ( a [ 1 ] ) S(\mathfrak{a}[1]) S ( a [ 1 ]) F 1 F_1 F 1 ν N \nu_N ν N N N N N N N ν N \nu_N ν N § 2 associates to it a coalgebra automorphism with identity linear part, and an L ∞ L_\infty L ∞ induced by a change of tubular neighbourhood .
When F 1 = id F_1=\operatorname{id} F 1 = id log F = ∑ k ≥ 1 ( − 1 ) k + 1 k ( F − id ) k \log F = \sum_{k\geq1}\frac{(-1)^{k+1}}{k}(F-\operatorname{id})^k log F = ∑ k ≥ 1 k ( − 1 ) k + 1 ( F − id ) k S ( a [ 1 ] ) S(\mathfrak{a}[1]) S ( a [ 1 ]) multiderivation is a graded symmetric multilinear map on a [ 1 ] \mathfrak{a}[1] a [ 1 ] a \mathfrak{a} a
Theorem A. Let ψ , ψ ′ \psi,\psi' ψ , ψ ′ L ∞ L_\infty L ∞ F : ( a , λ ψ ) → ( a , λ ψ ′ ) F\colon(\mathfrak{a},\lambda^\psi)\to(\mathfrak{a},\lambda^{\psi'}) F : ( a , λ ψ ) → ( a , λ ψ ′ ) F 1 = id F_1=\operatorname{id} F 1 = id log F \log F log F a \mathfrak{a} a
The inducing formal change is unique. The logarithm is essential, since the Taylor coefficients of F F F L ∞ L_\infty L ∞ [30 ] . Its proof, together with the comparison with the isomorphisms of Cattaneo and Schätz [8 ] , occupies § 2 .
We now return to a Radko surface ( Σ , σ ) (\Sigma,\sigma) ( Σ , σ ) D D D [15 , 14 ] . Near a component D j D_j D j D D D ( t , r ) ∈ S ℓ j 1 × ( − ϵ j , ϵ j ) (t,r)\in S^1_{\ell_j}\times(-\epsilon_j,\epsilon_j) ( t , r ) ∈ S ℓ j 1 × ( − ϵ j , ϵ j )
σ = r ∂ ∂ r ∧ ∂ ∂ t , S ℓ j 1 = R / ℓ j Z . \sigma=r\frac{\partial}{\partial r}\wedge\frac{\partial}{\partial t},\qquad S^1_{\ell_j}=\mathbb{R}/\ell_j\mathbb{Z}. σ = r ∂ r ∂ ∧ ∂ t ∂ , S ℓ j 1 = R / ℓ j Z . The positive number ℓ j \ell_j ℓ j D j D_j D j X v o l X_{\mathrm{vol}} X vol [34 ] of σ \sigma σ D j D_j D j [26 ] . Together with the oriented degeneracy locus and a regularized volume, the modular periods classify Radko structures [26 ] .
In a symplectic surface, the questions above have simple answers. The adjoint symplectic groupoid of a connected symplectic surface is its pair groupoid, two curves always have a clean homotopy fibre product, and every curve has a tubular neighbourhood whose homotopy Poisson algebra has no brackets of arity at least two. Along the degeneracy locus of a Radko surface, the rank of σ \sigma σ 2 2 2 0 0 0 D D D D D D
Let G Σ \mathcal{G}_\Sigma G Σ ( Σ , σ ) (\Sigma,\sigma) ( Σ , σ ) § 3 , we give explicit formulas for this groupoid and its symplectic form, using the affine local model and the integration constructions of Gualtieri and Li [14 ] . Over each component of Σ ∖ D \Sigma\setminus D Σ ∖ D D D D Aff + ( R ) = { ( a , b ) : a > 0 , b ∈ R } \operatorname{Aff}^+(\mathbb{R})=\{(a,b):a>0,\ b\in\mathbb{R}\} Aff + ( R ) = {( a , b ) : a > 0 , b ∈ R } ( a 2 , b 2 ) ( a 1 , b 1 ) = ( a 2 a 1 , b 1 + a 1 b 2 ) (a_2,b_2)(a_1,b_1)=(a_2a_1,b_1+a_1b_2) ( a 2 , b 2 ) ( a 1 , b 1 ) = ( a 2 a 1 , b 1 + a 1 b 2 ) D j D_j D j G Σ \mathcal{G}_\Sigma G Σ Aff + ( R ) \operatorname{Aff}^+(\mathbb{R}) Aff + ( R ) S ℓ j 1 × R S^1_{\ell_j}\times\mathbb{R} S ℓ j 1 × R ( a , b ) ⋅ ( t , r ) = ( t + b r , a r ) (a,b)\cdot(t,r)=(t+br,ar) ( a , b ) ⋅ ( t , r ) = ( t + b r , a r ) { ( 1 , b ) : b ∈ R } \{(1,b): b\in\mathbb{R}\} {( 1 , b ) : b ∈ R } translation subgroup of Aff + ( R ) \operatorname{Aff}^{+}(\mathbb{R}) Aff + ( R ) { ( a , 0 ) : a > 0 } \{(a,0):a>0\} {( a , 0 ) : a > 0 } dilation subgroups .
Throughout, a curve is a closed embedded one-dimensional submanifold. We call it admissible if each component is contained in Σ ∖ D \Sigma\setminus D Σ ∖ D D D D D D D N N N tangent to D D D x ∈ N ∩ D x\in N\cap D x ∈ N ∩ D T x N = T x D T_xN=T_xD T x N = T x D N N N D D D x x x N ∩ D N\cap D N ∩ D D D D D D D D D D D D D N ∩ D N\cap D N ∩ D D j D_j D j
Theorem B. Let N N N T N ∘ TN^\circ T N ∘ L ⇒ N L\Rightarrow N L ⇒ N G Σ \mathcal{G}_{\Sigma} G Σ p ∈ N ∩ D p\in N\cap D p ∈ N ∩ D N N N D D D p p p
The subgroupoid L L L N ∖ D N\setminus D N ∖ D L L L N ∖ D N\setminus D N ∖ D N ∩ D N\cap D N ∩ D T N ∘ TN^\circ T N ∘ [24 ] , and the Lagrangian subgroupoid of [5 ] lives in the source-simply-connected integration. Here it is embedded in the adjoint integration, which is Hausdorff, so that L L L
Given two admissible curves N 1 N_1 N 1 N 2 N_2 N 2 L 1 L_1 L 1 L 2 L_2 L 2 L 1 × G Σ h L 2 L_1\times^h_{\mathcal{G}_{\Sigma}}L_2 L 1 × G Σ h L 2 ( x , g , y ) (x,g,y) ( x , g , y ) x ∈ N 1 x\in N_1 x ∈ N 1 y ∈ N 2 y\in N_2 y ∈ N 2 g g g G Σ \mathcal{G}_{\Sigma} G Σ x x x y y y L 1 × G Σ h L 2 L_1\times^h_{\mathcal{G}_{\Sigma}}L_2 L 1 × G Σ h L 2 N 1 N_1 N 1 N 2 N_2 N 2 D D D
Theorem C. Let N 1 N_1 N 1 N 2 N_2 N 2 L 1 × G Σ h L 2 L_1\times^h_{\mathcal{G}_{\Sigma}}L_2 L 1 × G Σ h L 2 N 1 ∩ N 2 ∩ D N_1\cap N_2\cap D N 1 ∩ N 2 ∩ D D D D D D D
This condition is not ordinary transversality of the curves in Σ \Sigma Σ L × G Σ h L L\times^h_{\mathcal{G}_{\Sigma}}L L × G Σ h L N N N D D D D D D
When it is clean, the homotopy fibre product is Morita equivalent to a disjoint union of finitely many points and, for each common component D j D_j D j D D D ( S ℓ j 1 × R ) × R (S^1_{\ell_j}\times\mathbb{R})\times\mathbb{R} ( S ℓ j 1 × R ) × R [21 Corollary 4.12] restricts to a symplectic form on this cylinder, for which the Hamiltonian vector field of a canonical function projects to the modular vector field of σ \sigma σ D j D_j D j [3 Theorems 6.2 and 6.3] .
We next compute the homotopy Poisson algebra of a component N c N_c N c a c = C ∞ ( N c ) ⊕ C ∞ ( ν N c ) \mathfrak{a}_c=C^\infty(N_c)\oplus C^\infty(\nu_{N_c}) a c = C ∞ ( N c ) ⊕ C ∞ ( ν N c ) ( a c , λ 1 ) (\mathfrak{a}_c,\lambda_1) ( a c , λ 1 ) λ 1 \lambda_1 λ 1 ( x , y ) (x,y) ( x , y ) N c = { y = 0 } N_c=\{y=0\} N c = { y = 0 } e e e ν N c \nu_{N_c} ν N c y y y
σ ψ = ( ∑ k ≥ 0 ν k ( x ) y k ) ∂ ∂ x ∧ ∂ ∂ y . \sigma_\psi=\left(\sum_{k\geq0}\nu_k(x)y^k\right)\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}. σ ψ = ( k ≥ 0 ∑ ν k ( x ) y k ) ∂ x ∂ ∧ ∂ y ∂ . The higher brackets are expressed directly in terms of the coefficients υ k \upsilon_k υ k
λ n ψ ( f , g 1 e , … , g n − 1 e ) = ( n − 1 ) ! υ n − 1 ∂ f ∂ x g 1 ⋯ g n − 1 e , \lambda_n^\psi(f,g_1e,\ldots,g_{n-1}e)=(n-1)!\upsilon_{n-1}\frac{\partial f}{\partial x}g_1\cdots g_{n-1}e, λ n ψ ( f , g 1 e , … , g n − 1 e ) = ( n − 1 )! υ n − 1 ∂ x ∂ f g 1 ⋯ g n − 1 e , where f , g 1 , … , g n − 1 ∈ C ∞ ( N c ) f,g_1,\ldots,g_{n-1}\in C^\infty(N_c) f , g 1 , … , g n − 1 ∈ C ∞ ( N c ) n ≥ 1 n\ge1 n ≥ 1 λ 1 \lambda_1 λ 1 υ 0 \upsilon_0 υ 0 υ 1 \upsilon_1 υ 1
Suppose first that N c N_c N c D D D p ∈ N c ∩ D p\in N_c\cap D p ∈ N c ∩ D m p ∈ { 1 , 2 , … } ∪ { ∞ } m_p\in\{1,2,\ldots\}\cup\{\infty\} m p ∈ { 1 , 2 , … } ∪ { ∞ } N c N_c N c D D D λ 1 \lambda_1 λ 1 L ∞ L_\infty L ∞ L ∞ L_\infty L ∞ L ∞ L_\infty L ∞ L ∞ L_\infty L ∞ ( a c , λ ψ ) → ( a c , λ 1 ) (a_c,\lambda^\psi)\to(a_c,\lambda_1) ( a c , λ ψ ) → ( a c , λ 1 ) λ 1 \lambda_1 λ 1 ψ ′ ′ \psi'' ψ ′′ ψ \psi ψ ψ ′ ′ \psi'' ψ ′′
Theorem D. Let N c N_c N c N c ⊄ D N_c\not\subset D N c ⊂ D ψ \psi ψ N c N_c N c
(1) The algebra ( a c , λ ψ ) (a_c,\lambda^\psi) ( a c , λ ψ ) L ∞ L_\infty L ∞ ( a c , λ 1 ) (a_c,\lambda_1) ( a c , λ 1 ) L ∞ L_\infty L ∞ ∑ p m p ∈ { 0 , 1 , 2 , … } ∪ { ∞ } \sum_p m_p\in\{0,1,2,\ldots\}\cup\{\infty\} ∑ p m p ∈ { 0 , 1 , 2 , … } ∪ { ∞ }
(2) An L ∞ L_\infty L ∞ ( a c , λ ψ ) → ( a c , λ 1 ) (a_c,\lambda^\psi)\to(a_c,\lambda_1) ( a c , λ ψ ) → ( a c , λ 1 ) N c N_c N c D D D ψ \psi ψ λ 2 ψ ( f , v ) = X ( f ) v \lambda_2^\psi(f,v)=X(f)v λ 2 ψ ( f , v ) = X ( f ) v X X X N c N_c N c X ( p ) = X v o l ( p ) X(p)=X_{\mathrm{vol}}(p) X ( p ) = X vol ( p )
Here dim H 1 ( a c , λ 1 ) = 1 + ∑ p m p \dim H^1(a_c,\lambda_1)=1+\sum_p m_p dim H 1 ( a c , λ 1 ) = 1 + ∑ p m p H 1 H^1 H 1 λ n = 0 \lambda_n=0 λ n = 0 n ≥ 3 n\ge3 n ≥ 3 ∣ N c ∩ D ∣ = 2 k |N_c\cap D|=2k ∣ N c ∩ D ∣ = 2 k θ ∈ R / 2 π Z \theta\in\mathbb{R}/2\pi\mathbb{Z} θ ∈ R /2 π Z e e e λ 1 ( f ) = sin ( k θ ) ∂ f ∂ θ e \lambda_1(f)=\sin(k\theta)\frac{\partial f}{\partial\theta}e λ 1 ( f ) = sin ( k θ ) ∂ θ ∂ f e λ 1 ( f ) = ∂ f ∂ θ e \lambda_1(f)=\frac{\partial f}{\partial\theta}e λ 1 ( f ) = ∂ θ ∂ f e k = 0 k=0 k = 0 b b b [15 Theorem 27] and with the b b b [12 ] . At a tangential contact of finite multiplicity m p ≥ 2 m_p\ge2 m p ≥ 2 m p − 2 m_p-2 m p − 2 X X X p p p m p − 1 m_p-1 m p − 1
A component N c = D j N_c=D_j N c = D j [13 ] . Its differential λ 1 \lambda_1 λ 1 λ 2 ( f , v ) = X v o l ( f ) v \lambda_2(f,v)=X_{\mathrm{vol}}(f)v λ 2 ( f , v ) = X vol ( f ) v L ∞ L_\infty L ∞ L ∞ L_\infty L ∞ ( a c , λ ψ ) (a_c,\lambda^\psi) ( a c , λ ψ ) L ∞ L_\infty L ∞ L ∞ L_\infty L ∞ X v o l ∣ D j X_{\mathrm{vol}|D_j} X vol ∣ D j
For a connected admissible curve not contained in D D D L ∞ L_\infty L ∞ T N ∘ TN^\circ T N ∘ D D D Theorem A retain more, namely the modular vector field at the tangential contacts, which is invisible to the abstract classification but obstructs geometric linearization. The global structures retain the modular periods, through the canonical function on the shifted self-intersections of the degeneracy curves, whose Hamiltonian flow has period ℓ j \ell_j ℓ j
Organization of the paper. In § 2 , we recall the homotopy Poisson algebra of a coisotropic submanifold, study changes of tubular neighbourhood and prove Theorem A . In § 3 , we describe the adjoint symplectic groupoid of a Radko surface. In § 4 , we construct the Lagrangian subgroupoids of admissible curves and prove Theorem B . In § 5 , we determine the clean homotopy fibre products of these subgroupoids and their induced structures, and prove Theorem C . Finally, in § 6 , we compute and classify the homotopy Poisson algebras of curves and prove Theorem D.
Acknowledgments. Y.A.M. thanks the Institut des sciences mathématiques (ISM) for financial support. M.M. acknowledges support from a Discovery Grant (RGPIN-2023-04587) from the Natural Sciences and Engineering Research Council of Canada (NSERC).
Homotopy Poisson algebras and changes of tubular neighbourhood
In this section, we characterize the L ∞ L_\infty L ∞ Theorem A . Let ( M , σ ) (M,\sigma) ( M , σ ) N ⊂ M N \subset M N ⊂ M
ν = T M ∣ N / T N , a = C ∞ ( ⋀ ν ) , a [ 1 ] k = a k + 1 . \nu=TM|_N/TN,\qquad\mathfrak{a}=C^\infty(\bigwedge\nu),\qquad\mathfrak{a}[1]^k=\mathfrak{a}^{k+1}. ν = T M ∣ N / T N , a = C ∞ ( ⋀ ν ) , a [ 1 ] k = a k + 1 . A tubular neighbourhood of N N N a \mathfrak{a} a
Coalgebras and ∞ \infty ∞
All vector spaces are real and Z \mathbb{Z} Z [ T 1 , T 2 ] = T 1 ∘ T 2 − ( − 1 ) ∣ T 1 ∣ ∣ T 2 ∣ T 2 ∘ T 1 [T_1,T_2]=T_1\circ T_2-(-1)^{|T_1||T_2|}T_2\circ T_1 [ T 1 , T 2 ] = T 1 ∘ T 2 − ( − 1 ) ∣ T 1 ∣∣ T 2 ∣ T 2 ∘ T 1 V V V
S ( V ) = ⨁ n ≥ 0 S n V , S ≤ n V = ⨁ k = 0 n S k V . S(V)=\bigoplus_{n\geq0}S^nV,\qquad S^{\leq n}V=\bigoplus_{k=0}^{n}S^kV. S ( V ) = n ≥ 0 ⨁ S n V , S ≤ n V = k = 0 ⨁ n S k V . We equip S ( V ) S(V) S ( V ) Δ \Delta Δ V V V [18 , 20 ] . An element of S n V S^nV S n V length n n n .
Coalgebra morphisms will have degree zero and preserve the counit and coaugmentation. Such a morphism F : S ( V ) → S ( W ) F:S(V)\to S(W) F : S ( V ) → S ( W ) Q Q Q S ( V ) S(V) S ( V ) F n = pr W ∘ F ∣ S n V F_n=\operatorname{pr}_W\circ F|_{S^nV} F n = pr W ∘ F ∣ S n V Q n = pr V ∘ Q ∣ S n V Q_n=\operatorname{pr}_V\circ Q|_{S^nV} Q n = pr V ∘ Q ∣ S n V F 0 = 0 F_0=0 F 0 = 0 Q 0 ∈ V Q_0\in V Q 0 ∈ V F 0 = 0 F_0=0 F 0 = 0 [18 §2 and Lemma 2.4] , [17 §4.1] . Explicitly,
F ( v 1 ⋯ v n ) = ∑ B ε ( B ) F # B 1 ( v B 1 ) ⋯ F # B k ( v B k ) , (1) F(v_1\cdots v_n)=\sum_{\mathcal{B}}\varepsilon(\mathcal{B})F_{\#B_1}(v_{B_1})\cdots F_{\#B_k}(v_{B_k}), \tag*{(1)} F ( v 1 ⋯ v n ) = B ∑ ε ( B ) F # B 1 ( v B 1 ) ⋯ F # B k ( v B k ) , (1) where B = { B 1 , … , B k } \mathcal{B}=\{B_1,\ldots,B_k\} B = { B 1 , … , B k } { 1 , … , n } \{1,\ldots,n\} { 1 , … , n } v B v_B v B ε ( B ) \varepsilon(\mathcal{B}) ε ( B )
Q ( v 1 ⋯ v n ) = ∑ I ⊂ { 1 , … , n } ε ( I ) Q # I ( v I ) v I c , (2) Q(v_1\cdots v_n)=\sum_{I\subset\{1,\ldots,n\}}\varepsilon(I)Q_{\#I}(v_I)v_{I^c}, \tag*{(2)} Q ( v 1 ⋯ v n ) = I ⊂ { 1 , … , n } ∑ ε ( I ) Q # I ( v I ) v I c , (2) where ε ( I ) \varepsilon(I) ε ( I ) v 1 ⋯ v n ↦ v I v I c v_1\cdots v_n\mapsto v_Iv_{I^c} v 1 ⋯ v n ↦ v I v I c I = ∅ I=\varnothing I = ∅ Q 0 = Q 0 ( 1 ) Q_0=Q_0(1) Q 0 = Q 0 ( 1 )
We use the symmetric convention: an L ∞ L_\infty L ∞ V V V Q Q Q 1 1 1 Q ∘ Q = 0 Q\circ Q=0 Q ∘ Q = 0 Q 0 = 0 Q_0=0 Q 0 = 0 L ∞ L_\infty L ∞ F : ( V , Q ) → ( V ′ , Q ′ ) F:(V,Q)\to(V',Q') F : ( V , Q ) → ( V ′ , Q ′ ) L ∞ L_\infty L ∞ F 0 = 0 F_0=0 F 0 = 0 F ∘ Q = Q ′ ∘ F F\circ Q=Q'\circ F F ∘ Q = Q ′ ∘ F L ∞ L_\infty L ∞ F 1 F_1 F 1 [8 §2.1, Definition 2.2 and §3.1] .
When applying this convention to a graded algebra a \mathfrak{a} a S ( a [ 1 ] ) S(\mathfrak{a}[1]) S ( a [ 1 ]) a \mathfrak{a} a deg b \deg b deg b b ∈ a b\in\mathfrak{a} b ∈ a ∣ b ∣ = deg b − 1 |b|=\deg b-1 ∣ b ∣ = deg b − 1 a [ 1 ] \mathfrak{a}[1] a [ 1 ]
Definition 2.1. Let n ≥ 1 n\geq1 n ≥ 1 l : S n ( a [ 1 ] ) → a [ 1 ] l:S^n(\mathfrak{a}[1])\to\mathfrak{a}[1] l : S n ( a [ 1 ]) → a [ 1 ] d d d multiderivation of a \mathfrak{a} a b , c ∈ a b,c\in\mathfrak{a} b , c ∈ a a 2 , … , a n ∈ a [ 1 ] a_2,\ldots,a_n\in\mathfrak{a}[1] a 2 , … , a n ∈ a [ 1 ]
l ( b c , a 2 , … , a n ) = ( − 1 ) deg c ( ∣ a 2 ∣ + ⋯ + ∣ a n ∣ ) l ( b , a 2 , … , a n ) c + ( − 1 ) d deg b b l ( c , a 2 , … , a n ) . (3) l(bc,a_2,\ldots,a_n)=(-1)^{\deg c(|a_2|+\cdots+|a_n|)}l(b,a_2,\ldots,a_n)c+(-1)^{d\deg b}bl(c,a_2,\ldots,a_n). \tag*{(3)} l ( b c , a 2 , … , a n ) = ( − 1 ) d e g c ( ∣ a 2 ∣ + ⋯ + ∣ a n ∣ ) l ( b , a 2 , … , a n ) c + ( − 1 ) d d e g b b l ( c , a 2 , … , a n ) . (3) A flat L ∞ L_\infty L ∞ Q Q Q a [ 1 ] \mathfrak{a}[1] a [ 1 ] P ∞ P_\infty P ∞ a \mathfrak{a} a Q n Q_n Q n n ≥ 1 n\geq1 n ≥ 1
By graded symmetry, a multiderivation satisfies a Leibniz rule in each argument,
l ( … , b c , … ) = ± l ( … , b , … ) c ± b l ( … , c , … ) , (4) l(\ldots,bc,\ldots)=\pm l(\ldots,b,\ldots)c\pm bl(\ldots,c,\ldots), \tag*{(4)} l ( … , b c , … ) = ± l ( … , b , … ) c ± b l ( … , c , … ) , (4) with signs depending only on the degrees. For a = C ∞ ( ∧ ∙ ν ) a = C^\infty(\wedge^\bullet\nu) a = C ∞ ( ∧ ∙ ν ) x μ x^\mu x μ N N N ( e i ) (e_i) ( e i ) ν \nu ν x μ x^\mu x μ e i e_i e i
Let Aut 1 ( a ) \operatorname{Aut}_1(a) Aut 1 ( a ) F : S ( a [ 1 ] ) → S ( a [ 1 ] ) F:S(a[1])\to S(a[1]) F : S ( a [ 1 ]) → S ( a [ 1 ]) F 0 = 0 F_0=0 F 0 = 0 F 1 = id F_1=\operatorname{id} F 1 = id ∞ \infty ∞ [11 Chapter 3] , and Coder ≥ 2 ( a ) \operatorname{Coder}_{\geq2}(a) Coder ≥ 2 ( a ) δ \delta δ S ( a [ 1 ] ) S(a[1]) S ( a [ 1 ]) 0 0 0 δ 0 = δ 1 = 0 \delta_0=\delta_1=0 δ 0 = δ 1 = 0 (1) and (2) , F − id F-\operatorname{id} F − id δ \delta δ S n ( a [ 1 ] ) S^n(a[1]) S n ( a [ 1 ]) S ≤ n − 1 ( a [ 1 ] ) S^{\leq n-1}(a[1]) S ≤ n − 1 ( a [ 1 ]) S ≤ 1 ( a [ 1 ] ) S^{\leq1}(a[1]) S ≤ 1 ( a [ 1 ])
( F − id ) n ∣ S ≤ n ( a [ 1 ] ) = 0 , δ n ∣ S ≤ n ( a [ 1 ] ) = 0. (5) (F-\operatorname{id})^n\big|_{S^{\leq n}(a[1])}=0,\qquad\delta^n\big|_{S^{\leq n}(a[1])}=0. \tag*{(5)} ( F − id ) n S ≤ n ( a [ 1 ]) = 0 , δ n S ≤ n ( a [ 1 ]) = 0. (5) Hence the series log F = ∑ k ≥ 1 ( − 1 ) k + 1 k ( F − id ) k \log F=\sum_{k\geq1}\frac{(-1)^{k+1}}{k}(F-\operatorname{id})^k log F = ∑ k ≥ 1 k ( − 1 ) k + 1 ( F − id ) k e δ = ∑ k ≥ 0 1 k ! δ k e^\delta=\sum_{k\geq0}\frac{1}{k!}\delta^k e δ = ∑ k ≥ 0 k ! 1 δ k
The set Aut 1 ( a ) \operatorname{Aut}_1(a) Aut 1 ( a ) [11 Chapter 3] . To see that log F \log F log F
Δ ∘ ( F − id ) k = ( F ⊗ F − id ⊗ id ) k ∘ Δ , \Delta\circ(F-\operatorname{id})^k=(F\otimes F-\operatorname{id}\otimes\operatorname{id})^k\circ\Delta, Δ ∘ ( F − id ) k = ( F ⊗ F − id ⊗ id ) k ∘ Δ , log ( F ⊗ F ) = log ( F ⊗ id ) + log ( id ⊗ F ) . \log(F\otimes F)=\log(F\otimes\operatorname{id})+\log(\operatorname{id}\otimes F). log ( F ⊗ F ) = log ( F ⊗ id ) + log ( id ⊗ F ) . The second identity follows because F ⊗ id F\otimes\operatorname{id} F ⊗ id id ⊗ F \operatorname{id}\otimes F id ⊗ F
Δ ∘ log F = ( ( log F ) ⊗ id + id ⊗ ( log F ) ) ∘ Δ . \Delta\circ\log F=((\log F)\otimes\operatorname{id}+\operatorname{id}\otimes(\log F))\circ\Delta. Δ ∘ log F = (( log F ) ⊗ id + id ⊗ ( log F )) ∘ Δ. Thus log F ∈ Coder ≥ 2 ( a ) \log F\in\operatorname{Coder}_{\geq2}(a) log F ∈ Coder ≥ 2 ( a ) δ ∈ Coder ≥ 2 ( a ) \delta\in\operatorname{Coder}_{\geq2}(a) δ ∈ Coder ≥ 2 ( a ) Δ ∘ e δ = ( e δ ⊗ e δ ) ∘ Δ \Delta\circ e^\delta=(e^\delta\otimes e^\delta)\circ\Delta Δ ∘ e δ = ( e δ ⊗ e δ ) ∘ Δ e δ e^\delta e δ S ≤ 1 ( a [ 1 ] ) S^{\leq1}(a[1]) S ≤ 1 ( a [ 1 ]) Aut 1 ( a ) \operatorname{Aut}_1(a) Aut 1 ( a ) exp ∘ log = id \exp\circ\log=\operatorname{id} exp ∘ log = id log ∘ exp = id \log\circ\exp=\operatorname{id} log ∘ exp = id
log : Aut 1 ( a ) ⟶ Coder ≥ 2 ( a ) , log − 1 = exp . (6) \log:\operatorname{Aut}_1(a)\longrightarrow\operatorname{Coder}_{\geq2}(a),\qquad\log^{-1}=\exp. \tag*{(6)} log : Aut 1 ( a ) ⟶ Coder ≥ 2 ( a ) , log − 1 = exp . (6) Taking the components of length one in log F = ( F − id ) − 1 2 ( F − id ) 2 + ⋯ \log F=(F-\operatorname{id})-\frac{1}{2}(F-\operatorname{id})^2+\cdots log F = ( F − id ) − 2 1 ( F − id ) 2 + ⋯ (1) gives, for F ∈ Aut 1 ( a ) F\in\operatorname{Aut}_1(a) F ∈ Aut 1 ( a )
( log F ) 2 ( a 1 , a 2 ) = F 2 ( a 1 , a 2 ) , (7) (\log F)_2(a_1,a_2)=F_2(a_1,a_2), \tag*{(7)} ( log F ) 2 ( a 1 , a 2 ) = F 2 ( a 1 , a 2 ) , (7) ( log F ) 3 ( a 1 , a 2 , a 3 ) = F 3 ( a 1 , a 2 , a 3 ) − 1 2 ∑ ε F 2 ( F 2 ( a i , a j ) , a k ) , (8) (\log F)_3(a_1,a_2,a_3)=F_3(a_1,a_2,a_3)-\frac{1}{2}\sum_{\varepsilon}F_2(F_2(a_i,a_j),a_k), \tag*{(8)} ( log F ) 3 ( a 1 , a 2 , a 3 ) = F 3 ( a 1 , a 2 , a 3 ) − 2 1 ε ∑ F 2 ( F 2 ( a i , a j ) , a k ) , (8) where the sum runs over the decompositions { 1 , 2 , 3 } = { i < j } ⊔ { k } \{1,2,3\}=\{i<j\}\sqcup\{k\} { 1 , 2 , 3 } = { i < j } ⊔ { k } ε \varepsilon ε a 1 a 2 a 3 ↦ a i a j a k a_1a_2a_3\mapsto a_ia_ja_k a 1 a 2 a 3 ↦ a i a j a k
Since all series are finite on each S ≤ n ( a [ 1 ] ) S^{\leq n}(a[1]) S ≤ n ( a [ 1 ]) δ , δ ′ ∈ Coder ≥ 2 ( a ) \delta,\delta'\in\operatorname{Coder}_{\geq2}(a) δ , δ ′ ∈ Coder ≥ 2 ( a ) Q Q Q Q 0 = 0 Q_0=0 Q 0 = 0
e δ ∘ Q ∘ e − δ = ∑ k ≥ 0 1 k ! ad δ k ( Q ) , e δ ∘ e δ ′ = e BCH ( δ , δ ′ ) . (9) \begin{aligned}
e^\delta\circ Q\circ e^{-\delta}&=\sum_{k\geq0}\frac{1}{k!}\operatorname{ad}_\delta^k(Q),\\
e^\delta\circ e^{\delta'}&=e^{\operatorname{BCH}(\delta,\delta')}. \tag*{(9)}
\end{aligned} e δ ∘ Q ∘ e − δ e δ ∘ e δ ′ = k ≥ 0 ∑ k ! 1 ad δ k ( Q ) , = e BCH ( δ , δ ′ ) . (9) where BCH is the Baker–Campbell–Hausdorff series, and the series converge coefficientwise, since ad δ \operatorname{ad}_\delta ad δ
The homotopy Poisson algebra of a coisotropic submanifold
We now recall the higher-derived-bracket construction in the formal neighbourhood of N N N
Let I \mathcal I I ν \nu ν formal functions along N N N
O ^ = l i m ← k ≥ 1 C ∞ ( tot ( ν ) ) / I k . \widehat{\mathcal O}=\varprojlim_{k\geq1}C^\infty(\operatorname{tot}(\nu))/\mathcal I^k. O = k ≥ 1 lim C ∞ ( tot ( ν )) / I k . We use the same symbol I \mathcal I I O ^ → C ∞ ( N ) \widehat{\mathcal O}\to C^\infty(N) O → C ∞ ( N ) X ∙ ( ν ) \mathfrak X^\bullet(\nu) X ∙ ( ν ) ν \nu ν x μ x^\mu x μ N N N ( e i ) (e_i) ( e i ) ν \nu ν y i y^i y i y y y x x x N N N ν \nu ν X ( ν ) \mathfrak X(\nu) X ( ν ) O ^ \widehat{\mathcal O} O
We write h = X ( ν ) [ 1 ] \mathfrak h=\mathfrak X(\nu)[1] h = X ( ν ) [ 1 ] k k k ∣ u ∣ = k − 1 |u|=k-1 ∣ u ∣ = k − 1 h \mathfrak h h
[ X , f ] = X ( f ) , [ X , Y ] = X ∘ Y − Y ∘ X , (10) [X,f]=X(f),\qquad[X,Y]=X\circ Y-Y\circ X, \tag*{(10)} [ X , f ] = X ( f ) , [ X , Y ] = X ∘ Y − Y ∘ X , (10) for formal vector fields X , Y X,Y X , Y f f f [ X , u ] [X,u] [ X , u ] u u u X X X
[ u , f ] = ( − 1 ) ∣ u ∣ ι d f u . (11) [u,f]=(-1)^{|u|}\iota_{df}u. \tag*{(11)} [ u , f ] = ( − 1 ) ∣ u ∣ ι df u . (11) where ι d f ( X ∧ Y ) = X ( f ) Y − Y ( f ) X \iota_{df}(X\wedge Y)=X(f)Y-Y(f)X ι df ( X ∧ Y ) = X ( f ) Y − Y ( f ) X
Definition 2.2. Let E \mathcal{E} E ν \nu ν u u u w w w [ E , u ] = w u [\mathcal{E},u]=wu [ E , u ] = w u
In coordinates, x μ x^\mu x μ ∂ ∂ x μ \frac{\partial}{\partial x^\mu} ∂ x μ ∂ 0 0 0 y i y^i y i 1 1 1 ∂ ∂ y i \frac{\partial}{\partial y^i} ∂ y i ∂ − 1 -1 − 1 u ( w ) u_{(w)} u ( w ) w w w u u u X ≥ k ( ν ) \mathfrak{X}_{\geq k}(\nu) X ≥ k ( ν ) Z Z Z Z ( w ) = 0 Z_{(w)}=0 Z ( w ) = 0 w < k w<k w < k
A k k k ≥ − k \geq-k ≥ − k X ≥ 1 ( ν ) \mathfrak{X}_{\geq1}(\nu) X ≥ 1 ( ν ) ad Z \operatorname{ad}_Z ad Z Z ∈ X ≥ 1 ( ν ) Z\in\mathfrak{X}_{\geq1}(\nu) Z ∈ X ≥ 1 ( ν )
Z = Z μ ∂ ∂ x μ + Z i ∂ ∂ y i ∈ X ≥ 1 ( ν ) ⟺ Z μ ∈ I and Z i ∈ I 2 . (12) Z=Z^\mu\frac{\partial}{\partial x^\mu}+Z^i\frac{\partial}{\partial y^i}\in\mathfrak{X}_{\geq1}(\nu)\Longleftrightarrow Z^\mu\in\mathcal{I}\text{ and }Z^i\in\mathcal{I}^2. \tag*{(12)} Z = Z μ ∂ x μ ∂ + Z i ∂ y i ∂ ∈ X ≥ 1 ( ν ) ⟺ Z μ ∈ I and Z i ∈ I 2 . (12) Definition 2.3. A formal diffeomorphism φ \varphi φ ν \nu ν N N N R \mathbb{R} R φ ∗ \varphi^* φ ∗ O ^ \widehat{\mathcal{O}} O ( φ ∘ φ ′ ) ∗ = φ ′ ∗ ∘ φ ∗ (\varphi\circ\varphi')^*=\varphi'^*\circ\varphi^* ( φ ∘ φ ′ ) ∗ = φ ′ ∗ ∘ φ ∗ φ ∗ \varphi^* φ ∗ O ^ / I = C ∞ ( N ) \widehat{\mathcal{O}}/\mathcal{I}=C^\infty(N) O / I = C ∞ ( N ) I / I 2 = C ∞ ( ν ∗ ) \mathcal{I}/\mathcal{I}^2=C^\infty(\nu^*) I / I 2 = C ∞ ( ν ∗ ) Diff 1 ( ν ) \operatorname{Diff}_1(\nu) Diff 1 ( ν )
In coordinates, the condition reads φ ∗ ( x μ ) − x μ ∈ I \varphi^*(x^\mu)-x^\mu\in\mathcal{I} φ ∗ ( x μ ) − x μ ∈ I φ ∗ ( y i ) − y i ∈ I 2 \varphi^*(y^i)-y^i\in\mathcal{I}^2 φ ∗ ( y i ) − y i ∈ I 2 γ \gamma γ N N N ν \nu ν f ↦ f ∘ γ f\mapsto f\circ\gamma f ↦ f ∘ γ Diff 1 ( ν ) \operatorname{Diff}_1(\nu) Diff 1 ( ν ) expansion of γ \gamma γ φ ∈ Diff 1 ( ν ) \varphi\in\operatorname{Diff}_1(\nu) φ ∈ Diff 1 ( ν ) X ∙ ( ν ) \mathfrak{X}^\bullet(\nu) X ∙ ( ν ) φ ∗ \varphi^* φ ∗ φ ∗ X = φ ∗ ∘ X ∘ ( φ ∗ ) − 1 \varphi^*X=\varphi^*\circ X\circ(\varphi^*)^{-1} φ ∗ X = φ ∗ ∘ X ∘ ( φ ∗ ) − 1 γ \gamma γ γ ∗ = ( γ − 1 ) ∗ \gamma^*=(\gamma^{-1})_* γ ∗ = ( γ − 1 ) ∗ γ ∗ \gamma_* γ ∗
For Z ∈ X ≥ 1 ( ν ) Z\in\mathfrak{X}_{\geq1}(\nu) Z ∈ X ≥ 1 ( ν ) e Z e^Z e Z exp ( Z ) \exp(Z) exp ( Z ) log φ \log\varphi log φ log ( φ ∗ ) \log(\varphi^*) log ( φ ∗ )
Lemma 2.4. (1) For Z ∈ X ≥ 1 ( ν ) Z\in\mathfrak{X}_{\geq1}(\nu) Z ∈ X ≥ 1 ( ν ) e Z = ∑ k 1 k ! Z k e^Z=\sum_k\frac{1}{k!}Z^k e Z = ∑ k k ! 1 Z k O ^ \widehat{\mathcal{O}} O Diff 1 ( ν ) \operatorname{Diff}_1(\nu) Diff 1 ( ν )
(2) The map exp : X ≥ 1 ( ν ) → Diff 1 ( ν ) \exp:\mathfrak{X}_{\geq1}(\nu)\to\operatorname{Diff}_1(\nu) exp : X ≥ 1 ( ν ) → Diff 1 ( ν ) log \log log
(3) For φ = e Z \varphi=e^Z φ = e Z u ∈ X ∙ ( ν ) u\in\mathfrak{X}^\bullet(\nu) u ∈ X ∙ ( ν )
φ ∗ u = e ad Z u . (13) \varphi^*u=e^{\operatorname{ad}_Z}u. \tag*{(13)} φ ∗ u = e ad Z u . (13) Proof. By (12) , Z Z Z I w \mathcal{I}^w I w I w + 1 \mathcal{I}^{w+1} I w + 1 e Z e^Z e Z e − Z e^{-Z} e − Z Diff 1 ( ν ) \operatorname{Diff}_1(\nu) Diff 1 ( ν ) φ ∈ Diff 1 ( ν ) \varphi\in\operatorname{Diff}_1(\nu) φ ∈ Diff 1 ( ν ) f ∈ I w f\in\mathcal{I}^w f ∈ I w ∑ ∣ β ∣ = w y β f β \sum_{|\beta|=w}y^\beta f_\beta ∑ ∣ β ∣ = w y β f β φ ∗ − id \varphi^*-\operatorname{id} φ ∗ − id I w \mathcal{I}^w I w I w + 1 \mathcal{I}^{w+1} I w + 1 log φ ∗ \log\varphi^* log φ ∗ I k \mathcal{I}^k I k A = φ ∗ − id A=\varphi^*-\operatorname{id} A = φ ∗ − id ( id + A ) t (\operatorname{id}+A)^t ( id + A ) t t t t t t t t t t t = 0 t=0 t = 0 log φ ∗ \log\varphi^* log φ ∗ O ^ \widehat{\mathcal{O}} O x μ x^\mu x μ y i y^i y i X ≥ 1 ( ν ) \mathfrak{X}_{\geq1}(\nu) X ≥ 1 ( ν ) (12) . For (3), both sides are automorphisms of X ∙ ( ν ) \mathfrak{X}^\bullet(\nu) X ∙ ( ν ) (10) .
By Definition 2.3 , ( e Z ∘ e Z ′ ) ∗ = e Z ′ e Z (e^Z\circ e^{Z'})^*=e^{Z'}e^Z ( e Z ∘ e Z ′ ) ∗ = e Z ′ e Z O ^ \widehat{\mathcal{O}} O
e Z ∘ e Z ′ = e BCH ( Z ′ , Z ) . (14) e^Z\circ e^{Z'}=e^{\operatorname{BCH}(Z',Z)}. \tag*{(14)} e Z ∘ e Z ′ = e BCH ( Z ′ , Z ) . (14) and the Lie algebra of Diff 1 ( ν ) \operatorname{Diff}_1(\nu) Diff 1 ( ν ) X ≥ 1 ( ν ) \mathfrak{X}_{\geq1}(\nu) X ≥ 1 ( ν )
A tubular neighbourhood of N N N ψ \psi ψ M M M ν \nu ν N N N ν \nu ν σ ψ ∈ X 2 ( ν ) \sigma_\psi\in\mathfrak{X}^2(\nu) σ ψ ∈ X 2 ( ν ) N N N ψ ∗ σ = ( ψ − 1 ) ∗ σ \psi^*\sigma=(\psi^{-1})_*\sigma ψ ∗ σ = ( ψ − 1 ) ∗ σ transition φ ψ ψ ′ ∈ Diff 1 ( ν ) \varphi_{\psi\psi'}\in\operatorname{Diff}_1(\nu) φ ψ ψ ′ ∈ Diff 1 ( ν ) ψ − 1 ∘ ψ ′ \psi^{-1}\circ\psi' ψ − 1 ∘ ψ ′
φ ψ ψ ′ ∗ σ ψ = σ ψ ′ . (15) \varphi_{\psi\psi'}^*\sigma_\psi=\sigma_{\psi'}. \tag*{(15)} φ ψ ψ ′ ∗ σ ψ = σ ψ ′ . (15) We write
Diff 1 ( ψ , ψ ′ ) = { φ ∈ Diff 1 ( ν ) : φ ∗ σ ψ = σ ψ ′ } . \operatorname{Diff}_1(\psi,\psi')=\{\varphi\in\operatorname{Diff}_1(\nu):\varphi^*\sigma_\psi=\sigma_{\psi'}\}. Diff 1 ( ψ , ψ ′ ) = { φ ∈ Diff 1 ( ν ) : φ ∗ σ ψ = σ ψ ′ } . We now embed a [ 1 ] \mathfrak{a}[1] a [ 1 ] h \mathfrak{h} h [7 §2.6] and [8 §4.1] . A section a = a i 1 ⋯ i k ( x ) e i 1 ∧ ⋯ ∧ e i k a=a^{i_1\cdots i_k}(x)e_{i_1}\wedge\cdots\wedge e_{i_k} a = a i 1 ⋯ i k ( x ) e i 1 ∧ ⋯ ∧ e i k Λ k ν \Lambda^k\nu Λ k ν k k k a i 1 ⋯ i k ( x ) ∂ ∂ y i 1 ∧ ⋯ ∧ ∂ ∂ y i k a^{i_1\cdots i_k}(x)\frac{\partial}{\partial y^{i_1}}\wedge\cdots\wedge\frac{\partial}{\partial y^{i_k}} a i 1 ⋯ i k ( x ) ∂ y i 1 ∂ ∧ ⋯ ∧ ∂ y i k ∂ P : h → a [ 1 ] P:\mathfrak{h}\to\mathfrak{a}[1] P : h → a [ 1 ] N N N Λ ∙ ν \Lambda^\bullet\nu Λ ∙ ν P P P y = 0 y=0 y = 0 ∂ ∂ x μ \frac{\partial}{\partial x^\mu} ∂ x μ ∂ p = ker P \mathfrak{p}=\ker P p = ker P ( h , a [ 1 ] , P ) (\mathfrak{h},\mathfrak{a}[1],P) ( h , a [ 1 ] , P ) [8 Definition 2.3] : a [ 1 ] \mathfrak{a}[1] a [ 1 ] p \mathfrak{p} p h = a [ 1 ] ⊕ p \mathfrak{h}=\mathfrak{a}[1]\oplus\mathfrak{p} h = a [ 1 ] ⊕ p
Definition 2.5. For u ∈ h u\in\mathfrak{h} u ∈ h D u \mathcal{D}_u D u S ( a [ 1 ] ) S(\mathfrak{a}[1]) S ( a [ 1 ])
( D u ) 0 = P ( u ) , ( D u ) n ( a 1 , … , a n ) = P [ ⋯ [ [ u , a 1 ] , a 2 ] , … , a n ] , n ≥ 1. (\mathcal{D}_u)_0=P(u),\qquad(\mathcal{D}_u)_n(a_1,\ldots,a_n)=P[\cdots[[u,a_1],a_2],\ldots,a_n],\quad n\geq1. ( D u ) 0 = P ( u ) , ( D u ) n ( a 1 , … , a n ) = P [ ⋯ [[ u , a 1 ] , a 2 ] , … , a n ] , n ≥ 1. These coefficients are graded symmetric, since a [ 1 ] \mathfrak{a}[1] a [ 1 ] D u \mathcal{D}u D u ∣ u ∣ |u| ∣ u ∣ e i e_i e i x μ x^\mu x μ e i e_i e i x μ x^\mu x μ (11) ,
[ u , e i ] = − ∂ u ∂ y i , [ u , x μ ] = ( − 1 ) ∣ u ∣ ι d x μ u , (16) [u,e_i] = -\frac{\partial u}{\partial y^i}, \qquad[u,x^\mu] = (-1)^{|u|}\iota_{dx^\mu}u, \tag*{(16)} [ u , e i ] = − ∂ y i ∂ u , [ u , x μ ] = ( − 1 ) ∣ u ∣ ι d x μ u , (16) where ∂ u ∂ y i \frac{\partial u}{\partial y^i} ∂ y i ∂ u u u u u ∈ h u\in\mathfrak{h} u ∈ h [ D u , D u ] = D [ u , u ] [\mathcal{D}u,\mathcal{D}u]=\mathcal{D}[u,u] [ D u , D u ] = D [ u , u ] [32 Theorem 1] . For u , v ∈ p u,v\in\mathfrak{p} u , v ∈ p
D [ u , v ] = [ D u , D v ] (17) \mathcal{D}[u,v]=[\mathcal{D}u,\mathcal{D}v] \tag*{(17)} D [ u , v ] = [ D u , D v ] (17) [33 §5, Theorem 3] , since D u \mathcal{D}u D u ad u \operatorname{ad}_u ad u h \mathfrak{h} h p \mathfrak{p} p
We write Coder ( S ( a [ 1 ] ) ) \operatorname{Coder}(\mathcal{S}(\mathfrak{a}[1])) Coder ( S ( a [ 1 ])) S ( a [ 1 ] ) \mathcal{S}(\mathfrak{a}[1]) S ( a [ 1 ]) MultiDer ( a ) \operatorname{MultiDer}(\mathfrak{a}) MultiDer ( a ) 0 0 0 a [ 1 ] \mathfrak{a}[1] a [ 1 ] P P P ( D u ) n (\mathcal{D}u)_n ( D u ) n a \mathfrak{a} a ∣ u ∣ |u| ∣ u ∣ u ∈ h u\in\mathfrak{h} u ∈ h n ≥ 1 n\geq1 n ≥ 1 [7 Theorem 4.1 and Proposition 4.4] identifies formal multivectors with their higher derived brackets. In the form needed here, (16) shows that the coefficients of D u \mathcal{D}u D u u u u D : h → Coder ( S ( a [ 1 ] ) ) \mathcal{D}:\mathfrak{h}\to\operatorname{Coder}(\mathcal{S}(\mathfrak{a}[1])) D : h → Coder ( S ( a [ 1 ]))
D : X ( ν ) ⟶ MultiDer ( a ) . (18) \mathcal{D}:\mathfrak{X}(\nu)\longrightarrow\operatorname{MultiDer}(\mathfrak{a}). \tag*{(18)} D : X ( ν ) ⟶ MultiDer ( a ) . (18) With our conventions, (16) and [ Z , x μ ] = Z μ [Z,x^\mu]=Z^\mu [ Z , x μ ] = Z μ Z = Z μ ∂ ∂ x μ + Z i ∂ ∂ y i Z=Z^\mu\frac{\partial}{\partial x^\mu}+Z^i\frac{\partial}{\partial y^i} Z = Z μ ∂ x μ ∂ + Z i ∂ y i ∂
( D Z ) n ( e i 1 , … , e i n ) = ( − 1 ) n ∂ n Z i ∂ y i 1 ⋯ ∂ y i n ( x , 0 ) e i , (19) (\mathcal{D}Z)_n(e_{i_1},\ldots,e_{i_n})=(-1)^n\frac{\partial^n Z^i}{\partial y^{i_1}\cdots\partial y^{i_n}}(x,0)e_i, \tag*{(19)} ( D Z ) n ( e i 1 , … , e i n ) = ( − 1 ) n ∂ y i 1 ⋯ ∂ y i n ∂ n Z i ( x , 0 ) e i , (19) ( D Z ) n ( e i 1 , … , e i n − 1 , x μ ) = ( − 1 ) n − 1 ∂ n − 1 Z μ ∂ y i 1 ⋯ ∂ y i n − 1 ( x , 0 ) . (20) (\mathcal{D}Z)_n(e_{i_1},\ldots,e_{i_{n-1}},x^\mu)=(-1)^{n-1}\frac{\partial^{n-1}Z^\mu}{\partial y^{i_1}\cdots\partial y^{i_{n-1}}}(x,0). \tag*{(20)} ( D Z ) n ( e i 1 , … , e i n − 1 , x μ ) = ( − 1 ) n − 1 ∂ y i 1 ⋯ ∂ y i n − 1 ∂ n − 1 Z μ ( x , 0 ) . (20) Since a multiderivation is determined by its values on the x μ x^\mu x μ e i e_i e i MultiDer ( a ) \operatorname{MultiDer}(\mathfrak{a}) MultiDer ( a ) D Z \mathcal{D}Z D Z Z Z Z D \mathcal{D} D
The weights of u u u D u \mathcal{D}u D u
Lemma 2.6. Let u ∈ h u\in\mathfrak{h} u ∈ h d d d ( D u ) n = ( D u ( n − 1 − d ) ) n (\mathcal{D}u)_n=(\mathcal{D}u_{(n-1-d)})_n ( D u ) n = ( D u ( n − 1 − d ) ) n n ≥ 0 n\geq0 n ≥ 0
For a vector field Z Z Z ( D Z ) n (\mathcal{D}Z)_n ( D Z ) n Z ( n − 1 ) Z_{(n-1)} Z ( n − 1 ) X ≥ 1 ( ν ) = { Z ∈ X ( ν ) : ( D Z ) 0 = ( D Z ) 1 = 0 } \mathfrak{X}_{\geq1}(\nu)=\{Z\in\mathfrak{X}(\nu):(\mathcal{D}Z)_0=(\mathcal{D}Z)_1=0\} X ≥ 1 ( ν ) = { Z ∈ X ( ν ) : ( D Z ) 0 = ( D Z ) 1 = 0 } D \mathcal{D} D X ≥ 1 ( ν ) \mathfrak{X}_{\geq1}(\nu) X ≥ 1 ( ν ) Coder ≥ 2 ( a ) \operatorname{Coder}_{\geq2}(\mathfrak{a}) Coder ≥ 2 ( a ) MultiDer ( a ) ∩ Coder ≥ 2 ( a ) \operatorname{MultiDer}(\mathfrak{a})\cap\operatorname{Coder}_{\geq2}(\mathfrak{a}) MultiDer ( a ) ∩ Coder ≥ 2 ( a )
For a bivector π \pi π ( D π ) n (\mathcal{D}\pi)_n ( D π ) n π ( n − 2 ) \pi_{(n-2)} π ( n − 2 ) P ( π ) = 0 P(\pi)=0 P ( π ) = 0 π ( − 2 ) = 0 \pi_{(-2)}=0 π ( − 2 ) = 0 σ ψ \sigma_\psi σ ψ ≥ − 1 \geq-1 ≥ − 1 P ( σ ψ ) = 1 2 σ ψ ( d y j , d y k ) ∣ N e j ∧ e k = 0 P(\sigma_\psi)=\frac{1}{2}\sigma_\psi(dy^j,dy^k)|_N e_j\wedge e_k=0 P ( σ ψ ) = 2 1 σ ψ ( d y j , d y k ) ∣ N e j ∧ e k = 0 N N N [1 Lemma 5.17] .
Proof. A section of ⋀ k ν \bigwedge^k\nu ⋀ k ν k − 1 k-1 k − 1 − k -k − k P P P m − 1 m-1 m − 1 − m -m − m P [ ⋯ [ u ( w ) , a 1 ] , … , a n ] = 0 P[\cdots[u_{(w)},a_1],\ldots,a_n]=0 P [ ⋯ [ u ( w ) , a 1 ] , … , a n ] = 0 w = n − 1 − d w=n-1-d w = n − 1 − d d = 0 d=0 d = 0 ( D Z ) 0 (\mathcal{D}Z)_0 ( D Z ) 0 ( D Z ) 1 (\mathcal{D}Z)_1 ( D Z ) 1 Z ( − 1 ) Z_{(-1)} Z ( − 1 ) Z ( 0 ) Z_{(0)} Z ( 0 ) Z ∈ X ≥ 1 ( ν ) Z\in\mathfrak{X}_{\geq1}(\nu) Z ∈ X ≥ 1 ( ν ) (18) . Since X ≥ 1 ( ν ) ⊂ p \mathfrak{X}_{\geq1}(\nu)\subset\mathfrak{p} X ≥ 1 ( ν ) ⊂ p (12) , the rest follows from (17) and (18) . For (2), d = 1 d=1 d = 1 P ( π ) = π ( − 2 ) P(\pi)=\pi_{(-2)} P ( π ) = π ( − 2 )
In particular, D \mathcal{D} D X ≥ 1 ( ν ) \mathfrak{X}_{\geq1}(\nu) X ≥ 1 ( ν )
For a formal Poisson bivector π \pi π P ( π ) = 0 P(\pi)=0 P ( π ) = 0 Q π = D π Q_\pi=\mathcal{D}\pi Q π = D π Q ψ = Q σ ψ Q_\psi=Q_{\sigma_\psi} Q ψ = Q σ ψ Q π ∘ Q π = 1 2 D [ π , π ] = 0 Q_\pi\circ Q_\pi=\frac{1}{2}\mathcal{D}[\pi,\pi]=0 Q π ∘ Q π = 2 1 D [ π , π ] = 0 Q π Q_\pi Q π L ∞ L_\infty L ∞ a [ 1 ] \mathfrak{a}[1] a [ 1 ] P ∞ P_\infty P ∞ a \mathfrak{a} a ( D u ) n (\mathcal{D}u)_n ( D u ) n
λ n ψ ( a 1 , … , a n ) = ( − 1 ) 1 + ∑ i = 1 n ( n − i ) ∣ a i ∣ ( Q ψ ) n ( a 1 , … , a n ) . (21) \lambda^\psi_n(a_1,\ldots,a_n)=(-1)^{1+\sum_{i=1}^{n}(n-i)|a_i|}(Q_\psi)_n(a_1,\ldots,a_n). \tag*{(21)} λ n ψ ( a 1 , … , a n ) = ( − 1 ) 1 + ∑ i = 1 n ( n − i ) ∣ a i ∣ ( Q ψ ) n ( a 1 , … , a n ) . (21) These maps have degree 2 − n 2-n 2 − n ( a , λ ψ ) (\mathfrak{a},\lambda^\psi) ( a , λ ψ ) ( N , ψ ) (N,\psi) ( N , ψ ) λ n \lambda_n λ n λ n ψ \lambda^\psi_n λ n ψ ψ \psi ψ
The brackets are explicit in coordinates. Let { f , g } = σ ψ ( d f , d g ) \{f,g\}=\sigma_\psi(df,dg) { f , g } = σ ψ ( df , d g ) f , g ∈ C ∞ ( N ) f,g\in C^\infty(N) f , g ∈ C ∞ ( N ) (16) and (11) give
λ n ( e i 1 , … , e i n − 2 , f , g ) = ( − 1 ) n − 1 ∂ n − 2 { f , g } ∂ y i 1 ⋯ ∂ y i n − 2 ∣ y = 0 , n ≥ 2 , λ n ( e i 1 , … , e i n − 1 , f ) = ( − 1 ) n − 1 ∂ n − 1 { f , y j } ∂ y i 1 ⋯ ∂ y i n − 1 ∣ y = 0 e j , n ≥ 1 , λ n ( e i 1 , … , e i n ) = ( − 1 ) n − 1 2 ∂ n { y j , y k } ∂ y i 1 ⋯ ∂ y i n ∣ y = 0 e j ∧ e k , n ≥ 0. (22) \begin{aligned}
\lambda_n(e_{i_1},\ldots,e_{i_{n-2}},f,g)&=(-1)^{n-1}\left.\frac{\partial^{n-2}\{f,g\}}{\partial y^{i_1}\cdots\partial y^{i_{n-2}}}\right|_{y=0}, && n\geq2,\\
\lambda_n(e_{i_1},\ldots,e_{i_{n-1}},f)&=(-1)^{n-1}\left.\frac{\partial^{n-1}\{f,y^j\}}{\partial y^{i_1}\cdots\partial y^{i_{n-1}}}\right|_{y=0}e_j, && n\geq1,\\
\lambda_n(e_{i_1},\ldots,e_{i_n})&=\frac{(-1)^{n-1}}{2}\left.\frac{\partial^n\{y^j,y^k\}}{\partial y^{i_1}\cdots\partial y^{i_n}}\right|_{y=0}e_j\wedge e_k, && n\geq0. \tag*{(22)}
\end{aligned} λ n ( e i 1 , … , e i n − 2 , f , g ) λ n ( e i 1 , … , e i n − 1 , f ) λ n ( e i 1 , … , e i n ) = ( − 1 ) n − 1 ∂ y i 1 ⋯ ∂ y i n − 2 ∂ n − 2 { f , g } y = 0 , = ( − 1 ) n − 1 ∂ y i 1 ⋯ ∂ y i n − 1 ∂ n − 1 { f , y j } y = 0 e j , = 2 ( − 1 ) n − 1 ∂ y i 1 ⋯ ∂ y i n ∂ n { y j , y k } y = 0 e j ∧ e k , n ≥ 2 , n ≥ 1 , n ≥ 0. (22) These are the formulas of [7 Proposition 2.1] up to the sign ( − 1 ) m + 1 (-1)^{m+1} ( − 1 ) m + 1 m m m C ∞ ( N ) C^\infty(N) C ∞ ( N )
Remark 2.7. The differential λ 1 ψ \lambda^\psi_1 λ 1 ψ ψ \psi ψ − λ 1 ψ -\lambda^\psi_1 − λ 1 ψ T N ∘ TN^\circ T N ∘ § 3.1 [8 Remark 4.2] . Indeed, both are derivations, and for f ∈ C ∞ ( N ) f\in C^\infty(N) f ∈ C ∞ ( N ) ξ ∈ T N ∘ \xi\in TN^\circ ξ ∈ T N ∘ (22) gives ⟨ λ 1 ψ ( f ) , ξ ⟩ = σ ( d f , ξ ) = − σ ( ξ , d f ) \langle\lambda^\psi_1(f),\xi\rangle=\sigma(df,\xi)=-\sigma(\xi,df) ⟨ λ 1 ψ ( f ) , ξ ⟩ = σ ( df , ξ ) = − σ ( ξ , df ) e i e_i e i a ↦ ( − 1 ) deg a a a\mapsto(-1)^{\deg a}a a ↦ ( − 1 ) d e g a a a \mathfrak a a ( a , λ 1 ψ ) (\mathfrak a,\lambda^\psi_1) ( a , λ 1 ψ ) T N ∘ TN^\circ T N ∘
Changes of tubular neighbourhood
We now integrate D \mathcal D D Φ : Diff 1 ( ν ) → Aut 1 ( a ) \Phi:\operatorname{Diff}_1(\nu)\to\operatorname{Aut}_1(\mathfrak a) Φ : Diff 1 ( ν ) → Aut 1 ( a ) (14) , (6) and Lemma 2.4, we may set
Φ ( φ ) = exp ( D log φ ) ∈ Aut 1 ( a ) , that is, Φ ( e Z ) = e D Z . (23) \Phi(\varphi)=\exp(\mathcal D_{\log\varphi})\in\operatorname{Aut}_1(\mathfrak a),\qquad\text{that is,}\qquad\Phi(e^Z)=e^{\mathcal D_Z}. \tag*{(23)} Φ ( φ ) = exp ( D l o g φ ) ∈ Aut 1 ( a ) , that is, Φ ( e Z ) = e D Z . (23) Proposition 2.8. Let φ , φ ′ ∈ Diff 1 ( ν ) \varphi,\varphi'\in\operatorname{Diff}_1(\nu) φ , φ ′ ∈ Diff 1 ( ν ) π \pi π P ( π ) = 0 P(\pi)=0 P ( π ) = 0
Φ ( φ ) ∘ Q π ∘ Φ ( φ ) − 1 = Q φ ∗ π \Phi(\varphi)\circ Q_\pi\circ\Phi(\varphi)^{-1}=Q_{\varphi_*\pi} Φ ( φ ) ∘ Q π ∘ Φ ( φ ) − 1 = Q φ ∗ π
Φ ( φ ∘ φ ′ ) = Φ ( φ ′ ) ∘ Φ ( φ ) \Phi(\varphi\circ\varphi')=\Phi(\varphi')\circ\Phi(\varphi) Φ ( φ ∘ φ ′ ) = Φ ( φ ′ ) ∘ Φ ( φ )
Φ \Phi Φ
If φ ∈ Diff 1 ( ψ , ψ ′ ) \varphi\in\operatorname{Diff}_1(\psi,\psi') φ ∈ Diff 1 ( ψ , ψ ′ ) Φ ( φ ) \Phi(\varphi) Φ ( φ ) L ∞ L_\infty L ∞ ( a , λ ψ ) → ( a , λ ψ ′ ) (\mathfrak a,\lambda^\psi)\to(\mathfrak a,\lambda^{\psi'}) ( a , λ ψ ) → ( a , λ ψ ′ )
Proof. Let Z = log φ Z=\log\varphi Z = log φ Z ′ = log φ ′ Z'=\log\varphi' Z ′ = log φ ′ π \pi π ≥ − 1 \geq-1 ≥ − 1 ad Z k π \operatorname{ad}_Z^k\pi ad Z k π ≥ k − 1 \geq k-1 ≥ k − 1 D ad Z k π \mathcal D_{\operatorname{ad}_Z^k\pi} D ad Z k π ≤ k \leq k ≤ k Z , π ∈ p Z,\pi\in\mathfrak p Z , π ∈ p (9) , (17) and (13) give
Φ ( φ ) ∘ D π ∘ Φ ( φ ) − 1 = ∑ k ≥ 0 1 k ! ad D Z k ( D π ) = ∑ k ≥ 0 1 k ! D ad Z k π = D e ad Z π = D φ ∗ π . \begin{aligned}
\Phi(\varphi)\circ\mathcal D_\pi\circ\Phi(\varphi)^{-1}
&=\sum_{k\geq0}\frac{1}{k!}\operatorname{ad}_{\mathcal D_Z}^k(\mathcal D_\pi)\\
&=\sum_{k\geq0}\frac{1}{k!}\mathcal D_{\operatorname{ad}_Z^k\pi}\\
&=\mathcal D_{e^{\operatorname{ad}_Z}\pi}\\
&=\mathcal D_{\varphi_*\pi}.
\end{aligned} Φ ( φ ) ∘ D π ∘ Φ ( φ ) − 1 = k ≥ 0 ∑ k ! 1 ad D Z k ( D π ) = k ≥ 0 ∑ k ! 1 D ad Z k π = D e ad Z π = D φ ∗ π . and P ( φ ∗ π ) = 0 P(\varphi_*\pi)=0 P ( φ ∗ π ) = 0 φ ∗ π \varphi_*\pi φ ∗ π ≥ − 1 \geq-1 ≥ − 1 (14) , (9) and (17) give
Φ ( φ ′ ) ∘ Φ ( φ ) = e D Z ′ e D Z = e BCH ( D Z ′ , D Z ) = e D BCH ( Z ′ , Z ) = Φ ( φ ∘ φ ′ ) . \Phi(\varphi')\circ\Phi(\varphi)=e^{\mathcal D_{Z'}}e^{\mathcal D_Z}=e^{\operatorname{BCH}(\mathcal D_{Z'},\mathcal D_Z)}=e^{\mathcal D_{\operatorname{BCH}(Z',Z)}}=\Phi(\varphi\circ\varphi'). Φ ( φ ′ ) ∘ Φ ( φ ) = e D Z ′ e D Z = e BCH ( D Z ′ , D Z ) = e D BCH ( Z ′ , Z ) = Φ ( φ ∘ φ ′ ) . Part (3) follows from (6) , (18) and Lemma 2.4, and (4) from (1) with π = σ ψ \pi=\sigma_\psi π = σ ψ
Corollary 2.9. For tubular neighbourhoods ψ , ψ ′ , ψ ′ ′ \psi,\psi',\psi'' ψ , ψ ′ , ψ ′′ Φ ( φ ψ ψ ′ ) \Phi(\varphi_{\psi\psi'}) Φ ( φ ψ ψ ′ ) L ∞ L_\infty L ∞ ( a , λ ψ ) → ( a , λ ψ ′ ) (\mathfrak a,\lambda^\psi)\to(\mathfrak a,\lambda^{\psi'}) ( a , λ ψ ) → ( a , λ ψ ′ )
Φ ( φ ψ ′ ψ ′ ′ ) ∘ Φ ( φ ψ ψ ′ ) = Φ ( φ ψ ψ ′ ′ ) . \Phi(\varphi_{\psi'\psi''})\circ\Phi(\varphi_{\psi\psi'})=\Phi(\varphi_{\psi\psi''}). Φ ( φ ψ ′ ψ ′′ ) ∘ Φ ( φ ψ ψ ′ ) = Φ ( φ ψ ψ ′′ ) . Proof. This follows from (15) , Proposition 2.8(2) and (4), and φ ψ ψ ′ ′ = φ ψ ψ ′ ∘ φ ψ ′ ψ ′ ′ \varphi_{\psi\psi''}=\varphi_{\psi\psi'}\circ\varphi_{\psi'\psi''} φ ψ ψ ′′ = φ ψ ψ ′ ∘ φ ψ ′ ψ ′′
We next compare Φ \Phi Φ L ∞ L_\infty L ∞ [8 ] . A family ( φ τ ) τ ∈ [ 0 , 1 ] (\varphi_\tau)_{\tau\in[0,1]} ( φ τ ) τ ∈ [ 0 , 1 ] Diff 1 ( ν ) \operatorname{Diff}_1(\nu) Diff 1 ( ν ) φ ∗ ( x i ) \varphi_*(x^i) φ ∗ ( x i ) φ ∗ ( y i ) \varphi_*(y^i) φ ∗ ( y i ) ( τ , x ) (\tau,x) ( τ , x ) [28 ]
( d d τ e B τ ) ∘ e − B τ = ∑ k ≥ 0 1 ( k + 1 ) ! ad B τ k ( d d τ B τ ) . (24) \left(\frac{d}{d\tau}e^{B_\tau}\right)\circ e^{-B_\tau}
=\sum_{k\geq0}\frac{1}{(k+1)!}\operatorname{ad}_{B_\tau}^k\left(\frac{d}{d\tau}B_\tau\right). \tag*{(24)} ( d τ d e B τ ) ∘ e − B τ = k ≥ 0 ∑ ( k + 1 )! 1 ad B τ k ( d τ d B τ ) . (24) for a smooth family of operators B τ B_\tau B τ
Lemma 2.10. Let ( φ τ ) (\varphi_\tau) ( φ τ ) Diff 1 ( ν ) \operatorname{Diff}_1(\nu) Diff 1 ( ν ) φ 0 = id \varphi_0=\operatorname{id} φ 0 = id Z τ = log φ τ Z_\tau=\log\varphi_\tau Z τ = log φ τ Y τ = ( d d τ φ τ ∗ ) ∘ ( φ τ ∗ ) − 1 Y_\tau=\left(\frac{d}{d\tau}\varphi_\tau^*\right)\circ(\varphi_\tau^*)^{-1} Y τ = ( d τ d φ τ ∗ ) ∘ ( φ τ ∗ ) − 1
Y τ = ∑ k ≥ 0 1 ( k + 1 ) ! ad Z τ k ( d Z τ d τ ) ∈ X ≥ 1 ( ν ) , Y_\tau=\sum_{k\geq0}\frac{1}{(k+1)!}\operatorname{ad}_{Z_\tau}^{k}\left(\frac{dZ_\tau}{d\tau}\right)\in\mathfrak{X}_{\geq1}(\nu), Y τ = k ≥ 0 ∑ ( k + 1 )! 1 ad Z τ k ( d τ d Z τ ) ∈ X ≥ 1 ( ν ) , d d τ φ τ ∗ = ad Y τ φ τ ∗ on h . \frac{d}{d\tau}\varphi_\tau^*=\operatorname{ad}_{Y_\tau}\varphi_\tau^*\quad\text{on }\mathfrak{h}. d τ d φ τ ∗ = ad Y τ φ τ ∗ on h . Proof. Each weight component of log φ τ ∗ \log\varphi_\tau^* log φ τ ∗ φ τ ∗ ( x μ ) \varphi_\tau^*(x^\mu) φ τ ∗ ( x μ ) φ τ ∗ ( y i ) \varphi_\tau^*(y^i) φ τ ∗ ( y i ) Z τ Z_\tau Z τ τ \tau τ O ^ \widehat{\mathcal{O}} O φ τ ∗ = e Z τ \varphi_\tau^*=e^{Z_\tau} φ τ ∗ = e Z τ Z τ Z_\tau Z τ (24) gives Y τ ∈ X ≥ 1 ( ν ) Y_\tau\in\mathfrak{X}_{\geq1}(\nu) Y τ ∈ X ≥ 1 ( ν ) h \mathfrak{h} h φ τ ∗ = e ad Z τ \varphi_\tau^*=e^{\operatorname{ad}_{Z_\tau}} φ τ ∗ = e ad Z τ (13) , and (24) with B τ = ad Z τ B_\tau=\operatorname{ad}_{Z_\tau} B τ = ad Z τ ad Y τ \operatorname{ad}_{Y_\tau} ad Y τ ad \operatorname{ad} ad
Let ψ τ = ψ ∘ γ τ \psi_\tau=\psi\circ\gamma_\tau ψ τ = ψ ∘ γ τ γ 0 = id \gamma_0=\operatorname{id} γ 0 = id φ τ \varphi_\tau φ τ γ τ \gamma_\tau γ τ (15) and Lemma 2.10, the automorphisms φ τ ∗ \varphi_\tau^* φ τ ∗ σ ψ \sigma_\psi σ ψ σ ψ τ \sigma_{\psi_\tau} σ ψ τ ad Y τ \operatorname{ad}_{Y_\tau} ad Y τ [8 eq.(7)] . Since Y τ ∈ p Y_\tau\in\mathfrak{p} Y τ ∈ p ad Y τ \operatorname{ad}_{Y_\tau} ad Y τ p \mathfrak{p} p [8 ] . The Cauchy problem (10) of [8 ] has only the zero solution, since P [ Y τ , a ] = ( D Y τ ) 1 ( a ) = 0 P[Y_\tau,a]=(DY_\tau)_1(a)=0 P [ Y τ , a ] = ( D Y τ ) 1 ( a ) = 0 L ∞ L_\infty L ∞ [8 Theorems 3.2 and 4.3] attached to the isotopy is R 1 R_1 R 1 R τ R_\tau R τ [8 eq.(14)]
d d τ R τ = D Y τ ∘ R τ , R 0 = id . (25) \frac{d}{d\tau}R_\tau= D_{Y_\tau}\circ R_\tau,\qquad R_0=\operatorname{id}. \tag*{(25)} d τ d R τ = D Y τ ∘ R τ , R 0 = id . (25) Proposition 2.11. Let ( φ τ ) (\varphi_\tau) ( φ τ ) Diff 1 ( ν ) \operatorname{Diff}_1(\nu) Diff 1 ( ν ) φ 0 = id \varphi_0=\operatorname{id} φ 0 = id (25) is R τ = Φ ( φ τ ) R_\tau=\Phi(\varphi_\tau) R τ = Φ ( φ τ ) Φ ( φ 1 ) \Phi(\varphi_1) Φ ( φ 1 ) φ 1 \varphi_1 φ 1
Proof. By Lemma 2.10, (17) and Lemma 2.6,
D Y τ = ∑ k ≥ 0 1 ( k + 1 ) ! ad D Z τ k ( d d τ D Z τ ) . D_{Y_\tau}=\sum_{k\geq0}\frac{1}{(k+1)!}\operatorname{ad}_{D_{Z_\tau}}^{k}\left(\frac{d}{d\tau}D_{Z_\tau}\right). D Y τ = k ≥ 0 ∑ ( k + 1 )! 1 ad D Z τ k ( d τ d D Z τ ) . By (24) on S ( a [ 1 ] ) S(\mathfrak{a}[1]) S ( a [ 1 ]) B τ = D Z τ B_\tau=D_{Z_\tau} B τ = D Z τ ( d d τ e D Z τ ) ∘ e − D Z τ \left(\frac{d}{d\tau}e^{D_{Z_\tau}}\right)\circ e^{-D_{Z_\tau}} ( d τ d e D Z τ ) ∘ e − D Z τ Φ ( φ τ ) \Phi(\varphi_\tau) Φ ( φ τ ) (25) , and the solution is unique by [8 Proposition 3.4] . □
An L ∞ L_\infty L ∞ F : ( a , λ ψ ) → ( a , λ ψ ′ ) F:(\mathfrak{a},\lambda^\psi)\to(\mathfrak{a},\lambda^{\psi'}) F : ( a , λ ψ ) → ( a , λ ψ ′ ) F = Φ ( φ ) F=\Phi(\varphi) F = Φ ( φ ) φ ∈ Diff 1 ( ψ , ψ ′ ) \varphi\in\operatorname{Diff}_1(\psi,\psi') φ ∈ Diff 1 ( ψ , ψ ′ )
Proposition 2.12. Let F : ( a , λ ψ ) → ( a , λ ψ ′ ) F:(\mathfrak{a},\lambda^\psi)\to(\mathfrak{a},\lambda^{\psi'}) F : ( a , λ ψ ) → ( a , λ ψ ′ ) L ∞ L_\infty L ∞
F F F
F = e D Z F=e^{D_Z} F = e D Z Z ∈ X ≥ 1 ( ν ) Z\in\mathfrak{X}_{\geq1}(\nu) Z ∈ X ≥ 1 ( ν )
F 1 = id F_1=\operatorname{id} F 1 = id log F \log F log F a \mathfrak{a} a
In that case, the element φ ∈ Diff 1 ( ψ , ψ ′ ) \varphi\in\operatorname{Diff}_1(\psi,\psi') φ ∈ Diff 1 ( ψ , ψ ′ ) F = Φ ( φ ) F=\Phi(\varphi) F = Φ ( φ )
Proof. By definition, (1) implies (2), and (2) implies (3) by Lemma 2.6(1) and (6) . Assume (3). Then F ∈ Aut 1 ( a ) F\in\operatorname{Aut}_1(\mathfrak{a}) F ∈ Aut 1 ( a ) log F ∈ MultiDer ( a ) ∩ Coder ≥ 2 ( a ) \log F\in\operatorname{MultiDer}(\mathfrak{a})\cap\operatorname{Coder}_{\geq2}(\mathfrak{a}) log F ∈ MultiDer ( a ) ∩ Coder ≥ 2 ( a ) log F = D Z \log F=D_Z log F = D Z Z ∈ X ≥ 1 ( ν ) Z\in\mathfrak{X}_{\geq1}(\nu) Z ∈ X ≥ 1 ( ν ) φ = e Z \varphi=e^Z φ = e Z F = Φ ( φ ) F=\Phi(\varphi) F = Φ ( φ ) F F F L ∞ L_\infty L ∞ Proposition 2.8 (1) gives
D φ ∗ σ ψ = F ∘ Q ψ ∘ F − 1 = Q ψ ′ = D σ ψ ′ . D_{\varphi^*\sigma_\psi}=F\circ Q_\psi\circ F^{-1}=Q_{\psi'}=D_{\sigma_{\psi'}}. D φ ∗ σ ψ = F ∘ Q ψ ∘ F − 1 = Q ψ ′ = D σ ψ ′ . Injectivity in (18) therefore gives φ ∗ σ ψ = σ ψ ′ \varphi^*\sigma_\psi=\sigma_{\psi'} φ ∗ σ ψ = σ ψ ′ Proposition 2.8 (3). □
Proof of Theorem A. This is the equivalence of (1) and (3) in Proposition 2.12 . □
Remark 2.13. The element φ \varphi φ Proposition 2.12 need not be the transition φ ψ / ψ ′ \varphi_{\psi/\psi'} φ ψ / ψ ′ Diff 1 ( ψ , ψ ′ ) = φ ψ / ψ ′ ∘ Diff 1 ( ψ ′ , ψ ′ ) \operatorname{Diff}_1(\psi,\psi')=\varphi_{\psi/\psi'}\circ\operatorname{Diff}_1(\psi',\psi') Diff 1 ( ψ , ψ ′ ) = φ ψ / ψ ′ ∘ Diff 1 ( ψ ′ , ψ ′ ) (13) the stabilizer Diff 1 ( ψ ′ , ψ ′ ) \operatorname{Diff}_1(\psi',\psi') Diff 1 ( ψ ′ , ψ ′ ) e X e^X e X X ∈ X ≥ 1 ( ν ) X\in\mathfrak{X}_{\geq1}(\nu) X ∈ X ≥ 1 ( ν ) [ X , σ ψ ′ ] = 0 [X,\sigma_{\psi'}]=0 [ X , σ ψ ′ ] = 0 e ad X σ ψ ′ − σ ψ ′ e^{\operatorname{ad}_X}\sigma_{\psi'}-\sigma_{\psi'} e ad X σ ψ ′ − σ ψ ′ [ X , σ ψ ′ ] [X,\sigma_{\psi'}] [ X , σ ψ ′ ] g g g ≥ 2 \geq2 ≥ 2 X g = σ ψ ′ ♯ ( d g ) X_g=\sigma_{\psi'}^\sharp(dg) X g = σ ψ ′ ♯ ( d g ) X ≥ 1 ( ν ) \mathfrak{X}_{\geq1}(\nu) X ≥ 1 ( ν ) σ ψ ′ \sigma_{\psi'} σ ψ ′
Remark 2.14. An induced isomorphism need not have multiderivations as Taylor coefficients. Let M = R 4 M=\mathbb{R}^{4} M = R 4 σ = ∂ ∂ x 1 ∧ ∂ ∂ y 1 + ∂ ∂ x 2 ∧ ∂ ∂ y 2 \sigma=\frac{\partial}{\partial x_{1}}\wedge\frac{\partial}{\partial y_{1}}+\frac{\partial}{\partial x_{2}}\wedge\frac{\partial}{\partial y_{2}} σ = ∂ x 1 ∂ ∧ ∂ y 1 ∂ + ∂ x 2 ∂ ∧ ∂ y 2 ∂ N = { y = 0 } N=\{y=0\} N = { y = 0 } e i = ∂ ∂ y i e_i=\frac{\partial}{\partial y_i} e i = ∂ y i ∂ ψ \psi ψ Z = 1 2 ( ( y 1 ) 2 ∂ ∂ y 1 + ( y 2 ) 2 ∂ ∂ y 2 ) Z=\frac{1}{2}\left((y^{1})^{2}\frac{\partial}{\partial y_{1}}+(y^{2})^{2}\frac{\partial}{\partial y_{2}}\right) Z = 2 1 ( ( y 1 ) 2 ∂ y 1 ∂ + ( y 2 ) 2 ∂ y 2 ∂ ) γ ( x , y ) = ( x , y 1 1 − y 1 / 2 , y 2 1 − y 2 / 2 ) \gamma(x,y)=\left(x,\frac{y^{1}}{1-y^{1}/2},\frac{y^{2}}{1-y^{2}/2}\right) γ ( x , y ) = ( x , 1 − y 1 /2 y 1 , 1 − y 2 /2 y 2 ) ψ ′ = ψ ∘ γ \psi'=\psi\circ\gamma ψ ′ = ψ ∘ γ γ \gamma γ e Z = Φ ψ ψ ′ e^{Z}=\Phi_{\psi\psi'} e Z = Φ ψ ψ ′ F = Φ ( e Z ) F=\Phi(e^{Z}) F = Φ ( e Z ) (15) . Let δ = D Z \delta=D_Z δ = D Z Lemma 2.6 (1). The Lie derivative of ∂ ∂ y 1 ∧ ∂ ∂ y 2 \frac{\partial}{\partial y_{1}}\wedge\frac{\partial}{\partial y_{2}} ∂ y 1 ∂ ∧ ∂ y 2 ∂ Z Z Z − ( y 1 + y 2 ) ∂ ∂ y 1 ∧ ∂ ∂ y 2 -(y^{1}+y^{2})\frac{\partial}{\partial y_{1}}\wedge\frac{\partial}{\partial y_{2}} − ( y 1 + y 2 ) ∂ y 1 ∂ ∧ ∂ y 2 ∂ (19) and (16) give
δ 2 ( e 1 , e 1 ) = e 1 , δ 2 ( e 2 , e 2 ) = e 2 , δ 2 ( e 1 , e 2 ) = 0 , δ 2 ( e 1 ∧ e 2 , e j ) = e 1 ∧ e 2 . \delta_2(e_1,e_1)=e_1,\qquad\delta_2(e_2,e_2)=e_2,\qquad\delta_2(e_1,e_2)=0,\qquad\delta_2(e_1\wedge e_2,e_j)=e_1\wedge e_2. δ 2 ( e 1 , e 1 ) = e 1 , δ 2 ( e 2 , e 2 ) = e 2 , δ 2 ( e 1 , e 2 ) = 0 , δ 2 ( e 1 ∧ e 2 , e j ) = e 1 ∧ e 2 . By (7) and (8) with ( log F ) 3 = 0 (\log F)_3=0 ( log F ) 3 = 0 F 3 = 1 2 ∑ ε δ 2 ( δ 2 ( a i , a j ) , a k ) F_3=\frac{1}{2}\sum_{\varepsilon}\delta_2(\delta_2(a_i,a_j),a_k) F 3 = 2 1 ∑ ε δ 2 ( δ 2 ( a i , a j ) , a k ) + 1 +1 + 1 F 3 ( e 1 ∧ e 2 , e 1 , e 2 ) = e 1 ∧ e 2 F_3(e_1\wedge e_2,e_1,e_2)=e_1\wedge e_2 F 3 ( e 1 ∧ e 2 , e 1 , e 2 ) = e 1 ∧ e 2 F 3 ( e 1 , e 1 , e 2 ) = 1 2 δ 2 ( e 1 , e 2 ) = 0 F_3(e_1,e_1,e_2)=\frac{1}{2}\delta_2(e_1,e_2)=0 F 3 ( e 1 , e 1 , e 2 ) = 2 1 δ 2 ( e 1 , e 2 ) = 0 F 3 ( e 2 , e 1 , e 2 ) = 0 F_3(e_2,e_1,e_2)=0 F 3 ( e 2 , e 1 , e 2 ) = 0 F 3 F_3 F 3 (4) would give F 3 ( e 1 ∧ e 2 , e 1 , e 2 ) = 0 F_3(e_1\wedge e_2,e_1,e_2)=0 F 3 ( e 1 ∧ e 2 , e 1 , e 2 ) = 0
The adjoint symplectic groupoid of a Radko surface
In this section, we give an explicit coordinate description of the adjoint symplectic groupoid of a Radko surface ( Σ , σ ) (\Sigma,\sigma) ( Σ , σ ) [14 ] . This fixes the ambient symplectic groupoid G Σ \mathcal{G}_{\Sigma} G Σ
Conventions and Radko surfaces
We first fix our conventions and recall the normal form of a Radko surface near its degeneracy locus.
For a bivector π \pi π M M M
π : T ∗ M ⟶ T M , π ( ξ ) = ι ξ π = π ( ξ , ⋅ ) . \pi:T^{*}M\longrightarrow TM,\qquad\pi(\xi)=\iota_{\xi}\pi=\pi(\xi,\cdot). π : T ∗ M ⟶ T M , π ( ξ ) = ι ξ π = π ( ξ , ⋅ ) . If π \pi π { f , g } = π ( d f , d g ) \{f,g\}=\pi(df,dg) { f , g } = π ( df , d g ) X f = π ( d f ) X_f=\pi(df) X f = π ( df ) T π ∗ M T_{\pi}^{*}M T π ∗ M π \pi π [ d f , d g ] = d { f , g } [df,dg]=d\{f,g\} [ df , d g ] = d { f , g } ω \omega ω π \pi π inverse if
ι π ( ξ ) ω = ξ for every ξ ∈ T ∗ M . \iota_{\pi(\xi)}\omega=\xi\qquad\text{for every }\xi\in T^{*}M. ι π ( ξ ) ω = ξ for every ξ ∈ T ∗ M . Equivalently, their graphs agree:
Γ ω = { ( v , ι v ω ) : v ∈ T M } = { ( π ( ξ ) , ξ ) : ξ ∈ T ∗ M } . \Gamma_{\omega}=\{(v,\iota_v\omega):v\in TM\}=\{(\pi(\xi),\xi):\xi\in T^{*}M\}. Γ ω = {( v , ι v ω ) : v ∈ T M } = {( π ( ξ ) , ξ ) : ξ ∈ T ∗ M } . We then write π = ω − 1 \pi=\omega^{-1} π = ω − 1 ι X f ω = d f \iota_{X_f}\omega=df ι X f ω = df
Following [14 ] , we allow the arrow manifold of a Lie groupoid G ⇒ M G\Rightarrow M G ⇒ M g g g h h h g h gh g h s ( g ) = t ( h ) s(g)=t(h) s ( g ) = t ( h ) G x G_x G x x ∈ M x\in M x ∈ M U ⊂ M U\subset M U ⊂ M G ∣ U = s − 1 ( U ) ∩ t − 1 ( U ) G|_U=s^{-1}(U)\cap t^{-1}(U) G ∣ U = s − 1 ( U ) ∩ t − 1 ( U ) G ∘ G^{\circ} G ∘ x ∈ M x\in M x ∈ M s − 1 ( x ) s^{-1}(x) s − 1 ( x ) 1 x 1_x 1 x Lie ( G ) \operatorname{Lie}(G) Lie ( G ) ker d s \ker ds ker d s d t dt d t
A symplectic groupoid is a Lie groupoid equipped with a symplectic form Ω \Omega Ω multiplicative :
m ∗ Ω = pr 1 ∗ Ω + pr 2 ∗ Ω , m^{*}\Omega=\operatorname{pr}_{1}^{*}\Omega+\operatorname{pr}_{2}^{*}\Omega, m ∗ Ω = pr 1 ∗ Ω + pr 2 ∗ Ω , where m m m pr 1 , pr 2 \operatorname{pr}_{1},\operatorname{pr}_{2} pr 1 , pr 2 integrates ( M , π ) (M,\pi) ( M , π ) t : ( G , Ω − 1 ) → ( M , π ) t:(G,\Omega^{-1})\to(M,\pi) t : ( G , Ω − 1 ) → ( M , π )
Lie ( G ) ⟶ T π ∗ M , v ⟼ ι v Ω ∣ T M \operatorname{Lie}(G)\longrightarrow T_{\pi}^{*}M,\qquad v\longmapsto\iota_v\Omega|_{TM} Lie ( G ) ⟶ T π ∗ M , v ⟼ ι v Ω ∣ T M is an isomorphism of Lie algebroids, and the induced Dirac structure
{ ( d t ( v ) , ι v Ω ∣ T M ) : v ∈ Lie ( G ) } \left\{(dt(v),\iota_v\Omega|_{TM}):v\in\operatorname{Lie}(G)\right\} { ( d t ( v ) , ι v Ω ∣ T M ) : v ∈ Lie ( G ) } is the graph of π \pi π [21 ] , §§2.1 and 3.1, Lemma 3.1, Example 3.8 and Remark 4.9. Gualtieri and Li use the opposite convention, in which s s s [14 §1] .
A Radko surface is a compact oriented surface Σ \Sigma Σ σ \sigma σ D D D Σ ∖ D \Sigma\setminus D Σ ∖ D [26 ] . Writing σ = f σ 0 \sigma=f\sigma_0 σ = f σ 0 σ 0 \sigma_0 σ 0 d f ≠ 0 df\ne0 df = 0 D = f − 1 ( 0 ) D=f^{-1}(0) D = f − 1 ( 0 ) Σ \Sigma Σ
The degeneracy locus D = f − 1 ( 0 ) D=f^{-1}(0) D = f − 1 ( 0 ) Σ \Sigma Σ D 1 , … , D n D D_1,\ldots,D_{n_D} D 1 , … , D n D Figure 1 ). Its normal bundles ν D j \nu_{D_j} ν D j Σ \Sigma Σ D j D_j D j V i V_i V i Σ ∖ D \Sigma\setminus D Σ ∖ D ω i \omega_i ω i σ \sigma σ V i V_i V i V i V_i V i D D D D j D_j D j ( Σ , σ ) (\Sigma,\sigma) ( Σ , σ ) proper log symplectic surface in the sense of [14 Definition 1.10] .
Figure 1. A Radko surface and its degeneracy locus.
A log symplectic manifold is linearizable along its degeneracy locus [14 Theorem 1.12] , and the linear models of a proper one are classified in [14 Proposition 1.11] . In dimension two, the leaves in D j D_j D j ν D j \nu_{D_j} ν D j ℓ j > 0 \ell_j > 0 ℓ j > 0 modular period of D j D_j D j (27) below).
More explicitly, Radko’s normal form [26 §2.4, eq. (4)] , see also [26 Lemma 2] , gives coordinates
( θ , r ) ∈ ( R / 2 π Z ) × ( − c j , c j ) , D j = { r = 0 } , σ = 2 π ℓ j r ∂ ∂ r ∧ ∂ ∂ θ . (\theta,r) \in(\mathbb{R}/2\pi\mathbb{Z}) \times(-c_j,c_j), \qquad D_j = \{r=0\}, \qquad\sigma= \frac{2\pi}{\ell_j}r\frac{\partial}{\partial r}\wedge\frac{\partial}{\partial\theta}. ( θ , r ) ∈ ( R /2 π Z ) × ( − c j , c j ) , D j = { r = 0 } , σ = ℓ j 2 π r ∂ r ∂ ∧ ∂ θ ∂ . Set t = ℓ j θ / ( 2 π ) t=\ell_j\theta/(2\pi) t = ℓ j θ / ( 2 π ) S ℓ j 1 = R / ℓ j Z S^1_{\ell_j}=\mathbb{R}/\ell_j\mathbb{Z} S ℓ j 1 = R / ℓ j Z U j U_j U j D j D_j D j
( S ℓ j 1 × ( − c j , c j ) , r ∂ ∂ r ∧ ∂ ∂ t ) , \left(S^1_{\ell_j}\times(-c_j,c_j),\ r\frac{\partial}{\partial r}\wedge\frac{\partial}{\partial t}\right), ( S ℓ j 1 × ( − c j , c j ) , r ∂ r ∂ ∧ ∂ t ∂ ) , with D j = S ℓ j 1 × { 0 } D_j=S^1_{\ell_j}\times\{0\} D j = S ℓ j 1 × { 0 }
U ^ j = S ℓ j 1 × R , σ = r ∂ ∂ r ∧ ∂ ∂ t , (26) \widehat{U}_j=S^1_{\ell_j}\times\mathbb{R},\qquad\sigma=r\frac{\partial}{\partial r}\wedge\frac{\partial}{\partial t}, \tag*{(26)} U j = S ℓ j 1 × R , σ = r ∂ r ∂ ∧ ∂ t ∂ , (26) so that U j = S ℓ j 1 × ( − c j , c j ) U_j=S^1_{\ell_j}\times(-c_j,c_j) U j = S ℓ j 1 × ( − c j , c j ) U ^ j \widehat{U}_j U j D j D_j D j c j c_j c j U j U_j U j U j ∩ D = D j = { r = 0 } U_j\cap D=D_j=\{r=0\} U j ∩ D = D j = { r = 0 }
On { r ≠ 0 } \{r\ne0\} { r = 0 } ω = 1 r d t ∧ d r \omega=\frac{1}{r}\,dt\wedge dr ω = r 1 d t ∧ d r
σ ( d r ) = r ∂ ∂ t , ι σ ( d r ) ω = d r , σ ( d t ) = − r ∂ ∂ r , ι σ ( d t ) ω = d t . \sigma(dr)=r\frac{\partial}{\partial t},\qquad\iota_{\sigma(dr)}\omega=dr,\qquad\sigma(dt)=-r\frac{\partial}{\partial r},\qquad\iota_{\sigma(dt)}\omega=dt. σ ( d r ) = r ∂ t ∂ , ι σ ( d r ) ω = d r , σ ( d t ) = − r ∂ r ∂ , ι σ ( d t ) ω = d t . The normal-form identification is Poisson, so ω i = ω \omega_i=\omega ω i = ω U j ∩ V i U_j\cap V_i U j ∩ V i
Let vol \operatorname{vol} vol D j D_j D j X f = σ ( d f ) X_f=\sigma(df) X f = σ ( df ) X f ( g ) = σ ( d f , d g ) X_f(g)=\sigma(df,dg) X f ( g ) = σ ( df , d g ) modular vector field X vol X_{\operatorname{vol}} X vol σ \sigma σ [34 ] is defined by X vol ( f ) = div ( X f ) X_{\operatorname{vol}}(f)=\operatorname{div}(X_f) X vol ( f ) = div ( X f ) d ( ι X vol ) = div ( X ) vol d(\iota_X\operatorname{vol})=\operatorname{div}(X)\operatorname{vol} d ( ι X vol ) = div ( X ) vol
The restriction of X vol X_{\operatorname{vol}} X vol D j D_j D j [26 §2.4] , since changing vol \operatorname{vol} vol X vol X_{\operatorname{vol}} X vol D j D_j D j U j U_j U j vol = d r ∧ d t \operatorname{vol}=dr\wedge dt vol = d r ∧ d t X f = r ∂ f ∂ r ∂ ∂ t − r ∂ f ∂ t ∂ ∂ r X_f=r\frac{\partial f}{\partial r}\frac{\partial}{\partial t}-r\frac{\partial f}{\partial t}\frac{\partial}{\partial r} X f = r ∂ r ∂ f ∂ t ∂ − r ∂ t ∂ f ∂ r ∂ div ( X f ) = − ∂ f ∂ t \operatorname{div}(X_f)=-\frac{\partial f}{\partial t} div ( X f ) = − ∂ t ∂ f
X vol ∣ D j = − ∂ ∂ t . (27) X_{\operatorname{vol}}\big|_{D_j}=-\frac{\partial}{\partial t}. \tag*{(27)} X vol D j = − ∂ t ∂ . (27) In particular, the modular period ℓ j \ell_j ℓ j D j D_j D j
We next show that each leaf V i V_i V i U j U_j U j U j ∖ D j U_j\setminus D_j U j ∖ D j σ = f σ 0 \sigma=f\sigma_0 σ = f σ 0 f f f V i V_i V i U j U_j U j σ 0 = u ∂ ∂ r ∧ ∂ ∂ t \sigma_0=u\frac{\partial}{\partial r}\wedge\frac{\partial}{\partial t} σ 0 = u ∂ r ∂ ∧ ∂ t ∂ u u u f = r / u f=r/u f = r / u { r > 0 } \{r>0\} { r > 0 } { r < 0 } \{r<0\} { r < 0 } U j ∖ D j U_j\setminus D_j U j ∖ D j Σ ∖ D \Sigma\setminus D Σ ∖ D
V i ∩ U j ∈ { ∅ , S ℓ j 1 × ( 0 , c j ) , S ℓ j 1 × ( − c j , 0 ) } . (28) V_i\cap U_j\in\left\{\varnothing,\ S^1_{\ell_j}\times(0,c_j),\ S^1_{\ell_j}\times(-c_j,0)\right\}. \tag*{(28)} V i ∩ U j ∈ { ∅ , S ℓ j 1 × ( 0 , c j ) , S ℓ j 1 × ( − c j , 0 ) } . (28) If V i V_i V i U j U_j U j D D D ⋃ j U j \bigcup_j U_j ⋃ j U j D D D V i V_i V i Σ \Sigma Σ V i V_i V i Σ \Sigma Σ 2 n D 2n_D 2 n D U j ∖ D j U_j\setminus D_j U j ∖ D j V i V_i V i
Adjoint integrations
We recall the description of source-connected integrations used to identify G Σ \mathcal{G}_{\Sigma} G Σ [14 ] . Let E E E M M M E E E Gpd ( E ) \operatorname{Gpd}(E) Gpd ( E ) E E E [14 Definition 1.19 and the discussion following it] . The initial object is the source-simply-connected integration G s s c \mathcal{G}^{\mathrm{ssc}} G ssc adjoint integration G a d j \mathcal{G}^{\mathrm{adj}} G adj
Let Λ ( G s s c ) \Lambda(\mathcal{G}^{\mathrm{ssc}}) Λ ( G ssc ) N ⊂ G s s c \mathcal{N} \subset\mathcal{G}^{\mathrm{ssc}} N ⊂ G ssc discrete , meaning that s ∣ N s|_{\mathcal{N}} s ∣ N totally disconnected , meaning that s = t s=t s = t N \mathcal{N} N normal . By [14 Theorems 1.14 and 1.20] , the assignment
Λ ( G s s c ) ⟶ Gpd ( E ) , N ⟼ G s s c / N \Lambda(\mathcal{G}^{\mathrm{ssc}}) \longrightarrow\operatorname{Gpd}(E), \qquad\mathcal{N} \longmapsto\mathcal{G}^{\mathrm{ssc}}/\mathcal{N} Λ ( G ssc ) ⟶ Gpd ( E ) , N ⟼ G ssc / N is an equivalence. The quotient map is a surjective local diffeomorphism, and the quotient is Hausdorff if G s s c \mathcal{G}^{\mathrm{ssc}} G ssc N \mathcal{N} N N ∈ Λ ( G s s c ) \mathcal{N} \in\Lambda(\mathcal{G}^{\mathrm{ssc}}) N ∈ Λ ( G ssc ) G a d j = G s s c / N \mathcal{G}^{\mathrm{adj}}=\mathcal{G}^{\mathrm{ssc}}/\mathcal{N} G adj = G ssc / N
To pass from local to global integrations, an open cover O \mathcal{O} O M M M orbit cover if every orbit of E E E O , O ′ ∈ O O,O' \in\mathcal{O} O , O ′ ∈ O G ~ O \widetilde{\mathcal{G}}_{O} G O G ~ O O ′ \widetilde{\mathcal{G}}_{OO'} G O O ′ E ∣ O E|_{O} E ∣ O E ∣ O ∩ O ′ E|_{O\cap O'} E ∣ O ∩ O ′
res O , O ′ : Λ ( G ~ O ) ⟶ Λ ( G ~ O O ′ ) \operatorname{res}_{O,O'}:\Lambda(\widetilde{\mathcal{G}}_{O})\longrightarrow\Lambda(\widetilde{\mathcal{G}}_{OO'}) res O , O ′ : Λ ( G O ) ⟶ Λ ( G O O ′ ) sends N \mathcal{N} N N ∩ ( G ~ O ∣ O ∩ O ′ ) c \mathcal{N}\cap(\widetilde{\mathcal{G}}_{O|O\cap O'})^{c} N ∩ ( G O ∣ O ∩ O ′ ) c
G ~ O O ′ ⟶ ( G ~ O ∣ O ∩ O ′ ) c . \widetilde{\mathcal{G}}_{OO'}\longrightarrow(\widetilde{\mathcal{G}}_{O|O\cap O'})^{c}. G O O ′ ⟶ ( G O ∣ O ∩ O ′ ) c . A family ( N O ) O ∈ O (\mathcal{N}_{O})_{O\in\mathcal{O}} ( N O ) O ∈ O compatible if
res O , O ′ ( N O ) = res O ′ , O ( N O ′ ) for all O , O ′ ∈ O . \operatorname{res}_{O,O'}(\mathcal{N}_{O})=\operatorname{res}_{O',O}(\mathcal{N}_{O'})
\quad\text{for all }O,O'\in\mathcal{O}. res O , O ′ ( N O ) = res O ′ , O ( N O ′ ) for all O , O ′ ∈ O . For a locally finite orbit cover, compatible families glue to source-connected integrations G \mathcal{G} G ( G ∣ O ) c = G ~ O / N O (\mathcal{G}|_{O})^{c}=\widetilde{\mathcal{G}}_{O}/\mathcal{N}_{O} ( G ∣ O ) c = G O / N O Gpd ( E ) \operatorname{Gpd}(E) Gpd ( E ) [14 Theorem 3.4 and Corollary 3.7] . We will use the following gluing criterion : if each Λ ( G ~ O ) \Lambda(\widetilde{\mathcal{G}}_{O}) Λ ( G O ) N O \mathcal{N}_{O} N O E E E ( N O ) (\mathcal{N}_{O}) ( N O )
The action groupoid of Aff + ( R ) \operatorname{Aff}^{+}(\mathbb{R}) Aff + ( R )
We first describe the source-simply-connected symplectic integration of the cylinder ( U ^ j , σ ) (\widehat{U}_{j},\sigma) ( U j , σ ) (26) , and then take the quotient giving its adjoint integration. This is the surface case of the action-groupoid model in [14 §A.2] .
The cotangent Lie algebroid T σ ∗ U ^ j T^{*}_{\sigma}\widehat{U}_{j} T σ ∗ U j ( d r , d t ) (dr,dt) ( d r , d t )
σ ( d r ) = r ∂ ∂ t , σ ( d t ) = − r ∂ ∂ r , [ d r , d t ] = d ( σ ( d r , d t ) ) = d r . \sigma(dr)=r\frac{\partial}{\partial t},\qquad\sigma(dt)=-r\frac{\partial}{\partial r},\qquad[dr,dt]=d\bigl(\sigma(dr,dt)\bigr)=dr. σ ( d r ) = r ∂ t ∂ , σ ( d t ) = − r ∂ r ∂ , [ d r , d t ] = d ( σ ( d r , d t ) ) = d r . So its anchor is onto where r ≠ 0 r\ne0 r = 0 D j = { r = 0 } D_{j}=\{r=0\} D j = { r = 0 } T x ∗ U ^ j T^{*}_{x}\widehat{U}_{j} T x ∗ U j x ∈ D j x\in D_{j} x ∈ D j a f f ( R ) \mathfrak{aff}(\mathbb{R}) aff ( R ) Aff + ( R ) \operatorname{Aff}^{+}(\mathbb{R}) Aff + ( R )
The group Aff + ( R ) = { ( a , b ) : a > 0 , b ∈ R } \operatorname{Aff}^{+}(\mathbb{R})=\{(a,b):a>0,\ b\in\mathbb{R}\} Aff + ( R ) = {( a , b ) : a > 0 , b ∈ R }
( a 2 , b 2 ) ( a 1 , b 1 ) = ( a 2 a 1 , b 1 + a 1 b 2 ) (29) (a_{2},b_{2})(a_{1},b_{1})=(a_{2}a_{1},b_{1}+a_{1}b_{2}) \tag*{(29)} ( a 2 , b 2 ) ( a 1 , b 1 ) = ( a 2 a 1 , b 1 + a 1 b 2 ) (29) with unit ( 1 , 0 ) (1,0) ( 1 , 0 ) ( a , b ) − 1 = ( a − 1 , − b / a ) (a,b)^{-1}=(a^{-1},-b/a) ( a , b ) − 1 = ( a − 1 , − b / a ) R \mathbb{R} R ( a , b ) ↦ ( x ↦ ( x − b ) / a ) (a,b)\mapsto(x\mapsto(x-b)/a) ( a , b ) ↦ ( x ↦ ( x − b ) / a ) { ( 1 , b ) : b ∈ R } \{(1,b):b\in\mathbb{R}\} {( 1 , b ) : b ∈ R } translation subgroup , and the conjugates of { ( a , 0 ) : a > 0 } \{(a,0):a>0\} {( a , 0 ) : a > 0 } dilation subgroups . The group acts on U ^ j \widehat{U}_{j} U j ( a , b ) ⋅ ( t , r ) = ( t + b r , a r ) (a,b)\cdot(t,r)=(t+br,ar) ( a , b ) ⋅ ( t , r ) = ( t + b r , a r ) G j = Aff + ( R ) × U ^ j ⇉ U ^ j \mathcal{G}_{j}=\operatorname{Aff}^{+}(\mathbb{R})\times\widehat{U}_{j}\rightrightarrows\widehat{U}_{j} G j = Aff + ( R ) × U j ⇉ U j ( a , b , t , r ) (a,b,t,r) ( a , b , t , r )
s ( a , b , t , r ) = ( t , r ) , s(a,b,t,r)=(t,r), s ( a , b , t , r ) = ( t , r ) , t ( a , b , t , r ) = ( t + b r , a r ) , t(a,b,t,r)=(t+br,ar), t ( a , b , t , r ) = ( t + b r , a r ) , 1 ( t , r ) = ( 1 , 0 , t , r ) , 1_{(t,r)}=(1,0,t,r), 1 ( t , r ) = ( 1 , 0 , t , r ) , ( a 2 , b 2 , t + b 1 r , a 1 r ) ( a 1 , b 1 , t , r ) = ( a 2 a 1 , b 1 + a 1 b 2 , t , r ) . (a_{2},b_{2},t+b_{1}r,a_{1}r)(a_{1},b_{1},t,r)=(a_{2}a_{1},b_{1}+a_{1}b_{2},t,r). ( a 2 , b 2 , t + b 1 r , a 1 r ) ( a 1 , b 1 , t , r ) = ( a 2 a 1 , b 1 + a 1 b 2 , t , r ) . As a manifold, G j = Aff + ( R ) × U ^ j \mathcal{G}_j=\operatorname{Aff}^{+}(\mathbb{R})\times\widehat{U}_j G j = Aff + ( R ) × U j G j \mathcal{G}_j G j R 2 \mathbb{R}^2 R 2 Aff + ( R ) \operatorname{Aff}^{+}(\mathbb{R}) Aff + ( R ) α = log a \alpha=\log a α = log a a r ≠ 0 ar\ne0 a r = 0 r ≠ 0 r\ne0 r = 0 s − 1 ( { r ≠ 0 } ) = t − 1 ( { r ≠ 0 } ) s^{-1}(\{r\ne0\})=t^{-1}(\{r\ne0\}) s − 1 ({ r = 0 }) = t − 1 ({ r = 0 }) G j \mathcal{G}_j G j
Let ζ 1 , ζ 2 ∈ a f f ( R ) \zeta_1,\zeta_2\in\mathfrak{aff}(\mathbb{R}) ζ 1 , ζ 2 ∈ aff ( R ) 0 0 0 α ↦ ( e α , 0 ) \alpha\mapsto(e^\alpha,0) α ↦ ( e α , 0 ) b ↦ ( 1 , b ) b\mapsto(1,b) b ↦ ( 1 , b ) Lie ( G j ) \operatorname{Lie}(\mathcal{G}_j) Lie ( G j ) a f f ( R ) × U ^ j \mathfrak{aff}(\mathbb{R})\times\widehat{U}_j aff ( R ) × U j ζ 1 \zeta_1 ζ 1 ζ 2 \zeta_2 ζ 2 r ∂ ∂ r r\frac{\partial}{\partial r} r ∂ r ∂ r ∂ ∂ t r\frac{\partial}{\partial t} r ∂ t ∂ { r ≠ 0 } \{r\ne0\} { r = 0 } [ r ∂ ∂ r , r ∂ ∂ t ] = r ∂ ∂ t \left[r\frac{\partial}{\partial r},r\frac{\partial}{\partial t}\right]=r\frac{\partial}{\partial t} [ r ∂ r ∂ , r ∂ t ∂ ] = r ∂ t ∂ [ ζ 1 , ζ 2 ] = ζ 2 [\zeta_1,\zeta_2]=\zeta_2 [ ζ 1 , ζ 2 ] = ζ 2 Lie ( G j ) \operatorname{Lie}(\mathcal{G}_j) Lie ( G j )
χ : T ∗ U ^ j ⟶ Lie ( G j ) , χ ( d r ) = ζ 2 , χ ( d t ) = − ζ 1 . (30) \chi:T^*\widehat{U}_j\longrightarrow\operatorname{Lie}(\mathcal{G}_j),\qquad\chi(dr)=\zeta_2,\qquad\chi(dt)=-\zeta_1. \tag*{(30)} χ : T ∗ U j ⟶ Lie ( G j ) , χ ( d r ) = ζ 2 , χ ( d t ) = − ζ 1 . (30) The map χ \chi χ
[ χ ( d r ) , χ ( d t ) ] = [ ζ 2 , − ζ 1 ] = ζ 2 = χ ( [ d r , d t ] ) . [\chi(dr),\chi(dt)]=[\zeta_2,-\zeta_1]=\zeta_2=\chi([dr,dt]). [ χ ( d r ) , χ ( d t )] = [ ζ 2 , − ζ 1 ] = ζ 2 = χ ([ d r , d t ]) . The Leibniz rule then shows that χ \chi χ G j \mathcal{G}_j G j R 2 \mathbb{R}^2 R 2 G j \mathcal{G}_j G j T ∗ U ^ j T^*\widehat{U}_j T ∗ U j
Lemma 3.1. The 2 2 2
Ω = d t ∧ d α + d b ∧ d r + b d r ∧ d α + r d b ∧ d α (31) \Omega=dt\wedge d\alpha+db\wedge dr+b\,dr\wedge d\alpha+r\,db\wedge d\alpha \tag*{(31)} Ω = d t ∧ d α + d b ∧ d r + b d r ∧ d α + r d b ∧ d α (31) on G j \mathcal{G}_j G j ( G j , Ω ) (\mathcal{G}_j,\Omega) ( G j , Ω ) ( U ^ j , σ ) (\widehat{U}_j,\sigma) ( U j , σ ) v ↦ ι v Ω ∣ T U ^ j v\mapsto\iota_v\Omega|_{T\widehat{U}_j} v ↦ ι v Ω ∣ T U j χ \chi χ Ω = t ∗ ω − s ∗ ω \Omega=t^*\omega-s^*\omega Ω = t ∗ ω − s ∗ ω s − 1 ( { r ≠ 0 } ) s^{-1}(\{r\ne0\}) s − 1 ({ r = 0 })
Proof. Up to sign, Ω \Omega Ω [27 §5.2, (5–1)] , written there on ( R 2 , x ∂ ∂ x ∧ ∂ ∂ y ) (\mathbb{R}^2,x\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}) ( R 2 , x ∂ x ∂ ∧ ∂ y ∂ ) Ω ∧ Ω = 2 d t ∧ d α ∧ d b ∧ d r \Omega\wedge\Omega=2\,dt\wedge d\alpha\wedge db\wedge dr Ω ∧ Ω = 2 d t ∧ d α ∧ d b ∧ d r Ω \Omega Ω s − 1 ( { r ≠ 0 } ) s^{-1}(\{r\ne0\}) s − 1 ({ r = 0 }) d ( t + b r ) = d t + b d r + r d b d(t+br)=dt+b\,dr+r\,db d ( t + b r ) = d t + b d r + r d b d ( a r ) / a r = d α + d r / r d(ar)/ar=d\alpha+dr/r d ( a r ) / a r = d α + d r / r
t ∗ ω = ( d t + b d r + r d b ) ∧ ( d α + d r r ) = Ω + 1 r d t ∧ d r = Ω + s ∗ ω . \begin{aligned}
t^*\omega=(dt+b\,dr+r\,db)\wedge\left(d\alpha+\frac{dr}{r}\right) \\
&=\Omega+\frac{1}{r}dt\wedge dr \\
&=\Omega+s^*\omega.
\end{aligned} t ∗ ω = ( d t + b d r + r d b ) ∧ ( d α + r d r ) = Ω + r 1 d t ∧ d r = Ω + s ∗ ω . So Ω = ( t , s ) ∗ ( ω ⊕ ( − ω ) ) \Omega=(t,s)^*(\omega\oplus(-\omega)) Ω = ( t , s ) ∗ ( ω ⊕ ( − ω )) s − 1 ( { r ≠ 0 } ) s^{-1}(\{r\ne0\}) s − 1 ({ r = 0 }) Ω \Omega Ω ( t , s ) (t,s) ( t , s ) { r ≠ 0 } × { r ≠ 0 } \{r\ne0\}\times\{r\ne0\} { r = 0 } × { r = 0 } Ω − 1 \Omega^{-1} Ω − 1 ( t , s ) (t,s) ( t , s ) σ ⊕ ( − σ ) \sigma\oplus(-\sigma) σ ⊕ ( − σ ) Ω − 1 \Omega^{-1} Ω − 1 σ \sigma σ t t t G j \mathcal{G}_j G j (30) . At a unit, b = 0 b=0 b = 0 ζ 1 = ∂ ∂ α \zeta_1=\frac{\partial}{\partial\alpha} ζ 1 = ∂ α ∂ ζ 2 = ∂ ∂ b \zeta_2=\frac{\partial}{\partial b} ζ 2 = ∂ b ∂ (31) therefore gives
ι ζ 1 Ω ∣ T U ^ j = − d t = χ − 1 ( ζ 1 ) , \iota_{\zeta_1}\Omega|_{T\widehat{U}_j}=-dt=\chi^{-1}(\zeta_1), ι ζ 1 Ω ∣ T U j = − d t = χ − 1 ( ζ 1 ) , ι ζ 2 Ω ∣ T U ^ j = d r = χ − 1 ( ζ 2 ) . \iota_{\zeta_2}\Omega|_{T\widehat{U}_j}=dr=\chi^{-1}(\zeta_2). ι ζ 2 Ω ∣ T U j = d r = χ − 1 ( ζ 2 ) . The adjoint integration is obtained by quotienting G j \mathcal{G}_j G j D j D_j D j ℓ j \ell_j ℓ j
Lemma 3.2. The subset
N j = { ( 1 , 0 , t , 0 ) : t ∈ S ℓ j 1 } ⊔ { ( 1 , k ℓ j / r , t , r ) : k ∈ Z , t ∈ S ℓ j 1 , r ≠ 0 } N_j=\{(1,0,t,0):t\in S^1_{\ell_j}\}\ \sqcup\ \{(1,k\ell_j/r,t,r):k\in\mathbb{Z},\ t\in S^1_{\ell_j},\ r\ne0\} N j = {( 1 , 0 , t , 0 ) : t ∈ S ℓ j 1 } ⊔ {( 1 , k ℓ j / r , t , r ) : k ∈ Z , t ∈ S ℓ j 1 , r = 0 } is a closed wide Lie subgroupoid of G j \mathcal{G}_j G j N j , ( t , 0 ) = { ( 1 , 0 ) } N_{j,(t,0)}=\{(1,0)\} N j , ( t , 0 ) = {( 1 , 0 )} N j , ( t , r ) = { ( 1 , k ℓ j / r ) : k ∈ Z } = G j , ( t , r ) N_{j,(t,r)}=\{(1,k\ell_j/r):k\in\mathbb{Z}\}=\mathcal{G}_{j,(t,r)} N j , ( t , r ) = {( 1 , k ℓ j / r ) : k ∈ Z } = G j , ( t , r ) r ≠ 0 r\ne0 r = 0 s ∣ N j : N j → U ^ j s|_{N_j}:N_j\to\widehat{U}_j s ∣ N j : N j → U j N j N_j N j
Proof. For r ≠ 0 r\ne0 r = 0 k ∈ Z k\in\mathbb{Z} k ∈ Z g k ( t , r ) = ( 1 , k ℓ j / r , t , r ) g_k(t,r)=(1,k\ell_j/r,t,r) g k ( t , r ) = ( 1 , k ℓ j / r , t , r )
t ( g k ( t , r ) ) = ( t + k ℓ j , r ) = ( t , r ) , g k ( t , r ) g k ′ ( t , r ) = g k + k ′ ( t , r ) , g k ( t , r ) − 1 = g − k ( t , r ) . t(g_k(t,r))=(t+k\ell_j,r)=(t,r),\qquad g_k(t,r)g_{k'}(t,r)=g_{k+k'}(t,r),\qquad g_k(t,r)^{-1}=g_{-k}(t,r). t ( g k ( t , r )) = ( t + k ℓ j , r ) = ( t , r ) , g k ( t , r ) g k ′ ( t , r ) = g k + k ′ ( t , r ) , g k ( t , r ) − 1 = g − k ( t , r ) . Since N j N_j N j r = 0 r=0 r = 0 r ≠ 0 r\ne0 r = 0 ( a , b , t , r ) (a,b,t,r) ( a , b , t , r ) a r = r ar=r a r = r t + b r = t t+br=t t + b r = t S ℓ j 1 S^1_{\ell_j} S ℓ j 1 a = 1 a=1 a = 1 b ∈ ( ℓ j / r ) Z b\in(\ell_j/r)\mathbb{Z} b ∈ ( ℓ j / r ) Z n 0 = ( 1 , b 0 , t 0 , r 0 ) ∈ N j n_0=(1,b_0,t_0,r_0)\in\mathcal{N}_j n 0 = ( 1 , b 0 , t 0 , r 0 ) ∈ N j r 0 ≠ 0 r_0\ne0 r 0 = 0 k 0 ∈ Z k_0\in\mathbb{Z} k 0 ∈ Z b 0 = k 0 ℓ j / r 0 b_0=k_0\ell_j/r_0 b 0 = k 0 ℓ j / r 0 ε 0 = ℓ j / ( 3 ∣ r 0 ∣ ) \varepsilon_0=\ell_j/(3|r_0|) ε 0 = ℓ j / ( 3∣ r 0 ∣ )
B = { ( t , r ) : ∣ r − r 0 ∣ < ∣ r 0 ∣ / 2 } , B ~ = { ( a , b , t , r ) : ∣ a − 1 ∣ < 1 2 , ( t , r ) ∈ B , ∣ b − k 0 ℓ j r ∣ < ε 0 } . B=\{(t,r):|r-r_0|<|r_0|/2\},\qquad\widetilde{B}=\{(a,b,t,r):|a-1|<\tfrac{1}{2},\ (t,r)\in B,\ |b-\tfrac{k_0\ell_j}{r}|<\varepsilon_0\}. B = {( t , r ) : ∣ r − r 0 ∣ < ∣ r 0 ∣/2 } , B = {( a , b , t , r ) : ∣ a − 1∣ < 2 1 , ( t , r ) ∈ B , ∣ b − r k 0 ℓ j ∣ < ε 0 } . Then ( a , b , t , r ) ↦ ( a − 1 , b − k 0 ℓ j / r ) (a,b,t,r)\mapsto(a-1,b-k_0\ell_j/r) ( a , b , t , r ) ↦ ( a − 1 , b − k 0 ℓ j / r ) B ~ \widetilde{B} B n 0 n_0 n 0 R 2 \mathbb{R}^2 R 2 ( t , r ) ∈ B (t,r)\in B ( t , r ) ∈ B k ℓ j / r k\ell_j/r k ℓ j / r ℓ j / ∣ r ∣ > 2 ℓ j / ( 3 ∣ r 0 ∣ ) = 2 ε 0 \ell_j/|r|>2\ell_j/(3|r_0|)=2\varepsilon_0 ℓ j /∣ r ∣ > 2 ℓ j / ( 3∣ r 0 ∣ ) = 2 ε 0 N j ∩ B ~ = { ( 1 , k 0 ℓ j / r , t , r ) : ( t , r ) ∈ B } \mathcal{N}_j\cap\widetilde{B}=\{(1,k_0\ell_j/r,t,r):(t,r)\in B\} N j ∩ B = {( 1 , k 0 ℓ j / r , t , r ) : ( t , r ) ∈ B } r 0 = 0 r_0=0 r 0 = 0 B = { ( t , r ) : ∣ r ∣ < ℓ j / 2 } B=\{(t,r):|r|<\ell_j/2\} B = {( t , r ) : ∣ r ∣ < ℓ j /2 } B ~ = { ( a , b , t , r ) : ∣ a − 1 ∣ < 1 2 , ( t , r ) ∈ B , ∣ b ∣ < 1 } \widetilde{B}=\{(a,b,t,r):|a-1|<\tfrac{1}{2},\ (t,r)\in B,\ |b|<1\} B = {( a , b , t , r ) : ∣ a − 1∣ < 2 1 , ( t , r ) ∈ B , ∣ b ∣ < 1 } ( a , b , t , r ) ↦ ( a − 1 , b ) (a,b,t,r)\mapsto(a-1,b) ( a , b , t , r ) ↦ ( a − 1 , b ) ∣ k ℓ j / r ∣ > 2 |k\ell_j/r|>2 ∣ k ℓ j / r ∣ > 2 0 < ∣ r ∣ < ℓ j / 2 0<|r|<\ell_j/2 0 < ∣ r ∣ < ℓ j /2 k ≠ 0 k\ne0 k = 0 N j ∩ B ~ \mathcal{N}_j\cap\widetilde{B} N j ∩ B B B B N j \mathcal{N}_j N j N j ∩ B ~ \mathcal{N}_j\cap\widetilde{B} N j ∩ B s s s B B B s ∣ N j s|_{\mathcal{N}_j} s ∣ N j
For closedness, let ( 1 , b i , t i , r i ) ∈ N j (1,b_i,t_i,r_i)\in\mathcal{N}_j ( 1 , b i , t i , r i ) ∈ N j ( a , b , t , r ) (a,b,t,r) ( a , b , t , r ) a = 1 a=1 a = 1 r i = 0 r_i=0 r i = 0 i i i b i = 0 b_i=0 b i = 0 i i i ( a , b , t , r ) = ( 1 , 0 , t , 0 ) ∈ N j (a,b,t,r)=(1,0,t,0)\in\mathcal{N}_j ( a , b , t , r ) = ( 1 , 0 , t , 0 ) ∈ N j r i ≠ 0 r_i\ne0 r i = 0 b i = k i ℓ j / r i b_i=k_i\ell_j/r_i b i = k i ℓ j / r i k i ∈ Z k_i\in\mathbb{Z} k i ∈ Z r ≠ 0 r\ne0 r = 0 k i = b i r i / ℓ j → b r / ℓ j k_i=b_i r_i/\ell_j\to br/\ell_j k i = b i r i / ℓ j → b r / ℓ j k i = k k_i=k k i = k b = k ℓ j / r b=k\ell_j/r b = k ℓ j / r r = 0 r=0 r = 0 k i ℓ j = b i r i → 0 k_i\ell_j=b_i r_i\to0 k i ℓ j = b i r i → 0 k i = 0 k_i=0 k i = 0 b = 0 b=0 b = 0
Lemma 3.3. The subgroupoid N j \mathcal{N}_j N j Λ ( G j ) \Lambda(\mathcal{G}_j) Λ ( G j )
Proof. By Lemma 3.2, N j \mathcal{N}_j N j g = ( a , b , t , r ) g=(a,b,t,r) g = ( a , b , t , r ) x = ( t , r ) x=(t,r) x = ( t , r ) y = ( t + b r , a r ) y=(t+br,ar) y = ( t + b r , a r ) r ≠ 0 r\ne0 r = 0 a r ≠ 0 ar\ne0 a r = 0 N j , x = G j , x \mathcal{N}_{j,x}=\mathcal{G}_{j,x} N j , x = G j , x N j , y = G j , y \mathcal{N}_{j,y}=\mathcal{G}_{j,y} N j , y = G j , y g G j , x g − 1 = G j , y g\mathcal{G}_{j,x}g^{-1}=\mathcal{G}_{j,y} g G j , x g − 1 = G j , y r = 0 r=0 r = 0 y = x y=x y = x N j , x \mathcal{N}_{j,x} N j , x g N j , x g − 1 = N j , y g\mathcal{N}_{j,x}g^{-1}=\mathcal{N}_{j,y} g N j , x g − 1 = N j , y N j \mathcal{N}_j N j
Let N ′ ∈ Λ ( G j ) \mathcal{N}'\in\Lambda(\mathcal{G}_j) N ′ ∈ Λ ( G j ) s = t s=t s = t N ′ \mathcal{N}' N ′ N ′ = ⋃ x N x ′ \mathcal{N}'=\bigcup_x\mathcal{N}'_x N ′ = ⋃ x N x ′ N x ′ ⊂ G j , x \mathcal{N}'_x\subset\mathcal{G}_{j,x} N x ′ ⊂ G j , x s ∣ N ′ s|_{\mathcal{N}'} s ∣ N ′ N x ′ = ( s ∣ N ′ ) − 1 ( x ) \mathcal{N}'_x=(s|_{\mathcal{N}'})^{-1}(x) N x ′ = ( s ∣ N ′ ) − 1 ( x ) G j , x \mathcal{G}_{j,x} G j , x N ′ \mathcal{N}' N ′ x = ( t , r ) x=(t,r) x = ( t , r ) r ≠ 0 r\ne0 r = 0 N x ′ ⊂ G j , x = N j , x \mathcal{N}'_x\subset\mathcal{G}_{j,x}=\mathcal{N}_{j,x} N x ′ ⊂ G j , x = N j , x x = ( t , 0 ) x=(t,0) x = ( t , 0 ) N x ′ \mathcal{N}'_x N x ′ G j , x = Aff + ( R ) \mathcal{G}_{j,x}=\operatorname{Aff}^{+}(\mathbb{R}) G j , x = Aff + ( R ) N x ′ = { ( 1 , 0 ) } = N j , x \mathcal{N}'_x=\{(1,0)\}=\mathcal{N}_{j,x} N x ′ = {( 1 , 0 )} = N j , x N ′ ⊂ N j \mathcal{N}'\subset\mathcal{N}_j N ′ ⊂ N j
Proposition 3.4. The quotient G ‾ j = G j / N j ⇒ U ^ j \overline{\mathcal{G}}_j=\mathcal{G}_j/\mathcal{N}_j\Rightarrow\widehat{U}_j G j = G j / N j ⇒ U j T ∗ U ^ j T^*\widehat{U}_j T ∗ U j q j : G j → G ‾ j q_j:\mathcal{G}_j\to\overline{\mathcal{G}}_j q j : G j → G j 2 2 2 Ω ‾ j \overline{\Omega}_j Ω j G ‾ j \overline{\mathcal{G}}_j G j
q j ∗ Ω ‾ j = Ω . (32) q_j^*\overline{\Omega}_j=\Omega. \tag*{(32)} q j ∗ Ω j = Ω. (32) The pair ( G ‾ j , Ω ‾ j ) (\overline{\mathcal{G}}_j,\overline{\Omega}_j) ( G j , Ω j ) ( U ^ j , σ ) (\widehat{U}_j,\sigma) ( U j , σ )
Proof. By Lemmas 3.2 and 3.3 , N j \mathcal{N}_j N j Λ ( G j ) \Lambda(\mathcal{G}_j) Λ ( G j ) G j \mathcal{G}_j G j T ∗ U ^ j T^*\widehat{U}_j T ∗ U j (30) . Since G j \mathcal{G}_j G j [14 Theorems 1.14 and 1.20] show that G ‾ j \overline{\mathcal{G}}_j G j q j q_j q j
For k ∈ Z k\in\mathbb{Z} k ∈ Z R k ( g ) = g ⋅ ( 1 , k ℓ j / r , t , r ) R_k(g)=g\cdot(1,k\ell_j/r,t,r) R k ( g ) = g ⋅ ( 1 , k ℓ j / r , t , r ) s − 1 ( { r ≠ 0 } ) s^{-1}(\{r\ne0\}) s − 1 ({ r = 0 }) s ( g ) = ( t , r ) s(g)=(t,r) s ( g ) = ( t , r ) R k ( a , b , t , r ) = ( a , b + k ℓ j / r , t , r ) R_k(a,b,t,r)=(a,b+k\ell_j/r,t,r) R k ( a , b , t , r ) = ( a , b + k ℓ j / r , t , r ) d t ∧ d a d t\wedge d a d t ∧ d a d b ∧ d r d b\wedge d r d b ∧ d r (31) are unchanged, while
R k ∗ Ω − Ω = k ℓ j r d r ∧ d a + r ( − k ℓ j r 2 d r ) ∧ d a = 0. R_k^*\Omega-\Omega=\frac{k\ell_j}{r}\,dr\wedge da+r\left(-\frac{k\ell_j}{r^2}\,dr\right)\wedge da=0. R k ∗ Ω − Ω = r k ℓ j d r ∧ d a + r ( − r 2 k ℓ j d r ) ∧ d a = 0. To descend the form, let ϕ \phi ϕ G j \mathcal{G}_j G j q j ∘ ϕ = q j q_j\circ\phi=q_j q j ∘ ϕ = q j q j q_j q j N j \mathcal{N}_j N j ϕ ( g ) = g n ( g ) \phi(g)=g\,n(g) ϕ ( g ) = g n ( g ) n ( g ) = g − 1 ϕ ( g ) n(g)=g^{-1}\phi(g) n ( g ) = g − 1 ϕ ( g ) N j \mathcal{N}_j N j s ∘ n = s s\circ n=s s ∘ n = s g 0 g_0 g 0 s ( g 0 ) ∈ { r ≠ 0 } s(g_0)\in\{r\ne0\} s ( g 0 ) ∈ { r = 0 } k k k n ( g ) = ( 1 , k ℓ j / r , s ( g ) ) n(g)=(1,k\ell_j/r,s(g)) n ( g ) = ( 1 , k ℓ j / r , s ( g )) ϕ = R k \phi=R_k ϕ = R k g 0 g_0 g 0 s ( g 0 ) ∈ { r = 0 } s(g_0)\in\{r=0\} s ( g 0 ) ∈ { r = 0 } n ( g 0 ) n(g_0) n ( g 0 ) r 0 = 0 r_0=0 r 0 = 0 ϕ = id \phi=\operatorname{id} ϕ = id ϕ ∗ Ω = Ω \phi^*\Omega=\Omega ϕ ∗ Ω = Ω
For open sets O ⊂ G j O\subset\mathcal{G}_j O ⊂ G j q j q_j q j ( ( q j ∣ O ) − 1 ) ∗ Ω ((q_j|_O)^{-1})^*\Omega (( q j ∣ O ) − 1 ) ∗ Ω q j ( O ) q_j(O) q j ( O ) ϕ \phi ϕ 2 2 2 Ω ‾ j \overline{\Omega}_j Ω j (32) , which is unique.
Since q j q_j q j Ω ‾ j \overline{\Omega}_j Ω j G ‾ j \overline{\mathcal{G}}_j G j m ‾ \overline{m} m G ‾ j \overline{\mathcal{G}}_j G j q j × q j q_j\times q_j q j × q j
( q j × q j ) ∗ m ‾ ∗ Ω ‾ j = m ∗ Ω (q_j\times q_j)^*\overline{m}^*\overline{\Omega}_j=m^*\Omega ( q j × q j ) ∗ m ∗ Ω j = m ∗ Ω = pr 1 ∗ Ω + pr 2 ∗ Ω =\operatorname{pr}_1^*\Omega+\operatorname{pr}_2^*\Omega = pr 1 ∗ Ω + pr 2 ∗ Ω = ( q j × q j ) ∗ ( pr 1 ∗ Ω ‾ j + pr 2 ∗ Ω ‾ j ) . =(q_j\times q_j)^*(\operatorname{pr}_1^*\overline{\Omega}_j+\operatorname{pr}_2^*\overline{\Omega}_j). = ( q j × q j ) ∗ ( pr 1 ∗ Ω j + pr 2 ∗ Ω j ) . Hence Ω ‾ j \overline{\Omega}_j Ω j
The gluing in §3.4 uses the restrictions of these groupoids to the open sets U I = S ℓ j 1 × I ⊂ U ^ j U_I=S^1_{\ell_j}\times I\subset\widehat{U}_j U I = S ℓ j 1 × I ⊂ U j I ⊂ R I\subset\mathbb{R} I ⊂ R ( a , b , t , r ) (a,b,t,r) ( a , b , t , r ) ( t + b r , a r ) (t+br,ar) ( t + b r , a r ) G j ∣ U I = { ( a , b , t , r ) : r ∈ I , a r ∈ I } \mathcal{G}_j|_{U_I}=\{(a,b,t,r):r\in I,\ ar\in I\} G j ∣ U I = {( a , b , t , r ) : r ∈ I , a r ∈ I }
Lemma 3.5. Let I ⊂ R I\subset\mathbb{R} I ⊂ R G j ∣ U I \mathcal{G}_j|_{U_I} G j ∣ U I T σ ∗ U I T^*_\sigma U_I T σ ∗ U I N j ∩ G j ∣ U I \mathcal{N}_j\cap\mathcal{G}_j|_{U_I} N j ∩ G j ∣ U I Λ ( G j ∣ U I ) \Lambda(\mathcal{G}_j|_{U_I}) Λ ( G j ∣ U I ) G ‾ j ∣ U I \overline{\mathcal{G}}_j|_{U_I} G j ∣ U I G j ∣ U I / ( N j ∩ G j ∣ U I ) \mathcal{G}_j|_{U_I}/(\mathcal{N}_j\cap\mathcal{G}_j|_{U_I}) G j ∣ U I / ( N j ∩ G j ∣ U I ) T σ ∗ U I T^*_\sigma U_I T σ ∗ U I Ω ‾ j \overline{\Omega}_j Ω j
Proof. The restriction G j ∣ U I \mathcal{G}_j|_{U_I} G j ∣ U I T σ ∗ U I T^*_\sigma U_I T σ ∗ U I ( t , r ) ∈ U I (t,r)\in U_I ( t , r ) ∈ U I { a > 0 : a r ∈ I } × R \{a>0:ar\in I\}\times\mathbb{R} { a > 0 : a r ∈ I } × R 1 1 1 R 2 \mathbb{R}^2 R 2 N j ∩ G j ∣ U I \mathcal{N}_j\cap\mathcal{G}_j|_{U_I} N j ∩ G j ∣ U I U I U_I U I
Since q j q_j q j s s s t t t q j − 1 ( G ‾ j ∣ U I ) = G j ∣ U I q_j^{-1}(\overline{\mathcal{G}}_j|_{U_I})=\mathcal{G}_j|_{U_I} q j − 1 ( G j ∣ U I ) = G j ∣ U I q j : G j ∣ U I → G ‾ j ∣ U I q_j:\mathcal{G}_j|_{U_I}\to\overline{\mathcal{G}}_j|_{U_I} q j : G j ∣ U I → G j ∣ U I N j ∩ G j ∣ U I \mathcal{N}_j\cap\mathcal{G}_j|_{U_I} N j ∩ G j ∣ U I G ‾ j ∣ U I \overline{\mathcal{G}}_j|_{U_I} G j ∣ U I G j ∣ U I \mathcal{G}_j|_{U_I} G j ∣ U I [14 Theorems 1.14 and 1.20] . The last claim holds since G ‾ j ∣ U I \overline{\mathcal{G}}_j|_{U_I} G j ∣ U I G ‾ j \overline{\mathcal{G}}_j G j
The adjoint symplectic groupoid G Σ \mathcal{G}_\Sigma G Σ
We now recover the adjoint integration of the surface by gluing these coordinate models. We apply the criterion of § 3.2 to E = T σ ∗ Σ E=T^*_\sigma\Sigma E = T σ ∗ Σ ( Σ , σ ) (\Sigma,\sigma) ( Σ , σ ) [14 Theorem 2.15] . Since σ \sigma σ Σ ∖ D \Sigma\setminus D Σ ∖ D Lie ( G ) → T σ ∗ Σ \operatorname{Lie}(\mathcal{G})\to T^*_\sigma\Sigma Lie ( G ) → T σ ∗ Σ
The cover { V i } i ∪ { U j } j \{V_i\}_i\cup\{U_j\}_j { V i } i ∪ { U j } j V i V_i V i D D D V i V_i V i σ \sigma σ T σ ∗ V i T^*_\sigma V_i T σ ∗ V i T V i TV_i T V i Π 1 ( V i ) \Pi_1(V_i) Π 1 ( V i ) Λ ( Π 1 ( V i ) ) \Lambda(\Pi_1(V_i)) Λ ( Π 1 ( V i )) N V i \mathcal{N}_{V_i} N V i Π 1 ( V i ) / N V i = V i × V i \Pi_1(V_i)/\mathcal{N}_{V_i}=V_i\times V_i Π 1 ( V i ) / N V i = V i × V i [14 Example 1.23] . Over U j U_j U j I = ( − c j , c j ) I=(-c_j,c_j) I = ( − c j , c j ) G j ∣ U j \mathcal{G}_j|_{U_j} G j ∣ U j N j ∩ G j ∣ U j \mathcal{N}_j\cap\mathcal{G}_j|_{U_j} N j ∩ G j ∣ U j Λ ( G j ∣ U j ) \Lambda(\mathcal{G}_j|_{U_j}) Λ ( G j ∣ U j ) G ‾ j ∣ U j \overline{\mathcal{G}}_j|_{U_j} G j ∣ U j
The family ( N V i ) i ∪ ( N j ∩ G j ∣ U j ) j (\mathcal{N}_{V_i})_i\cup(\mathcal{N}_j\cap\mathcal{G}_j|_{U_j})_j ( N V i ) i ∪ ( N j ∩ G j ∣ U j ) j V i V_i V i U j U_j U j (28) , the remaining overlaps are the components U U U U j ∖ D j U_j\setminus D_j U j ∖ D j U ⊂ V i U\subset V_i U ⊂ V i i i i U = U I U=U_I U = U I I = ( 0 , c j ) I=(0,c_j) I = ( 0 , c j ) I = ( − c j , 0 ) I=(-c_j,0) I = ( − c j , 0 ) T σ ∗ U ≃ T U T^*_\sigma U\simeq TU T σ ∗ U ≃ T U U U U Π 1 ( U ) \Pi_1(U) Π 1 ( U ) Iso ( Π 1 ( U ) ) \operatorname{Iso}(\Pi_1(U)) Iso ( Π 1 ( U )) N V i \mathcal{N}_{V_i} N V i Π 1 ( U ) → ( Π 1 ( V i ) ) U \Pi_1(U)\to(\Pi_1(V_i))_U Π 1 ( U ) → ( Π 1 ( V i ) ) U res V i , U ( N V i ) = Iso ( Π 1 ( U ) ) \operatorname{res}_{V_i,U}(\mathcal{N}_{V_i})=\operatorname{Iso}(\Pi_1(U)) res V i , U ( N V i ) = Iso ( Π 1 ( U )) G j ∣ U ≃ Π 1 ( U ) \mathcal{G}_j|_U\simeq\Pi_1(U) G j ∣ U ≃ Π 1 ( U ) N j ∩ G j ∣ U \mathcal{N}_j\cap\mathcal{G}_j|_U N j ∩ G j ∣ U G j ∣ U \mathcal{G}_j|_U G j ∣ U res U , V i ( N j ∩ G j ∣ U j ) = Iso ( Π 1 ( U ) ) \operatorname{res}_{U,V_i}(\mathcal{N}_j\cap\mathcal{G}_j|_{U_j})=\operatorname{Iso}(\Pi_1(U)) res U , V i ( N j ∩ G j ∣ U j ) = Iso ( Π 1 ( U ))
Proposition 3.6. The family ( N V i ) i ∪ ( N j ∩ G j ∣ U j ) j (\mathcal{N}_{V_i})_i\cup(\mathcal{N}_j\cap\mathcal{G}_j|_{U_j})_j ( N V i ) i ∪ ( N j ∩ G j ∣ U j ) j G Σ \mathcal{G}_\Sigma G Σ T σ ∗ Σ T^*_\sigma\Sigma T σ ∗ Σ G Σ \mathcal{G}_\Sigma G Σ 2 2 2 Ω Σ \Omega_\Sigma Ω Σ G Σ \mathcal{G}_\Sigma G Σ
G Σ ∣ V i = V i × V i , Ω Σ ∣ V i × V i = t ∗ ω i − s ∗ ω i , \mathcal{G}_\Sigma|_{V_i}=V_i\times V_i,
\qquad\Omega_\Sigma|_{V_i\times V_i}=t^*\omega_i-s^*\omega_i, G Σ ∣ V i = V i × V i , Ω Σ ∣ V i × V i = t ∗ ω i − s ∗ ω i , G Σ ∣ U j = G ‾ j ∣ U j , Ω Σ ∣ G ‾ j ∣ U j = Ω ‾ j , \mathcal{G}_\Sigma|_{U_j}=\overline{\mathcal{G}}_j|_{U_j},
\qquad\Omega_\Sigma|_{\overline{\mathcal{G}}_j|_{U_j}}=\overline{\Omega}_j, G Σ ∣ U j = G j ∣ U j , Ω Σ ∣ G j ∣ U j = Ω j , and ( G Σ , Ω Σ ) (\mathcal{G}_\Sigma,\Omega_\Sigma) ( G Σ , Ω Σ ) ( Σ , σ ) (\Sigma,\sigma) ( Σ , σ ) U j U_j U j Lie ( G Σ ) → T σ ∗ Σ \operatorname{Lie}(\mathcal{G}_\Sigma)\to T^*_\sigma\Sigma Lie ( G Σ ) → T σ ∗ Σ v ↦ ι v Ω Σ v\mapsto\iota_v\Omega_\Sigma v ↦ ι v Ω Σ q j q_j q j χ \chi χ
Proof. By compatibility and the gluing criterion of § 3.2 , the family glues to the adjoint integration G Σ \mathcal{G}_\Sigma G Σ ( G Σ ∣ V i ) c = V i × V i (\mathcal{G}_\Sigma|_{V_i})^c=V_i\times V_i ( G Σ ∣ V i ) c = V i × V i ( G Σ ∣ U j ) c = G ‾ j ∣ U j (\mathcal{G}_\Sigma|_{U_j})^c=\overline{\mathcal{G}}_j|_{U_j} ( G Σ ∣ U j ) c = G j ∣ U j G Σ \mathcal{G}_\Sigma G Σ x ∈ V i x\in V_i x ∈ V i G Σ ∣ V i \mathcal{G}_\Sigma|_{V_i} G Σ ∣ V i x x x s − 1 ( x ) s^{-1}(x) s − 1 ( x ) G Σ ∣ V i = V i × V i \mathcal{G}_\Sigma|_{V_i}=V_i\times V_i G Σ ∣ V i = V i × V i t t t s − 1 ( x ) s^{-1}(x) s − 1 ( x ) V i V_i V i x ∈ U j x\in U_j x ∈ U j G Σ ∣ U j \mathcal{G}_\Sigma|_{U_j} G Σ ∣ U j x x x s − 1 ( x ) ∩ t − 1 ( U j ) ≃ V i ∩ U j s^{-1}(x)\cap t^{-1}(U_j)\simeq V_i\cap U_j s − 1 ( x ) ∩ t − 1 ( U j ) ≃ V i ∩ U j (28) . If x ∈ D j x\in D_j x ∈ D j s − 1 ( x ) s^{-1}(x) s − 1 ( x ) x x x { x } \{x\} { x } G Σ ∣ U j = G ‾ j ∣ U j \mathcal{G}_\Sigma|_{U_j}=\overline{\mathcal{G}}_j|_{U_j} G Σ ∣ U j = G j ∣ U j
The open subgroupoids G Σ ∣ V i \mathcal{G}_\Sigma|_{V_i} G Σ ∣ V i G Σ ∣ U j \mathcal{G}_\Sigma|_{U_j} G Σ ∣ U j G Σ \mathcal{G}_\Sigma G Σ ( g , h ) (g,h) ( g , h ) V i V_i V i s ( g ) = t ( h ) ∈ V i s(g)=t(h)\in V_i s ( g ) = t ( h ) ∈ V i G ‾ j ∣ U j \overline{\mathcal{G}}_j|_{U_j} G j ∣ U j s ( g ) ∈ D j s(g)\in D_j s ( g ) ∈ D j G Σ ∣ U \mathcal{G}_\Sigma|_U G Σ ∣ U U U U ( t , s ) (t,s) ( t , s ) t ∗ ω i − s ∗ ω i t^*\omega_i-s^*\omega_i t ∗ ω i − s ∗ ω i Ω ‾ j \overline{\Omega}_j Ω j t ∗ ω − s ∗ ω t^*\omega-s^*\omega t ∗ ω − s ∗ ω (32) and ω i = ω \omega_i=\omega ω i = ω U U U 2 2 2 Ω Σ \Omega_\Sigma Ω Σ
Closedness, nondegeneracy, multiplicativity and the Poisson property of t t t U j U_j U j V i V_i V i t ∗ ω i − s ∗ ω i = ( t , s ) ∗ ( ω i ⊕ ( − ω i ) ) t^*\omega_i-s^*\omega_i=(t,s)^*(\omega_i\oplus(-\omega_i)) t ∗ ω i − s ∗ ω i = ( t , s ) ∗ ( ω i ⊕ ( − ω i )) t = pr 1 ∘ ( t , s ) t=\operatorname{pr}_1\circ(t,s) t = pr 1 ∘ ( t , s ) (32) .
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For the Hausdorff property, let g ≠ g ′ g \ne g' g = g ′ G Σ G_\Sigma G Σ ( t , s ) ( g ) ≠ ( t , s ) ( g ′ ) (t,s)(g) \ne(t,s)(g') ( t , s ) ( g ) = ( t , s ) ( g ′ ) ( t , s ) (t,s) ( t , s ) g g g g ′ g' g ′ s ( g ) = s ( g ′ ) = x s(g) = s(g') = x s ( g ) = s ( g ′ ) = x t ( g ) = t ( g ′ ) = y t(g) = t(g') = y t ( g ) = t ( g ′ ) = y x ∈ V i x \in V_i x ∈ V i g = g ′ = ( y , x ) g = g' = (y,x) g = g ′ = ( y , x ) V i × V i V_i \times V_i V i × V i x ∈ D j x \in D_j x ∈ D j j j j g , g ′ g,g' g , g ′ G j ∣ U j \mathcal{G}_j|_{U_j} G j ∣ U j
Gualtieri and Li also obtain the adjoint integration of a proper log symplectic manifold as a symplectic pair groupoid [14 Theorem 2.15 and Corollary 2.19] , which is isomorphic to G Σ G_\Sigma G Σ x ∈ D j x \in D_j x ∈ D j N j \mathcal{N}_j N j { r = 0 } \{r=0\} { r = 0 } q j q_j q j G j , x = Aff + ( R ) \mathcal{G}_{j,x} = \operatorname{Aff}^{+}(\mathbb{R}) G j , x = Aff + ( R ) G Σ , x G_{\Sigma,x} G Σ , x G Σ , x G_{\Sigma,x} G Σ , x Aff + ( R ) \operatorname{Aff}^{+}(\mathbb{R}) Aff + ( R )
Since G Σ G_\Sigma G Σ t ( g ) t(g) t ( g ) s ( g ) s(g) s ( g ) { s ( g ) } \{s(g)\} { s ( g )} s ( g ) ∈ D s(g) \in D s ( g ) ∈ D V i V_i V i s ( g ) ∈ V i s(g) \in V_i s ( g ) ∈ V i g ∈ G Σ ∣ V i = V i × V i g \in G_\Sigma|_{V_i} = V_i \times V_i g ∈ G Σ ∣ V i = V i × V i g g g G Σ G_\Sigma G Σ
t ( g ) = s ( g ) if s ( g ) ∈ D , (33) t(g) = s(g) \qquad\text{if } s(g) \in D, \tag*{(33)} t ( g ) = s ( g ) if s ( g ) ∈ D , (33) g = ( t ( g ) , s ( g ) ) ∈ V i × V i if s ( g ) ∈ V i . (34) g = (t(g),s(g)) \in V_i \times V_i \qquad\text{if } s(g) \in V_i. \tag*{(34)} g = ( t ( g ) , s ( g )) ∈ V i × V i if s ( g ) ∈ V i . (34) Lagrangian subgroupoids of admissible curves
In this section, we construct the Lagrangian integrations of curves inside the adjoint symplectic groupoid G Σ G_\Sigma G Σ N ⊂ Σ N \subset\Sigma N ⊂ Σ T N ∘ TN^\circ T N ∘ T ∗ Σ T^*\Sigma T ∗ Σ G Σ G_\Sigma G Σ § 5 and § 6 .
Admissible curves
We first fix the types of components of N N N N ∩ D N \cap D N ∩ D N N N N c N_c N c
Definition 4.1. A curve N N N N c N_c N c
N c ⊂ Σ ∖ D N_c \subset\Sigma\setminus D N c ⊂ Σ ∖ D
N c = D j N_c = D_j N c = D j j j j
N c ∩ D N_c \cap D N c ∩ D
The curve N N N tangent to D D D x ∈ N ∩ D x \in N \cap D x ∈ N ∩ D T x N = T x D T_xN = T_xD T x N = T x D N c N_c N c contact at x ∈ N c ∩ D x \in N_c \cap D x ∈ N c ∩ D tangential if T x N = T x D T_xN = T_xD T x N = T x D transverse otherwise.
Figure 2 shows a component with two transverse contacts p p p q q q p p p N N N N c N_c N c C c = N c ∩ D C_c = N_c \cap D C c = N c ∩ D N 0 N_0 N 0 N D N_D N D C C C C c C_c C c N c N_c N c ∣ C c ∣ ≥ 1 |C_c| \ge1 ∣ C c ∣ ≥ 1 C c C_c C c N c N_c N c ∣ C c ∣ |C_c| ∣ C c ∣ Σ ∖ D \Sigma\setminus D Σ ∖ D K \mathcal{K} K
Figure 2. Transverse and tangential contacts.
N = N 0 ⊔ N D ⊔ ( ⨆ K ∈ K K ) ⊔ C . (35) N = N_0 \sqcup N_D \sqcup\left(\bigsqcup_{K \in\mathcal{K}} K\right) \sqcup C. \tag*{(35)} N = N 0 ⊔ N D ⊔ ( K ∈ K ⨆ K ) ⊔ C . (35) A box around p ∈ D j p \in D_j p ∈ D j W = J × I ⊂ U j W = J \times I \subset U_j W = J × I ⊂ U j p p p J ⊂ S ℓ j 1 J \subset S^1_{\ell_j} J ⊂ S ℓ j 1 ℓ j \ell_j ℓ j I ⊂ ( − c j , c j ) I \subset(-c_j,c_j) I ⊂ ( − c j , c j ) J ~ ⊂ R \widetilde{J} \subset\mathbb{R} J ⊂ R J J J J J J t + b r t + br t + b r t − γ ( r ) t - \gamma(r) t − γ ( r ) t + b r ∈ J ~ t + br \in\widetilde{J} t + b r ∈ J ℓ j \ell_j ℓ j
Lemma 4.2. Let N c N_c N c p ∈ N c ∩ D j p \in N_c \cap D_j p ∈ N c ∩ D j t t t p = ( 0 , 0 ) p=(0,0) p = ( 0 , 0 ) W = J × I W=J\times I W = J × I p p p γ : I → J \gamma:I\to J γ : I → J γ ( 0 ) = 0 \gamma(0)=0 γ ( 0 ) = 0 ( s , r ) (s,r) ( s , r ) W W W s = t − γ ( r ) s=t-\gamma(r) s = t − γ ( r )
σ = r ∂ ∂ r ∧ ∂ ∂ s , D j ∩ W = { r = 0 } , N c ∩ W = { s = 0 } = { ( γ ( r ) , r ) : r ∈ I } . \sigma=r\frac{\partial}{\partial r}\wedge\frac{\partial}{\partial s},\qquad D_j\cap W=\{r=0\},\qquad N_c\cap W=\{s=0\}=\{(\gamma(r),r):r\in I\}. σ = r ∂ r ∂ ∧ ∂ s ∂ , D j ∩ W = { r = 0 } , N c ∩ W = { s = 0 } = {( γ ( r ) , r ) : r ∈ I } . Lemma 4.3. Let N c N_c N c p ∈ N c ∩ D j p\in N_c\cap D_j p ∈ N c ∩ D j t t t p = ( 0 , 0 ) p=(0,0) p = ( 0 , 0 ) W = J × I W=J\times I W = J × I p p p I = ( − ε , ε ) I=(-\varepsilon,\varepsilon) I = ( − ε , ε ) h : J → I h:J\to I h : J → I h ( 0 ) = h ′ ( 0 ) = 0 h(0)=h'(0)=0 h ( 0 ) = h ′ ( 0 ) = 0 h ( t ) ≠ 0 h(t)\ne0 h ( t ) = 0 t ≠ 0 t\ne0 t = 0 (26)
D j ∩ W = { r = 0 } , N c ∩ W = { ( t , h ( t ) ) : t ∈ J } . D_j\cap W=\{r=0\},\qquad N_c\cap W=\{(t,h(t)):t\in J\}. D j ∩ W = { r = 0 } , N c ∩ W = {( t , h ( t )) : t ∈ J } . Proof of Lemmas 4.2 and 4.3. Since T p D j = ⟨ ∂ ∂ t ⟩ T_pD_j=\langle\frac{\partial}{\partial t}\rangle T p D j = ⟨ ∂ t ∂ ⟩ d r dr d r T p N c T_pN_c T p N c d t dt d t N p ⊂ N c N_p\subset N_c N p ⊂ N c p p p r r r t t t
At a transverse contact, write this graph as t = γ ( r ) t=\gamma(r) t = γ ( r ) r ∈ I 0 r\in I_0 r ∈ I 0 γ ( 0 ) = 0 \gamma(0)=0 γ ( 0 ) = 0 N c ∖ N p N_c\setminus N_p N c ∖ N p p p p J × I 1 J\times I_1 J × I 1 p p p I 1 ⊂ I 0 I_1\subset I_0 I 1 ⊂ I 0 I 1 I_1 I 1 I ∋ 0 I\ni0 I ∋ 0 γ ( I ) ⊂ J \gamma(I)\subset J γ ( I ) ⊂ J W = J × I W=J\times I W = J × I N c ∩ W = { ( γ ( r ) , r ) : r ∈ I } N_c\cap W=\{(\gamma(r),r):r\in I\} N c ∩ W = {( γ ( r ) , r ) : r ∈ I } s = t − γ ( r ) s=t-\gamma(r) s = t − γ ( r )
∂ ∂ r ∧ ∂ ∂ s = ∂ ∂ r ∧ ∂ ∂ t , \frac{\partial}{\partial r}\wedge\frac{\partial}{\partial s}=\frac{\partial}{\partial r}\wedge\frac{\partial}{\partial t}, ∂ r ∂ ∧ ∂ s ∂ = ∂ r ∂ ∧ ∂ t ∂ , where the first ∂ ∂ r \frac{\partial}{\partial r} ∂ r ∂ ( s , r ) (s,r) ( s , r )
At a tangential contact, write the graph as r = h ( t ) r=h(t) r = h ( t ) t ∈ J 0 t\in J_0 t ∈ J 0
⟨ ∂ ∂ t + h ′ ( 0 ) ∂ ∂ r ⟩ = T p N c = ⟨ ∂ ∂ t ⟩ , \left\langle\frac{\partial}{\partial t}+h'(0)\frac{\partial}{\partial r}\right\rangle=T_pN_c=\left\langle\frac{\partial}{\partial t}\right\rangle, ⟨ ∂ t ∂ + h ′ ( 0 ) ∂ r ∂ ⟩ = T p N c = ⟨ ∂ t ∂ ⟩ , so h ( 0 ) = h ′ ( 0 ) = 0 h(0)=h'(0)=0 h ( 0 ) = h ′ ( 0 ) = 0 J 1 × ( − ε , ε ) J_1\times(-\varepsilon,\varepsilon) J 1 × ( − ε , ε ) N c ∖ N p N_c\setminus N_p N c ∖ N p J 1 ⊂ J 0 J_1\subset J_0 J 1 ⊂ J 0 N c ∩ D N_c\cap D N c ∩ D J 1 J_1 J 1 J ∋ 0 J\ni0 J ∋ 0
h ( J ) ⊂ I = ( − ε , ε ) , h − 1 ( 0 ) = { 0 } . h(J)\subset I=(-\varepsilon,\varepsilon),\qquad h^{-1}(0)=\{0\}. h ( J ) ⊂ I = ( − ε , ε ) , h − 1 ( 0 ) = { 0 } . For W = J × I W=J\times I W = J × I N c ∩ W = { ( t , h ( t ) ) : t ∈ J } N_c\cap W=\{(t,h(t)):t\in J\} N c ∩ W = {( t , h ( t )) : t ∈ J }
From now on, ∂ ∂ r \frac{\partial}{\partial r} ∂ r ∂ r r r
Local Lagrangian subgroupoids
We construct the integration separately over the pieces of (35) . Away from the finite contact set C C C (34) throughout.
Let K ⊂ Σ ∖ D K\subset\Sigma\setminus D K ⊂ Σ ∖ D V i V_i V i G Σ = V i × V i G_\Sigma=V_i\times V_i G Σ = V i × V i t ∗ ω i − s ∗ ω i t^*\omega_i-s^*\omega_i t ∗ ω i − s ∗ ω i K K K V i V_i V i σ : T ∗ V i → T V i \sigma:T^*V_i\to TV_i σ : T ∗ V i → T V i T K ∘ TK^\circ T K ∘ T K TK T K K × K K\times K K × K T K ∘ TK^\circ T K ∘
( t ∗ ω i − s ∗ ω i ) ∣ K × K = t ∗ ( ω i ∣ K ) − s ∗ ( ω i ∣ K ) = 0. (t^*\omega_i-s^*\omega_i)|_{K\times K}=t^*(\omega_i|_K)-s^*(\omega_i|_K)=0. ( t ∗ ω i − s ∗ ω i ) ∣ K × K = t ∗ ( ω i ∣ K ) − s ∗ ( ω i ∣ K ) = 0. Near D j D_j D j G j G_j G j q j q_j q j G Σ G_\Sigma G Σ embedding criterion . Let O ⊂ G j O\subset G_j O ⊂ G j L ⊂ O L\subset O L ⊂ O
q j − 1 ( q j ( g ) ) ∩ O = { g } for every g ∈ L . (36) q_j^{-1}(q_j(g))\cap O=\{g\}\qquad\text{for every }g\in L. \tag*{(36)} q j − 1 ( q j ( g )) ∩ O = { g } for every g ∈ L . (36) Then q j ∣ L q_j|_L q j ∣ L q j ( L ∩ O ′ ) = q j ( L ) ∩ q j ( O ′ ) q_j(L\cap O')=q_j(L)\cap q_j(O') q j ( L ∩ O ′ ) = q j ( L ) ∩ q j ( O ′ ) O ′ ⊂ O O'\subset O O ′ ⊂ O q j ∣ L q_j|_L q j ∣ L q j ( L ) q_j(L) q j ( L ) G j G_j G j q j ∣ L q_j|_L q j ∣ L [19 ] , Proposition 5.2].
The fibres of q j q_j q j N j N_j N j g ≠ g ′ g\ne g' g = g ′ G j G_j G j q j ( g ) = q j ( g ′ ) q_j(g)=q_j(g') q j ( g ) = q j ( g ′ ) g g g g ′ g' g ′ a a a t t t r ≠ 0 r\ne0 r = 0 b b b k ℓ j / r k\ell_j/r k ℓ j / r k ∈ Z ∖ { 0 } k\in\mathbb{Z}\setminus\{0\} k ∈ Z ∖ { 0 } q j q_j q j G j ∣ W G_j|_W G j ∣ W W W W r ∈ I ∖ { 0 } r\in I\setminus\{0\} r ∈ I ∖ { 0 } ( a , b , t , r ) (a,b,t,r) ( a , b , t , r ) ( a , b + k ℓ j / r , t , r ) (a,b+k\ell_j/r,t,r) ( a , b + k ℓ j / r , t , r ) G j ∣ W G_j|_W G j ∣ W q j q_j q j J ~ \widetilde J J
Lemma 4.4. Let W = J × I W=J\times I W = J × I D j D_j D j
G j , W l o c = { ( a , b , t , r ) ∈ G j : t ∈ J ~ , t + b r ∈ J ~ , r ∈ I , a r ∈ I } \mathcal{G}^{\mathrm{loc}}_{j,W}=\{(a,b,t,r)\in\mathcal{G}_j:t\in\widetilde{J},\ t+br\in\widetilde{J},\ r\in I,\ ar\in I\} G j , W loc = {( a , b , t , r ) ∈ G j : t ∈ J , t + b r ∈ J , r ∈ I , a r ∈ I } is an open subset of G j ∣ W \mathcal{G}_j|_W G j ∣ W G j , x \mathcal{G}_{j,x} G j , x x ∈ D j ∩ W x\in D_j\cap W x ∈ D j ∩ W q j q_j q j G j , W l o c \mathcal{G}^{\mathrm{loc}}_{j,W} G j , W loc G ‾ j ∣ W \overline{\mathcal{G}}_j|_W G j ∣ W
Proof. The conditions are open, and they hold on G j , x \mathcal{G}_{j,x} G j , x r = 0 r=0 r = 0 g ≠ g ′ g\ne g' g = g ′ G j , W l o c \mathcal{G}^{\mathrm{loc}}_{j,W} G j , W loc r ≠ 0 r\ne0 r = 0 t + b r t+br t + b r g g g g ′ g' g ′ k ℓ j k\ell_j k ℓ j k ≠ 0 k\ne0 k = 0 J ~ \widetilde{J} J q j q_j q j G j , W l o c \mathcal{G}^{\mathrm{loc}}_{j,W} G j , W loc G ‾ j ∣ W \overline{\mathcal{G}}_j|_W G j ∣ W q j ( a , b , t , r ) q_j(a,b,t,r) q j ( a , b , t , r ) t ∈ J ~ t\in\widetilde{J} t ∈ J r ∈ I r\in I r ∈ I a r ∈ I ar\in I a r ∈ I t + b r ∈ J ~ + ℓ j Z t+br\in\widetilde{J}+\ell_j\mathbb{Z} t + b r ∈ J + ℓ j Z r = 0 r=0 r = 0 t + b r = t ∈ J ~ t+br=t\in\widetilde{J} t + b r = t ∈ J r ≠ 0 r\ne0 r = 0 t + b r − k ℓ j ∈ J ~ t+br-k\ell_j\in\widetilde{J} t + b r − k ℓ j ∈ J k ∈ Z k\in\mathbb{Z} k ∈ Z ( a , b − k ℓ j / r , t , r ) = ( a , b , t , r ) ( 1 , − k ℓ j / r , t , r ) (a,b-k\ell_j/r,t,r)=(a,b,t,r)(1,-k\ell_j/r,t,r) ( a , b − k ℓ j / r , t , r ) = ( a , b , t , r ) ( 1 , − k ℓ j / r , t , r ) G j , W l o c \mathcal{G}^{\mathrm{loc}}_{j,W} G j , W loc q j q_j q j G j , W l o c \mathcal{G}^{\mathrm{loc}}_{j,W} G j , W loc G ‾ j ∣ W \overline{\mathcal{G}}_j|_W G j ∣ W q j q_j q j
Proposition 4.5. Let N c = D j N_c=D_j N c = D j L ^ c = { ( 1 , b , t , 0 ) : b ∈ R , t ∈ S ℓ j 1 } ⊂ G j \widehat{L}_c=\{(1,b,t,0):b\in\mathbb{R},\ t\in S^1_{\ell_j}\}\subset\mathcal{G}_j L c = {( 1 , b , t , 0 ) : b ∈ R , t ∈ S ℓ j 1 } ⊂ G j L c = q j ( L ^ c ) L_c=q_j(\widehat{L}_c) L c = q j ( L c ) G Σ \mathcal{G}_\Sigma G Σ D j D_j D j T D j ∘ TD_j^\circ T D j ∘ R \mathbb{R} R
Proof. In the coordinates (26) , we have T D j ∘ = ⟨ d r ⟩ TD_j^\circ=\langle dr\rangle T D j ∘ = ⟨ d r ⟩ χ ( d r ) = ζ 2 \chi(dr)=\zeta_2 χ ( d r ) = ζ 2 (30) . All arrows of L ^ c \widehat{L}_c L c ( 1 , b 2 , t , 0 ) ( 1 , b 1 , t , 0 ) = ( 1 , b 1 + b 2 , t , 0 ) (1,b_2,t,0)(1,b_1,t,0)=(1,b_1+b_2,t,0) ( 1 , b 2 , t , 0 ) ( 1 , b 1 , t , 0 ) = ( 1 , b 1 + b 2 , t , 0 ) L ^ c \widehat{L}_c L c R \mathbb{R} R ⟨ ζ 2 ⟩ = χ ( T D j ∘ ) \langle\zeta_2\rangle=\chi(TD_j^\circ) ⟨ ζ 2 ⟩ = χ ( T D j ∘ )
Since r = 0 r=0 r = 0 L ^ c \widehat{L}_c L c g ∈ L ^ c g\in\widehat{L}_c g ∈ L c q j q_j q j (36) therefore applies with O = G j O=\mathcal{G}_j O = G j (31) contains d α d\alpha d α d r dr d r Ω ∣ L ^ c = 0 \Omega|_{\widehat{L}_c}=0 Ω ∣ L c = 0 L ^ c \widehat{L}_c L c 2 2 2 L ^ c ⊂ G j ∣ U j \widehat{L}_c\subset\mathcal{G}_j|_{U_j} L c ⊂ G j ∣ U j (32) give L c ⊂ G Σ L_c\subset\mathcal{G}_\Sigma L c ⊂ G Σ Ω Σ ∣ L c = 0 \Omega_\Sigma|_{L_c}=0 Ω Σ ∣ L c = 0
At a transverse contact, consider arrows ( a , b , γ ( r ) , r ) (a,b,\gamma(r),r) ( a , b , γ ( r ) , r ) r ≠ 0 r\ne0 r = 0 r , a r ∈ I r,ar\in I r , a r ∈ I γ ( r ) + b r \gamma(r)+br γ ( r ) + b r J J J N c ∩ W N_c\cap W N c ∩ W γ ( r ) + b r = γ ( a r ) \gamma(r)+br=\gamma(ar) γ ( r ) + b r = γ ( a r ) b b b β ( a , r ) \beta(a,r) β ( a , r ) r = 0 r=0 r = 0 G j \mathcal{G}_j G j
Lemma 4.6. Let W = J × I W=J\times I W = J × I γ \gamma γ
β ( a , r ) = { γ ( a r ) − γ ( r ) r , r ≠ 0 , ( a − 1 ) γ ′ ( 0 ) , r = 0. \beta(a,r)=
\begin{cases}
\dfrac{\gamma(ar)-\gamma(r)}{r}, & r\ne0,\\
(a-1)\gamma'(0), & r=0.
\end{cases} β ( a , r ) = ⎩ ⎨ ⎧ r γ ( a r ) − γ ( r ) , ( a − 1 ) γ ′ ( 0 ) , r = 0 , r = 0. Then β \beta β { ( a , r ) : a > 0 , r ∈ I , a r ∈ I } \{(a,r):a>0,\ r\in I,\ ar\in I\} {( a , r ) : a > 0 , r ∈ I , a r ∈ I } a a a b ′ = b − β ( a , r ) b'=b-\beta(a,r) b ′ = b − β ( a , r ) s = t − γ ( r ) s=t-\gamma(r) s = t − γ ( r ) r r r O W = { ( a , b , t , r ) ∈ G j : t ∈ J , r ∈ I , a r ∈ I } O_W=\{(a,b,t,r)\in\mathcal{G}_j:t\in J,\ r\in I,\ ar\in I\} O W = {( a , b , t , r ) ∈ G j : t ∈ J , r ∈ I , a r ∈ I }
s ( a , b ′ , s , r ) = ( s , r ) , t ( a , b ′ , s , r ) = ( s + b ′ r , a r ) , s(a,b',s,r)=(s,r),\qquad t(a,b',s,r)=(s+b'r,ar), s ( a , b ′ , s , r ) = ( s , r ) , t ( a , b ′ , s , r ) = ( s + b ′ r , a r ) , ( a 2 , b 2 ′ , s + b 1 ′ r , a 1 r ) ( a 1 , b 1 ′ , s , r ) = ( a 2 a 1 , b 1 ′ + a 1 b 2 ′ , s , r ) , (a_2,b'_2,s+b'_1r,a_1r)(a_1,b'_1,s,r)=(a_2a_1,b'_1+a_1b'_2,s,r), ( a 2 , b 2 ′ , s + b 1 ′ r , a 1 r ) ( a 1 , b 1 ′ , s , r ) = ( a 2 a 1 , b 1 ′ + a 1 b 2 ′ , s , r ) , Ω = d s ∧ d α + d b ′ ∧ d r + b ′ d r ∧ d α + r d b ′ ∧ d α . \Omega=ds\wedge d\alpha+db'\wedge dr+b'\,dr\wedge d\alpha+r\,db'\wedge d\alpha. Ω = d s ∧ d α + d b ′ ∧ d r + b ′ d r ∧ d α + r d b ′ ∧ d α . Here ( s + b ′ r , a r ) (s+b'r,ar) ( s + b ′ r , a r ) ( s + b ′ r + γ ( a r ) , a r ) (s+b'r+\gamma(ar),ar) ( s + b ′ r + γ ( a r ) , a r ) U ~ j \widetilde{U}_j U j ℓ j \ell_j ℓ j
Proof. Hadamard’s lemma gives γ ( u ) = γ ′ ( 0 ) u + u 2 γ 2 ( u ) \gamma(u)=\gamma'(0)u+u^2\gamma_2(u) γ ( u ) = γ ′ ( 0 ) u + u 2 γ 2 ( u ) γ 2 \gamma_2 γ 2 I I I
β ( a , r ) = γ ′ ( 0 ) ( a − 1 ) + r ( a 2 γ 2 ( a r ) − γ 2 ( r ) ) \beta(a,r)=\gamma'(0)(a-1)+r\bigl(a^2\gamma_2(ar)-\gamma_2(r)\bigr) β ( a , r ) = γ ′ ( 0 ) ( a − 1 ) + r ( a 2 γ 2 ( a r ) − γ 2 ( r ) ) for every r r r
( a , b ′ , s , r ) ⟼ ( a , b ′ + β ( a , r ) , s + γ ( r ) , r ) . (a,b',s,r)\longmapsto(a,b'+\beta(a,r),s+\gamma(r),r). ( a , b ′ , s , r ) ⟼ ( a , b ′ + β ( a , r ) , s + γ ( r ) , r ) . The identity r β ( a , r ) = γ ( a r ) − γ ( r ) r\beta(a,r)=\gamma(ar)-\gamma(r) r β ( a , r ) = γ ( a r ) − γ ( r ) t + b r = ( s + b ′ r ) + γ ( a r ) t+br=(s+b'r)+\gamma(ar) t + b r = ( s + b ′ r ) + γ ( a r ) b i = b i ′ + β ( a i , r i ) b_i=b'_i+\beta(a_i,r_i) b i = b i ′ + β ( a i , r i ) r 1 = r r_1=r r 1 = r r 2 = a 1 r r_2=a_1r r 2 = a 1 r (29) reduces the multiplication formula to β ( a 1 , r ) + a 1 β ( a 2 , a 1 r ) = β ( a 2 a 1 , r ) \beta(a_1,r)+a_1\beta(a_2,a_1r)=\beta(a_2a_1,r) β ( a 1 , r ) + a 1 β ( a 2 , a 1 r ) = β ( a 2 a 1 , r ) r ≠ 0 r\ne0 r = 0 r = 0 r=0 r = 0
( a 1 − 1 ) γ ′ ( 0 ) + a 1 ( a 2 − 1 ) γ ′ ( 0 ) = ( a 2 a 1 − 1 ) γ ′ ( 0 ) . (a_1-1)\gamma'(0)+a_1(a_2-1)\gamma'(0)=(a_2a_1-1)\gamma'(0). ( a 1 − 1 ) γ ′ ( 0 ) + a 1 ( a 2 − 1 ) γ ′ ( 0 ) = ( a 2 a 1 − 1 ) γ ′ ( 0 ) . Since d t = d s + γ ′ ( r ) d r dt=ds+\gamma'(r)\,dr d t = d s + γ ′ ( r ) d r ω = 1 r d s ∧ d r \omega=\frac{1}{r}ds\wedge dr ω = r 1 d s ∧ d r { r ≠ 0 } \{r\ne0\} { r = 0 } ( t , b ) (t,b) ( t , b ) ( s , b ′ ) (s,b') ( s , b ′ ) Ω = t ∗ ω − s ∗ ω \Omega=t^*\omega-s^*\omega Ω = t ∗ ω − s ∗ ω { r ≠ 0 } \{r\ne0\} { r = 0 } O W O_W O W
Proposition 4.7. Let p ∈ N c ∩ D j p \in N_c \cap D_j p ∈ N c ∩ D j transverse contact , with W = J × I W = J \times I W = J × I γ \gamma γ ( a , b ′ , s , r ) (a,b',s,r) ( a , b ′ , s , r ) Lemmas 4.2 and 4.6 . Let L ^ W = { ( a , 0 , 0 , r ) : a > 0 , r ∈ I , a r ∈ I } ⊂ O W \widehat{L}_W = \{(a,0,0,r): a > 0,\ r \in I,\ ar \in I\} \subset O_W L W = {( a , 0 , 0 , r ) : a > 0 , r ∈ I , a r ∈ I } ⊂ O W L W = q j ( L ^ W ) L_W = q_j(\widehat{L}_W) L W = q j ( L W ) G Σ G_\Sigma G Σ N c ∩ W N_c \cap W N c ∩ W T N ∘ TN^\circ T N ∘ R > 0 \mathbb{R}_{>0} R > 0 p p p L W L_W L W p p p
{ ( ( γ ( r ′ ) , r ′ ) , ( γ ( r ) , r ) ) : r , r ′ ∈ I ∖ { 0 } , sign r = sign r ′ } (37) \left\{((\gamma(r'),r'),(\gamma(r),r)): r,r' \in I \setminus\{0\},\ \operatorname{sign} r = \operatorname{sign} r'\right\} \tag*{(37)} { (( γ ( r ′ ) , r ′ ) , ( γ ( r ) , r )) : r , r ′ ∈ I ∖ { 0 } , sign r = sign r ′ } (37) that is, the pairs of points of ( N c ∩ W ) ∖ { p } (N_c \cap W) \setminus\{p\} ( N c ∩ W ) ∖ { p } p p p Figure 3 ).
Figure 3. A transverse contact.
Proof. The set L ^ W = { b ′ = s = 0 } \widehat{L}_W=\{b'=s=0\} L W = { b ′ = s = 0 } O W O_W O W ( a , r ) (a,r) ( a , r ) Lemma 4.6 ,
s ( a , 0 , 0 , r ) = ( 0 , r ) , t ( a , 0 , 0 , r ) = ( 0 , a r ) , ( a 2 , 0 , 0 , a 1 r ) ( a 1 , 0 , 0 , r ) = ( a 2 a 1 , 0 , 0 , r ) . s(a,0,0,r)=(0,r),\qquad t(a,0,0,r)=(0,ar),\qquad(a_2,0,0,a_1r)(a_1,0,0,r)=(a_2a_1,0,0,r). s ( a , 0 , 0 , r ) = ( 0 , r ) , t ( a , 0 , 0 , r ) = ( 0 , a r ) , ( a 2 , 0 , 0 , a 1 r ) ( a 1 , 0 , 0 , r ) = ( a 2 a 1 , 0 , 0 , r ) . So L ^ W \widehat{L}_W L W { s = 0 } = N c ∩ W \{s=0\}=N_c\cap W { s = 0 } = N c ∩ W { a > 0 : a r ∈ I } \{a>0: ar\in I\} { a > 0 : a r ∈ I } ( 0 , r ) (0,r) ( 0 , r ) R > 0 \mathbb{R}_{>0} R > 0 p p p
In the coordinates ( a , b , t , r ) (a,b,t,r) ( a , b , t , r ) α ↦ ( e α , β ( e α , r ) , γ ( r ) , r ) \alpha\mapsto(e^\alpha,\beta(e^\alpha,r),\gamma(r),r) α ↦ ( e α , β ( e α , r ) , γ ( r ) , r ) L ^ W \widehat{L}_W L W 1 ( γ ( r ) , r ) 1_{(\gamma(r),r)} 1 ( γ ( r ) , r ) α = 0 \alpha=0 α = 0 ζ 1 + γ ′ ( r ) ζ 2 \zeta_1+\gamma'(r)\zeta_2 ζ 1 + γ ′ ( r ) ζ 2 ∂ β ∂ a ( a , r ) = γ ′ ( a r ) \frac{\partial\beta}{\partial a}(a,r)=\gamma'(ar) ∂ a ∂ β ( a , r ) = γ ′ ( a r ) r r r N c ∩ W N_c\cap W N c ∩ W
χ ( T N ∘ ) = χ ⟨ d t − γ ′ ( r ) d r ⟩ = ⟨ ζ 1 + γ ′ ( r ) ζ 2 ⟩ . \chi(TN^\circ)=\chi\langle dt-\gamma'(r)\,dr\rangle=\langle\zeta_1+\gamma'(r)\zeta_2\rangle. χ ( T N ∘ ) = χ ⟨ d t − γ ′ ( r ) d r ⟩ = ⟨ ζ 1 + γ ′ ( r ) ζ 2 ⟩ . Thus L ^ W \widehat{L}_W L W T N ∘ TN^\circ T N ∘ N c ∩ W N_c\cap W N c ∩ W Ω \Omega Ω Lemma 4.6 contains d s ds d s d b ′ db' d b ′ b ′ b' b ′ Ω ∣ L ^ W = 0 \Omega|_{\widehat{L}_W}=0 Ω ∣ L W = 0 L ^ W \widehat{L}_W L W
Let r ˉ = sup I ∣ r ∣ \bar r=\sup_I|r| r ˉ = sup I ∣ r ∣ O = O W ∩ { ∣ b ′ ∣ < ℓ j / ( 2 r ˉ ) } ⊃ L ^ W O=O_W\cap\{|b'|<\ell_j/(2\bar r)\}\supset\widehat{L}_W O = O W ∩ { ∣ b ′ ∣ < ℓ j / ( 2 r ˉ )} ⊃ L W O O O q j q_j q j a a a s s s r r r b ′ b' b ′ ℓ j / r ˉ \ell_j/\bar r ℓ j / r ˉ L W L_W L W G j ∣ U j ⊂ G Σ \mathcal{G}_j|_{U_j}\subset G_\Sigma G j ∣ U j ⊂ G Σ Ω Σ ∣ L W = 0 \Omega_\Sigma|_{L_W}=0 Ω Σ ∣ L W = 0
For g = ( a , 0 , 0 , r ) g=(a,0,0,r) g = ( a , 0 , 0 , r ) r ≠ 0 r\ne0 r = 0 s ( g ) = ( γ ( r ) , r ) s(g)=(\gamma(r),r) s ( g ) = ( γ ( r ) , r ) t ( g ) = ( γ ( a r ) , a r ) t(g)=(\gamma(ar),ar) t ( g ) = ( γ ( a r ) , a r ) { a r : a > 0 , a r ∈ I } = I ∩ r R > 0 \{ar:a>0,\ ar\in I\}=I\cap r\mathbb{R}_{>0} { a r : a > 0 , a r ∈ I } = I ∩ r R > 0 □ \square □
At a tangential contact, an arrow ( a , b , t , h ( t ) ) (a,b,t,h(t)) ( a , b , t , h ( t )) ( t + b h ( t ) , a h ( t ) ) (t+bh(t),ah(t)) ( t + bh ( t ) , ah ( t )) t + b h ( t ) t+bh(t) t + bh ( t ) J J J N c ∩ W N_c\cap W N c ∩ W a h ( t ) = h ( t + b h ( t ) ) ah(t)=h(t+bh(t)) ah ( t ) = h ( t + bh ( t )) t ≠ 0 t\ne0 t = 0 a a a ( t , b ) (t,b) ( t , b ) t = 0 t=0 t = 0 τ ( t , b ) \tau(t,b) τ ( t , b ) a a a η ( t , b ) \eta(t,b) η ( t , b )
Lemma 4.8. Let p p p W = J × I W=J\times I W = J × I h h h Lemma 4.3 . Write h ( t ) = t h 1 ( t ) h(t)=t h_1(t) h ( t ) = t h 1 ( t ) h 1 h_1 h 1
τ ( t , b ) = t + b h ( t ) , S = { ( t , b ) ∈ J × R : τ ( t , b ) ∈ J , 1 + b h 1 ( t ) > 0 } . \tau(t,b)=t+bh(t),\qquad S=\{(t,b)\in J\times\mathbb{R}:\tau(t,b)\in J,\ 1+bh_1(t)>0\}. τ ( t , b ) = t + bh ( t ) , S = {( t , b ) ∈ J × R : τ ( t , b ) ∈ J , 1 + b h 1 ( t ) > 0 } . For t ∈ J t \in J t ∈ J S t = { b ∈ R : ( t , b ) ∈ S } S_t=\{b\in\mathbb{R}:(t,b)\in S\} S t = { b ∈ R : ( t , b ) ∈ S } t ≠ 0 t\ne0 t = 0 J t = { u ∈ J : u t > 0 } J_t=\{u\in J:ut>0\} J t = { u ∈ J : u t > 0 } S S S S 0 = R S_0=\mathbb{R} S 0 = R t ≠ 0 t\ne0 t = 0 S t S_t S t 0 0 0 b ↦ τ ( t , b ) b\mapsto\tau(t,b) b ↦ τ ( t , b ) J t J_t J t η : S → R > 0 \eta:S\to\mathbb{R}_{>0} η : S → R > 0 η ( 0 , ⋅ ) ≡ 1 \eta(0,\cdot)\equiv1 η ( 0 , ⋅ ) ≡ 1
η ( t , b ) h ( t ) = h ( τ ( t , b ) ) , ( t , b ) ∈ S . (38) \eta(t,b)h(t)=h(\tau(t,b)),\qquad(t,b)\in S. \tag*{(38)} η ( t , b ) h ( t ) = h ( τ ( t , b )) , ( t , b ) ∈ S . (38) Proof. The set S S S S 0 = R S_0=\mathbb{R} S 0 = R τ ( 0 , b ) = 0 ∈ J \tau(0,b)=0\in J τ ( 0 , b ) = 0 ∈ J h 1 ( 0 ) = h ′ ( 0 ) = 0 h_1(0)=h'(0)=0 h 1 ( 0 ) = h ′ ( 0 ) = 0 t ≠ 0 t\ne0 t = 0 τ ( t , b ) = t ( 1 + b h 1 ( t ) ) \tau(t,b)=t(1+bh_1(t)) τ ( t , b ) = t ( 1 + b h 1 ( t )) h 1 ( t ) ≠ 0 h_1(t)\ne0 h 1 ( t ) = 0 b ↦ τ ( t , b ) b\mapsto\tau(t,b) b ↦ τ ( t , b ) R \mathbb{R} R 1 + b h 1 ( t ) > 0 1+bh_1(t)>0 1 + b h 1 ( t ) > 0 t τ ( t , b ) > 0 t\tau(t,b)>0 t τ ( t , b ) > 0 S t S_t S t J t J_t J t S t S_t S t 0 0 0 τ ( t , 0 ) = t ∈ J t \tau(t,0)=t\in J_t τ ( t , 0 ) = t ∈ J t
To construct η \eta η S ′ = { ( t , b ) ∈ J × R : τ ( t , b ) ∈ J } ⊃ S S'=\{(t,b)\in J\times\mathbb{R}:\tau(t,b)\in J\}\supset S S ′ = {( t , b ) ∈ J × R : τ ( t , b ) ∈ J } ⊃ S
η ( t , b ) = 1 + b ∫ 0 1 h ′ ( t + u b h ( t ) ) d u . \eta(t,b)=1+b\int_0^1h'(t+ubh(t))\,du. η ( t , b ) = 1 + b ∫ 0 1 h ′ ( t + u bh ( t )) d u . This is smooth, since t + u b h ( t ) = ( 1 − u ) t + u τ ( t , b ) ∈ J t+ubh(t)=(1-u)t+u\tau(t,b)\in J t + u bh ( t ) = ( 1 − u ) t + uτ ( t , b ) ∈ J u ↦ h ( t + u b h ( t ) ) u\mapsto h(t+ubh(t)) u ↦ h ( t + u bh ( t )) (38) , and η ( 0 , ⋅ ) ≡ 1 \eta(0,\cdot)\equiv1 η ( 0 , ⋅ ) ≡ 1 h ′ ( 0 ) = 0 h'(0)=0 h ′ ( 0 ) = 0
If t ≠ 0 t\ne0 t = 0 η ( t , b ) = 0 \eta(t,b)=0 η ( t , b ) = 0 h ( τ ( t , b ) ) = 0 h(\tau(t,b))=0 h ( τ ( t , b )) = 0 (38) , so τ ( t , b ) = 0 \tau(t,b)=0 τ ( t , b ) = 0 1 + b h 1 ( t ) = 0 1+bh_1(t)=0 1 + b h 1 ( t ) = 0 S S S η \eta η S t S_t S t η > 0 \eta>0 η > 0 η ( t , 0 ) = 1 \eta(t,0)=1 η ( t , 0 ) = 1
Proposition 4.9. Let p p p W = J × I W=J\times I W = J × I h h h τ \tau τ η \eta η S S S L ^ W = { g ( t , b ) = ( η ( t , b ) , b , t , h ( t ) ) : ( t , b ) ∈ S } ⊂ G j \widehat{L}_W=\{g(t,b)=(\eta(t,b),b,t,h(t)):(t,b)\in S\}\subset G_j L W = { g ( t , b ) = ( η ( t , b ) , b , t , h ( t )) : ( t , b ) ∈ S } ⊂ G j L W = q j ( L ^ W ) L_W=q_j(\widehat{L}_W) L W = q j ( L W ) G Σ G_\Sigma G Σ N c ∘ ∩ W N_c^\circ\cap W N c ∘ ∩ W T N ∘ TN^\circ T N ∘ R \mathbb{R} R p p p L W L_W L W p p p
{ ( ( t ′ , h ( t ′ ) ) , ( t , h ( t ) ) ) : t ∈ J ∖ { 0 } , t ′ ∈ J t } , (39) \left\{\big((t',h(t')),(t,h(t))\big):t\in J\setminus\{0\},\ t'\in J_t\right\}, \tag*{(39)} { ( ( t ′ , h ( t ′ )) , ( t , h ( t )) ) : t ∈ J ∖ { 0 } , t ′ ∈ J t } , (39) that is, the pairs of points of ( N c ∩ W ) ∖ { p } (N_c\cap W)\setminus\{p\} ( N c ∩ W ) ∖ { p } p p p Figure 4 ).
Figure 4. A tangential contact.
Proof. The set L ^ W \widehat{L}_W L W G j G_j G j S S S ( t , b ) ↦ ( η ( t , b ) , h ( t ) ) (t,b)\mapsto(\eta(t,b),h(t)) ( t , b ) ↦ ( η ( t , b ) , h ( t )) (38) , g ( t , b ) g(t,b) g ( t , b ) ( t , h ( t ) ) (t,h(t)) ( t , h ( t )) ( τ ( t , b ) , h ( τ ( t , b ) ) ) (\tau(t,b),h(\tau(t,b))) ( τ ( t , b ) , h ( τ ( t , b ))) N c ∩ W N_c\cap W N c ∩ W g ( t , 0 ) g(t,0) g ( t , 0 )
As J J J ℓ j \ell_j ℓ j L ^ W \widehat{L}_W L W g ( t 1 , b 2 ) g(t_1,b_2) g ( t 1 , b 2 ) g ( t , b 1 ) g(t,b_1) g ( t , b 1 ) t 1 = τ ( t , b 1 ) t_1=\tau(t,b_1) t 1 = τ ( t , b 1 ) η 1 = η ( t , b 1 ) \eta_1=\eta(t,b_1) η 1 = η ( t , b 1 )
τ ( t , b 1 + η 1 b 2 ) = t 1 + b 2 η 1 h ( t ) = t 1 + b 2 h ( t 1 ) = τ ( t 1 , b 2 ) , \tau(t,b_1+\eta_1b_2)=t_1+b_2\eta_1h(t)=t_1+b_2h(t_1)=\tau(t_1,b_2), τ ( t , b 1 + η 1 b 2 ) = t 1 + b 2 η 1 h ( t ) = t 1 + b 2 h ( t 1 ) = τ ( t 1 , b 2 ) , η ( t , b 1 + η 1 b 2 ) = h ( τ ( t 1 , b 2 ) ) h ( t 1 ) h ( t 1 ) h ( t ) = η ( t 1 , b 2 ) η 1 ( t ≠ 0 ) , \eta(t,b_1+\eta_1b_2)=\frac{h(\tau(t_1,b_2))}{h(t_1)}\frac{h(t_1)}{h(t)}=\eta(t_1,b_2)\eta_1\qquad(t\ne0), η ( t , b 1 + η 1 b 2 ) = h ( t 1 ) h ( τ ( t 1 , b 2 )) h ( t ) h ( t 1 ) = η ( t 1 , b 2 ) η 1 ( t = 0 ) , and both sides of the last identity equal 1 1 1 t = 0 t=0 t = 0 ( t , b 1 + η 1 b 2 ) ∈ S (t,b_1+\eta_1b_2)\in S ( t , b 1 + η 1 b 2 ) ∈ S τ ( t 1 , b 2 ) ∈ J t 1 = J t \tau(t_1,b_2)\in J_{t_1}=J_t τ ( t 1 , b 2 ) ∈ J t 1 = J t t ≠ 0 t\ne0 t = 0 (29) gives
g ( t 1 , b 2 ) g ( t , b 1 ) = g ( t , b 1 + η 1 b 2 ) , g ( t , b 1 ) − 1 = g ( t 1 , − b 1 / η 1 ) . g(t_1,b_2)g(t,b_1)=g(t,b_1+\eta_1b_2),\qquad g(t,b_1)^{-1}=g(t_1,-b_1/\eta_1). g ( t 1 , b 2 ) g ( t , b 1 ) = g ( t , b 1 + η 1 b 2 ) , g ( t , b 1 ) − 1 = g ( t 1 , − b 1 / η 1 ) . The parameter of the inverse belongs to S S S τ ( t 1 , − b 1 / η 1 ) = t 1 − b 1 h ( t ) = t \tau(t_1,-b_1/\eta_1)=t_1-b_1h(t)=t τ ( t 1 , − b 1 / η 1 ) = t 1 − b 1 h ( t ) = t L ^ W \widehat{L}_W L W S t S_t S t { ( 1 , b , 0 ) : b ∈ R } ≅ R \{(1,b,0):b\in\mathbb{R}\}\cong\mathbb{R} {( 1 , b , 0 ) : b ∈ R } ≅ R p p p
At x = ( t 0 , h ( t 0 ) ) x=(t_0,h(t_0)) x = ( t 0 , h ( t 0 )) g ( t 0 , 0 ) g(t_0,0) g ( t 0 , 0 ) d d b ∣ b = 0 g ( t 0 , b ) = ∂ η ∂ b ( t 0 , 0 ) ζ 1 + ζ 2 \left.\frac{d}{db}\right|_{b=0}g(t_0,b)=\frac{\partial\eta}{\partial b}(t_0,0)\,\zeta_1+\zeta_2 d b d b = 0 g ( t 0 , b ) = ∂ b ∂ η ( t 0 , 0 ) ζ 1 + ζ 2 (38) in b b b t t t
∂ η ∂ b h ( t ) = h ′ ( τ ) h ( t ) , \frac{\partial\eta}{\partial b}h(t)=h'(\tau)h(t), ∂ b ∂ η h ( t ) = h ′ ( τ ) h ( t ) , ∂ η ∂ t h + η h ′ = h ′ ( τ ) ( 1 + b h ′ ) , (40) \frac{\partial\eta}{\partial t}h+\eta h'=h'(\tau)(1+bh'), \tag*{(40)} ∂ t ∂ η h + η h ′ = h ′ ( τ ) ( 1 + b h ′ ) , (40) where h h h h ′ h' h ′ t t t ∂ η ∂ b = h ′ ( τ ) \frac{\partial\eta}{\partial b}=h'(\tau) ∂ b ∂ η = h ′ ( τ ) t ≠ 0 t\ne0 t = 0 ∂ η ∂ b ( t 0 , 0 ) = h ′ ( t 0 ) \frac{\partial\eta}{\partial b}(t_0,0)=h'(t_0) ∂ b ∂ η ( t 0 , 0 ) = h ′ ( t 0 )
T x N c ∘ = ⟨ h ′ ( t 0 ) d t − d r ⟩ , χ ( h ′ ( t 0 ) d t − d r ) = − h ′ ( t 0 ) ζ 1 − ζ 2 . T_xN_c^\circ=\langle h'(t_0)\,dt-dr\rangle,\qquad\chi(h'(t_0)\,dt-dr)=-h'(t_0)\,\zeta_1-\zeta_2. T x N c ∘ = ⟨ h ′ ( t 0 ) d t − d r ⟩ , χ ( h ′ ( t 0 ) d t − d r ) = − h ′ ( t 0 ) ζ 1 − ζ 2 . Thus the tangent line to the source fibre is χ ( T x N c ∘ ) \chi(T_xN_c^\circ) χ ( T x N c ∘ ) L ^ W \widehat{L}_W L W T N ∘ TN^\circ T N ∘ N c ∩ W N_c\cap W N c ∩ W
Along L ^ W \widehat{L}_W L W d r = h ′ d t dr=h'\,dt d r = h ′ d t η d α = ∂ η ∂ t d t + ∂ η ∂ b d b \eta\,d\alpha=\frac{\partial\eta}{\partial t}dt+\frac{\partial\eta}{\partial b}db η d α = ∂ t ∂ η d t + ∂ b ∂ η d b (31) and (40) , the coefficient C C C d t ∧ d b dt\wedge db d t ∧ d b Ω ∣ L ^ W \Omega|_{\widehat{L}_W} Ω ∣ L W
η C = ∂ η ∂ b ( 1 + b h ′ ) − ( ∂ η ∂ t h + η h ′ ) = h ′ ( τ ) ( 1 + b h ′ ) − h ′ ( τ ) ( 1 + b h ′ ) = 0. \begin{aligned}
\eta C=\frac{\partial\eta}{\partial b}(1+bh')-\left(\frac{\partial\eta}{\partial t}h+\eta h'\right) \\
&=h'(\tau)(1+bh')-h'(\tau)(1+bh') \\
&=0.
\end{aligned} η C = ∂ b ∂ η ( 1 + b h ′ ) − ( ∂ t ∂ η h + η h ′ ) = h ′ ( τ ) ( 1 + b h ′ ) − h ′ ( τ ) ( 1 + b h ′ ) = 0. So Ω ∣ L ^ W = 0 \left.\Omega\right|_{\widehat{L}_W}=0 Ω ∣ L W = 0 L ^ W \widehat{L}_W L W L ^ W \widehat{L}_W L W G j , W l o c G_{j,W}^{\mathrm{loc}} G j , W loc τ ( t , b ) ∈ J ~ \tau(t,b)\in\widetilde{J} τ ( t , b ) ∈ J η ( t , b ) h ( t ) = h ( τ ( t , b ) ) ∈ I \eta(t,b)h(t)=h(\tau(t,b))\in I η ( t , b ) h ( t ) = h ( τ ( t , b )) ∈ I q j q_j q j G j , W l o c G_{j,W}^{\mathrm{loc}} G j , W loc (36) shows that L W L_W L W G ‾ j ∣ U j ⊂ G Σ \overline{G}_{j|U_j}\subset G_\Sigma G j ∣ U j ⊂ G Σ Ω Σ ∣ L W = 0 \left.\Omega_\Sigma\right|_{L_W}=0 Ω Σ ∣ L W = 0 (32) . For t ≠ 0 t\ne0 t = 0 τ ( t , ⋅ ) \tau(t,\mathord\cdot) τ ( t , ⋅ ) S t S_t S t J t J_t J t (39) by (34) . □
The Lagrangian subgroupoid of an admissible curve
We now assemble the local models of §4.2 into a subgroupoid over the whole curve. The only point requiring care is embeddedness near the isotropy groups at the contacts. Enumerate the points of C C C p 1 , … , p n C p_1,\ldots,p_{n_C} p 1 , … , p n C i i i D j D_j D j N c N_c N c D D D N N N p i p_i p i W i W_i W i p i p_i p i N N N N c N_c N c p i p_i p i I I I J J J N ∩ W i = N c ∩ W i N\cap W_i=N_c\cap W_i N ∩ W i = N c ∩ W i N ∩ D ∩ W i = { p i } N\cap D\cap W_i=\{p_i\} N ∩ D ∩ W i = { p i }
Let L W i L_{W_i} L W i N c ∩ W i N_c\cap W_i N c ∩ W i L p i i s o \mathcal{L}^{\mathrm{iso}}_{p_i} L p i iso L W i L_{W_i} L W i p i p_i p i q j q_j q j { ( a , ( a − 1 ) γ ′ ( 0 ) , 0 , 0 ) : a > 0 } ≅ R > 0 \{(a,(a-1)\gamma'(0),0,0):a>0\}\cong\mathbb{R}_{>0} {( a , ( a − 1 ) γ ′ ( 0 ) , 0 , 0 ) : a > 0 } ≅ R > 0 { ( 1 , b , 0 , 0 ) : b ∈ R } ≅ R \{(1,b,0,0):b\in\mathbb{R}\}\cong\mathbb{R} {( 1 , b , 0 , 0 ) : b ∈ R } ≅ R Figure 5 shows the pieces of L \mathcal{L} L p i p_i p i
Figure 5. The pieces of L \mathcal{L} L p i p_i p i
The following lemma shows that, near each point of L p i i s o \mathcal{L}^{\mathrm{iso}}_{p_i} L p i iso L W i L_{W_i} L W i
Lemma 4.10. Let K i − , K i + ∈ K K_i^-,K_i^+\in\mathcal{K} K i − , K i + ∈ K ( N c ∩ W i ) ∖ { p i } (N_c\cap W_i)\setminus\{p_i\} ( N c ∩ W i ) ∖ { p i }
A i = ( K i − × K i − ) ∪ ( K i + × K i + ) ∪ L p i i s o ⊂ G Σ . A_i=(K_i^-\times K_i^-)\cup(K_i^+\times K_i^+)\cup\mathcal{L}^{\mathrm{iso}}_{p_i}\subset G_\Sigma. A i = ( K i − × K i − ) ∪ ( K i + × K i + ) ∪ L p i iso ⊂ G Σ . (1) If p i p_i p i K i − ≠ K i + K_i^- \ne K_i^+ K i − = K i + A i ∩ G ‾ j ∣ W i = L W i A_i \cap\overline{G}_{j|W_i} = L_{W_i} A i ∩ G j ∣ W i = L W i
(2) In general, every g 0 ∈ L p i i s o g_0 \in L_{p_i}^{\mathrm{iso}} g 0 ∈ L p i iso O g 0 ⊂ G ‾ j ∣ W i O_{g_0} \subset\overline{G}_{j|W_i} O g 0 ⊂ G j ∣ W i A i ∩ O g 0 = L W i ∩ O g 0 A_i \cap O_{g_0} = L_{W_i} \cap O_{g_0} A i ∩ O g 0 = L W i ∩ O g 0
Proof. Let g ∈ L W i g \in L_{W_i} g ∈ L W i N c ∩ W i N_c \cap W_i N c ∩ W i g ∈ G ‾ j ∣ W i g \in\overline{G}_{j|W_i} g ∈ G j ∣ W i s ( g ) = t ( g ) = p i \mathrm{s}(g) = \mathrm{t}(g) = p_i s ( g ) = t ( g ) = p i g ∈ L p i i s o g \in L_{p_i}^{\mathrm{iso}} g ∈ L p i iso s ( g ) \mathrm{s}(g) s ( g ) t ( g ) \mathrm{t}(g) t ( g ) p i p_i p i (37) and (39) , hence in the same arc K i ± K_i^\pm K i ± L W i ⊂ A i ∩ G ‾ j ∣ W i L_{W_i} \subset A_i \cap\overline{G}_{j|W_i} L W i ⊂ A i ∩ G j ∣ W i
(1) Let g ∈ ( A i ∩ G ‾ j ∣ W i ) ∖ L p i i s o g \in(A_i \cap\overline{G}_{j|W_i}) \setminus L_{p_i}^{\mathrm{iso}} g ∈ ( A i ∩ G j ∣ W i ) ∖ L p i iso g ∈ K i ± × K i ± g \in K_i^\pm\times K_i^\pm g ∈ K i ± × K i ± ( N c ∩ W i ) ∖ { p i } (N_c \cap W_i) \setminus\{p_i\} ( N c ∩ W i ) ∖ { p i } (37) and (39) , it suffices to show that they lie on the same side of p i p_i p i ( a , b ′ , s , r ) ∈ O W i (a,b',s,r) \in O_{W_i} ( a , b ′ , s , r ) ∈ O W i g g g s = 0 s=0 s = 0 N c ∩ W i = { s = 0 } N_c \cap W_i=\{s=0\} N c ∩ W i = { s = 0 } s ( g ) = ( γ ( r ) , r ) \mathrm{s}(g)=(\gamma(r),r) s ( g ) = ( γ ( r ) , r ) t ( g ) = ( γ ( a r ) , a r ) \mathrm{t}(g)=(\gamma(ar),ar) t ( g ) = ( γ ( a r ) , a r ) r ≠ 0 r\ne0 r = 0 sign ( a r ) = sign ( r ) \operatorname{sign}(ar)=\operatorname{sign}(r) sign ( a r ) = sign ( r ) K i − ≠ K i + K_i^- \ne K_i^+ K i − = K i + K i − ∩ W i K_i^- \cap W_i K i − ∩ W i { ( t , h ( t ) ) : t < 0 } \{(t,h(t)):t<0\} {( t , h ( t )) : t < 0 } { ( t , h ( t ) ) : t > 0 } \{(t,h(t)):t>0\} {( t , h ( t )) : t > 0 } s ( g ) = ( t , h ( t ) ) \mathrm{s}(g)=(t,h(t)) s ( g ) = ( t , h ( t )) t ( g ) = ( t ′ , h ( t ′ ) ) \mathrm{t}(g)=(t',h(t')) t ( g ) = ( t ′ , h ( t ′ )) t t ′ > 0 tt'>0 t t ′ > 0
(2) By (1), we may assume that p i p_i p i K i − = K i + = K K_i^-=K_i^+=K K i − = K i + = K q j q_j q j G j , W i l o c G_{j,W_i}^{\mathrm{loc}} G j , W i loc G ‾ j ∣ W i \overline{G}_{j|W_i} G j ∣ W i g 0 = q j ( g ^ 0 ) g_0=q_j(\widehat{g}_0) g 0 = q j ( g 0 ) g ^ 0 = ( 1 , b ‾ , 0 , 0 ) ∈ G j , W i l o c \widehat{g}_0=(1,\overline{b},0,0)\in G_{j,W_i}^{\mathrm{loc}} g 0 = ( 1 , b , 0 , 0 ) ∈ G j , W i loc h ( t ) / t → h ′ ( 0 ) = 0 h(t)/t\to h'(0)=0 h ( t ) / t → h ′ ( 0 ) = 0 t → 0 t\to0 t → 0 h ( t ) ≠ 0 h(t)\ne0 h ( t ) = 0 t ≠ 0 t\ne0 t = 0 ε 1 > 0 \varepsilon_1>0 ε 1 > 0
∣ t ∣ ∣ h ( t ) ∣ > ∣ b ‾ ∣ + 1 for t ∈ J , 0 < ∣ t ∣ < ε 1 . (41) \frac{|t|}{|h(t)|}>|\overline{b}|+1\quad\text{for }t\in J,\ 0<|t|<\varepsilon_1. \tag*{(41)} ∣ h ( t ) ∣ ∣ t ∣ > ∣ b ∣ + 1 for t ∈ J , 0 < ∣ t ∣ < ε 1 . (41) Since q j q_j q j O g 0 = q j ( { ( a , b , t , r ) ∈ G j , W i l o c : ∣ t ∣ < ε 1 , ∣ b − b ‾ ∣ < 1 } ) O_{g_0}=q_j(\{(a,b,t,r)\in G_{j,W_i}^{\mathrm{loc}}:|t|<\varepsilon_1,\ |b-\overline{b}|<1\}) O g 0 = q j ({( a , b , t , r ) ∈ G j , W i loc : ∣ t ∣ < ε 1 , ∣ b − b ∣ < 1 }) g 0 g_0 g 0 G ‾ j ∣ W i \overline{G}_{j|W_i} G j ∣ W i
Let g ∈ A i ∩ O g 0 g\in A_i\cap O_{g_0} g ∈ A i ∩ O g 0 ( a , b , t , r ) (a,b,t,r) ( a , b , t , r ) s ( g ) ∈ N c ∩ W i \mathrm{s}(g)\in N_c\cap W_i s ( g ) ∈ N c ∩ W i r = h ( t ) r=h(t) r = h ( t ) t = 0 t=0 t = 0 s ( g ) = t ( g ) = p i \mathrm{s}(g)=\mathrm{t}(g)=p_i s ( g ) = t ( g ) = p i (33) , so g ∈ L p i i s o g\in L_{p_i}^{\mathrm{iso}} g ∈ L p i iso K K K D D D t ≠ 0 t\ne0 t = 0 g ∈ K × K g\in K\times K g ∈ K × K t ( g ) = ( t ′ , h ( t ′ ) ) \mathrm{t}(g)=(t',h(t')) t ( g ) = ( t ′ , h ( t ′ )) t ′ = t + b h ( t ) ∈ J ∖ { 0 } t'=t+bh(t)\in J\setminus\{0\} t ′ = t + bh ( t ) ∈ J ∖ { 0 } t t ′ < 0 tt'<0 t t ′ < 0 ∣ t ′ − t ∣ > ∣ t ∣ |t'-t|>|t| ∣ t ′ − t ∣ > ∣ t ∣ (41) gives ∣ b ∣ = ∣ t ′ − t ∣ / ∣ h ( t ) ∣ > ∣ t ∣ / ∣ h ( t ) ∣ > ∣ b ‾ ∣ + 1 |b|=|t'-t|/|h(t)|>|t|/|h(t)|>|\overline{b}|+1 ∣ b ∣ = ∣ t ′ − t ∣/∣ h ( t ) ∣ > ∣ t ∣/∣ h ( t ) ∣ > ∣ b ∣ + 1 ∣ b − b ‾ ∣ > 1 |b-\overline{b}|>1 ∣ b − b ∣ > 1 t ′ ∈ J t t'\in J_t t ′ ∈ J t g ∈ L W i g\in L_{W_i} g ∈ L W i (39) . □ \square □
Remark 4.11. If C c = { p i } C_c=\{p_i\} C c = { p i } K = N c ∖ { p i } K=N_c\setminus\{p_i\} K = N c ∖ { p i } Σ ∖ D \Sigma\setminus D Σ ∖ D K ∩ W i K\cap W_i K ∩ W i D j D_j D j (28) . Hence the contact at p i p_i p i N c ∩ W i N_c\cap W_i N c ∩ W i h ( t ) h ( t ′ ) > 0 h(t)h(t')>0 h ( t ) h ( t ′ ) > 0 t , t ′ ∈ J ∖ { 0 } t,t'\in J\setminus\{0\} t , t ′ ∈ J ∖ { 0 } t t ′ < 0 tt'<0 t t ′ < 0
( h ( t ′ ) h ( t ) , t ′ − t h ( t ) , t , h ( t ) ) ∈ G ‾ j ∣ W i \left(\frac{h(t')}{h(t)},\frac{t'-t}{h(t)},t,h(t)\right)\in\overline{G}_{j|W_i} ( h ( t ) h ( t ′ ) , h ( t ) t ′ − t , t , h ( t ) ) ∈ G j ∣ W i an arrow from ( t , h ( t ) ) (t,h(t)) ( t , h ( t )) ( t ′ , h ( t ′ ) ) (t',h(t')) ( t ′ , h ( t ′ )) ( K × K ) ∩ G ‾ j ∣ W i (K\times K)\cap\overline{G}_{j|W_i} ( K × K ) ∩ G j ∣ W i L W i L_{W_i} L W i (39) , so the equality of Lemma 4.10(1) fails. For h ( t ) = t 2 h(t)=t^2 h ( t ) = t 2 t ′ = − t t'=-t t ′ = − t ( 1 , − 2 / t + k ℓ j / t 2 , t , t 2 ) (1,-2/t+k\ell_j/t^2,t,t^2) ( 1 , − 2/ t + k ℓ j / t 2 , t , t 2 ) k ∈ Z k\in\mathbb{Z} k ∈ Z ∣ b ∣ ≥ 2 / ∣ t ∣ |b|\ge2/|t| ∣ b ∣ ≥ 2/∣ t ∣ 0 < ∣ t ∣ ≤ ℓ j / 4 0<|t|\le\ell_j/4 0 < ∣ t ∣ ≤ ℓ j /4 G Σ G_\Sigma G Σ t → 0 t\to0 t → 0 q j q_j q j W i W_i W i
Proposition 4.12. The subset of G Σ G_\Sigma G Σ
L = ( ∐ N c ⊂ Σ ∖ D N c × N c ) ∪ ( ∐ N c ⊂ D L c ) ∪ ( ∐ K ∈ K K × K ) ∪ ( ∐ i = 1 n c L p i i s o ) \mathcal{L}=\left(\coprod_{N_c\subset\Sigma\setminus D}N_c\times N_c\right)\cup\left(\coprod_{N_c\subset D}\mathcal{L}_c\right)\cup\left(\coprod_{K\in\mathcal{K}}K\times K\right)\cup\left(\coprod_{i=1}^{n_c}L_{p_i}^{\mathrm{iso}}\right) L = N c ⊂ Σ ∖ D ∐ N c × N c ∪ ( N c ⊂ D ∐ L c ) ∪ ( K ∈ K ∐ K × K ) ∪ ( i = 1 ∐ n c L p i iso ) is an embedded, source-connected Lagrangian subgroupoid of G Σ G_\Sigma G Σ N N N T N ∘ TN^\circ T N ∘
Proof. Each piece of L \mathcal{L} L N 0 N_0 N 0 N D N_D N D K ∈ K K\in\mathcal{K} K ∈ K C C C (35) . So two composable arrows of L \mathcal{L} L L \mathcal{L} L N N N L c \mathcal{L}_c L c
Let g 0 ∈ L g_0\in\mathcal{L} g 0 ∈ L g 0 g_0 g 0 p i ∈ C p_i\in C p i ∈ C g 0 ∈ L p i i s o g_0\in L_{p_i}^{\mathrm{iso}} g 0 ∈ L p i iso (33) , applied to g 0 g_0 g 0 g 0 − 1 g_0^{-1} g 0 − 1 D D D C C C x x x U x U_x U x Σ \Sigma Σ U x ∩ N U_x\cap N U x ∩ N N 0 N_0 N 0 N D N_D N D K \mathcal{K} K ( s , t ) − 1 ( U s ( g 0 ) × U t ( g 0 ) ) (s,t)^{-1}(U_{s(g_0)}\times U_{t(g_0)}) ( s , t ) − 1 ( U s ( g 0 ) × U t ( g 0 ) ) L \mathcal{L} L N c × N c N_c\times N_c N c × N c L c \mathcal{L}_c L c K × K K\times K K × K
If g 0 ∈ L p i i s o g_0\in L_{p_i}^{\mathrm{iso}} g 0 ∈ L p i iso L ∩ G ‾ j ∣ W i = A i ∩ G ‾ j ∣ W i \mathcal{L}\cap\overline{G}_{j|W_i}=A_i\cap\overline{G}_{j|W_i} L ∩ G j ∣ W i = A i ∩ G j ∣ W i N c ∩ W i ⊂ K i − ∪ K i + ∪ { p i } N_c\cap W_i\subset K_i^-\cup K_i^+\cup\{p_i\} N c ∩ W i ⊂ K i − ∪ K i + ∪ { p i } L ∩ O g 0 = L W i ∩ O g 0 \mathcal{L}\cap O_{g_0}=L_{W_i}\cap O_{g_0} L ∩ O g 0 = L W i ∩ O g 0
So L \mathcal{L} L N c × N c N_c\times N_c N c × N c L c \mathcal{L}_c L c K × K K\times K K × K L W i L_{W_i} L W i T N ∘ TN^\circ T N ∘ N N N § 4.2 and Propositions 4.5, 4.7 and 4.9). Hence L \mathcal{L} L s ∣ L s|_{\mathcal{L}} s ∣ L N c N_c N c R \mathbb{R} R K K K L p i i s o L_{p_i}^{\mathrm{iso}} L p i iso □ \square □
It remains to identify the isotropy groups at x ∈ N ∩ D x\in N\cap D x ∈ N ∩ D G Σ , x ≅ Aff + ( R ) G_\Sigma,x\cong\operatorname{Aff}^+(\mathbb{R}) G Σ , x ≅ Aff + ( R ) (29) , a dilation subgroup has the form
{ ( a , ( a − 1 ) b 0 ) : a > 0 } = ( 1 , b 0 ) { ( a , 0 ) : a > 0 } ( 1 , b 0 ) − 1 . (42) \{(a,(a-1)b_0):a>0\}=(1,b_0)\{(a,0):a>0\}(1,b_0)^{-1}. \tag*{(42)} {( a , ( a − 1 ) b 0 ) : a > 0 } = ( 1 , b 0 ) {( a , 0 ) : a > 0 } ( 1 , b 0 ) − 1 . (42) and its Lie algebra is ⟨ ζ 1 + b 0 ζ 2 ⟩ \langle\zeta_1+b_0\zeta_2\rangle ⟨ ζ 1 + b 0 ζ 2 ⟩ b 0 b_0 b 0 aff ( R ) \operatorname{aff}(\mathbb{R}) aff ( R ) ⟨ ζ 2 ⟩ \langle\zeta_2\rangle ⟨ ζ 2 ⟩ Aff + ( R ) \operatorname{Aff}^{+}(\mathbb{R}) Aff + ( R ) Aff + ( R ) \operatorname{Aff}^{+}(\mathbb{R}) Aff + ( R )
Proof of Theorem B. The subgroupoid L \mathcal{L} L [21 Example 3.8] , since Ω Σ ∣ L = 0 \Omega_\Sigma|_{\mathcal{L}}=0 Ω Σ ∣ L = 0 Lie ( L ) = T N ∘ \operatorname{Lie}(\mathcal{L})=T N^\circ Lie ( L ) = T N ∘
The arrows of L \mathcal{L} L N ∖ D N\setminus D N ∖ D N ∖ D N\setminus D N ∖ D N c ⊂ N 0 N_c\subset N_0 N c ⊂ N 0 K ∈ K K\in\mathcal{K} K ∈ K N ∩ D N\cap D N ∩ D (33) .
Let x ∈ N ∩ D j x\in N\cap D_j x ∈ N ∩ D j D j ⊂ N D_j\subset N D j ⊂ N x = p i x=p_i x = p i T x N = T x D T_xN=T_xD T x N = T x D L x \mathcal{L}_x L x { ( 1 , b , t , 0 ) : b ∈ R } \{(1,b,t,0):b\in\mathbb{R}\} {( 1 , b , t , 0 ) : b ∈ R } { ( 1 , b , 0 , 0 ) : b ∈ R } \{(1,b,0,0):b\in\mathbb{R}\} {( 1 , b , 0 , 0 ) : b ∈ R } x = p i x=p_i x = p i T x N ≠ T x D T_xN\ne T_xD T x N = T x D L x \mathcal{L}_x L x { ( a , ( a − 1 ) γ ′ ( 0 ) , 0 , 0 ) : a > 0 } \{(a,(a-1)\gamma'(0),0,0):a>0\} {( a , ( a − 1 ) γ ′ ( 0 ) , 0 , 0 ) : a > 0 } b 0 = γ ′ ( 0 ) b_0=\gamma'(0) b 0 = γ ′ ( 0 )
Homotopy fibre products of Lagrangian subgroupoids
In this section, we study the shifted intersections of two admissible curves. Let N 1 , N 2 N_1,N_2 N 1 , N 2 L 1 ⇒ N 1 \mathcal{L}_1\Rightarrow N_1 L 1 ⇒ N 1 L 2 ⇒ N 2 \mathcal{L}_2\Rightarrow N_2 L 2 ⇒ N 2 D D D D D D
Cleanness
We first describe the homotopy fibre product and the tangent-space condition defining cleanness. Let H = L 1 × G Σ h L 2 \mathcal{H}=\mathcal{L}_1\times^h_{\mathcal{G}_\Sigma}\mathcal{L}_2 H = L 1 × G Σ h L 2 L 1 \mathcal{L}_1 L 1 L 2 \mathcal{L}_2 L 2 [21 §4.2] . It is a groupoid whose space of objects is
H 0 = { ( x 1 , g , x 2 ) ∈ N 1 × G Σ × N 2 : s ( g ) = x 1 , t ( g ) = x 2 } . \mathcal{H}^0=\{(x_1,g,x_2)\in N_1\times\mathcal{G}_\Sigma\times N_2:s(g)=x_1,\ t(g)=x_2\}. H 0 = {( x 1 , g , x 2 ) ∈ N 1 × G Σ × N 2 : s ( g ) = x 1 , t ( g ) = x 2 } . An arrow with source ( x 1 , g , x 2 ) (x_1,g,x_2) ( x 1 , g , x 2 ) ( l , l ~ ) ∈ L 1 × L 2 (l,\widetilde{l})\in\mathcal{L}_1\times\mathcal{L}_2 ( l , l ) ∈ L 1 × L 2 s ( l ) = x 1 s(l)=x_1 s ( l ) = x 1 s ( l ~ ) = x 2 s(\widetilde{l})=x_2 s ( l ) = x 2 ( t ( l ) , l ~ g l − 1 , t ( l ~ ) ) (t(l),\widetilde{l}gl^{-1},t(\widetilde{l})) ( t ( l ) , l g l − 1 , t ( l ))
We write pr 1 \operatorname{pr}_1 pr 1 pr 0 \operatorname{pr}_0 pr 0 pr 2 \operatorname{pr}_2 pr 2 H 0 \mathcal{H}^0 H 0 N 1 N_1 N 1 G Σ \mathcal{G}_\Sigma G Σ N 2 N_2 N 2 ( x 1 , g , x 2 ) (x_1,g,x_2) ( x 1 , g , x 2 )
T ( x 1 , g , x 2 ) e x p = { ( u , w , v ) ∈ T x 1 N 1 × T g G Σ × T x 2 N 2 : d s g ( w ) = u , d t g ( w ) = v } . T^{\mathrm{exp}}_{(x_1,g,x_2)}=\{(u,w,v)\in T_{x_1}N_1\times T_g\mathcal{G}_\Sigma\times T_{x_2}N_2:d s_g(w)=u,\ d t_g(w)=v\}. T ( x 1 , g , x 2 ) exp = {( u , w , v ) ∈ T x 1 N 1 × T g G Σ × T x 2 N 2 : d s g ( w ) = u , d t g ( w ) = v } . The homotopy fibre product H \mathcal{H} H H 0 \mathcal{H}^0 H 0 N 1 × G Σ × N 2 N_1\times\mathcal{G}_\Sigma\times N_2 N 1 × G Σ × N 2 [21 §§2.3 and 4.2] . When the object is fixed, we abbreviate this space to T e x p T^{\mathrm{exp}} T exp
The map g ↦ ( s ( g ) , g , t ( g ) ) g\mapsto(s(g),g,t(g)) g ↦ ( s ( g ) , g , t ( g )) G Σ \mathcal{G}_\Sigma G Σ Σ × G Σ × Σ \Sigma\times\mathcal{G}_\Sigma\times\Sigma Σ × G Σ × Σ ( s , t ) − 1 ( N 1 × N 2 ) (s,t)^{-1}(N_1\times N_2) ( s , t ) − 1 ( N 1 × N 2 ) H 0 \mathcal{H}^0 H 0 H 0 \mathcal{H}^0 H 0 ( s , t ) − 1 ( N 1 × N 2 ) ⊂ G Σ (s,t)^{-1}(N_1\times N_2)\subset\mathcal{G}_\Sigma ( s , t ) − 1 ( N 1 × N 2 ) ⊂ G Σ
T e x p = { w ∈ T g G Σ : d s g ( w ) ∈ T x 1 N 1 , d t g ( w ) ∈ T x 2 N 2 } . T^{\mathrm{exp}}=\{w\in T_g\mathcal{G}_\Sigma:d s_g(w)\in T_{x_1}N_1,\ d t_g(w)\in T_{x_2}N_2\}. T exp = { w ∈ T g G Σ : d s g ( w ) ∈ T x 1 N 1 , d t g ( w ) ∈ T x 2 N 2 } . For a 2-form ω \omega ω Γ ω \Gamma_\omega Γ ω L 1 \mathcal{L}_1 L 1 L 2 \mathcal{L}_2 L 2 L N 1 = Γ 0 = T N 1 \mathbb{L}_{N_1}=\Gamma_0=T N_1 L N 1 = Γ 0 = T N 1 L N 2 = Γ 0 = T N 2 \mathbb{L}_{N_2}=\Gamma_0=T N_2 L N 2 = Γ 0 = T N 2 [21 §3.2, Example 3.8 and Remark 4.9] . For graphs, the pullback and sum of Dirac structures satisfy
f ∗ Γ ω = Γ f ∗ ω , Γ ω + Γ η = Γ ω + η . f^*\Gamma_\omega=\Gamma_{f^*\omega},\qquad\Gamma_\omega+\Gamma_\eta=\Gamma_{\omega+\eta}. f ∗ Γ ω = Γ f ∗ ω , Γ ω + Γ η = Γ ω + η . If H \mathcal{H} H
L = pr 1 ∗ L N 1 + pr 2 ∗ L N 2 − pr 0 ∗ Ω Σ = Γ − pr 0 ∗ Ω Σ . \begin{aligned}
\mathbb{L}=\operatorname{pr}_1^*\mathbb{L}_{N_1}+\operatorname{pr}_2^*\mathbb{L}_{N_2}-\operatorname{pr}_0^*\Omega_\Sigma\\
&=\Gamma_{-\operatorname{pr}_0^*\Omega_\Sigma}.
\end{aligned} L = pr 1 ∗ L N 1 + pr 2 ∗ L N 2 − pr 0 ∗ Ω Σ = Γ − pr 0 ∗ Ω Σ . It is smooth because it is the graph of a smooth 2-form. Thus it defines a 0-shifted Poisson structure on H \mathcal{H} H [21 Corollary 4.12] .
We separate the objects lying over the symplectic locus from those lying over D D D
H g e n 0 = { ( x 1 , g , x 2 ) ∈ H 0 : x 1 ∉ D } , \mathcal{H}^0_{\mathrm{gen}}=\{(x_1,g,x_2)\in\mathcal{H}^0:x_1\notin D\}, H gen 0 = {( x 1 , g , x 2 ) ∈ H 0 : x 1 ∈ / D } , H i s o 0 = { ( x , g , x ) ∈ H 0 : x ∈ D ∩ N 1 ∩ N 2 } . \mathcal{H}^0_{\mathrm{iso}}=\{(x,g,x)\in\mathcal{H}^0:x\in D\cap N_1\cap N_2\}. H iso 0 = {( x , g , x ) ∈ H 0 : x ∈ D ∩ N 1 ∩ N 2 } . By (33) , applied to g g g g − 1 g^{-1} g − 1 x 1 ∈ D x_1 \in D x 1 ∈ D x 2 ∈ D x_2 \in D x 2 ∈ D x 2 = x 1 x_2=x_1 x 2 = x 1 H 0 = H g e n 0 ⊔ H i s o 0 \mathcal{H}^0=\mathcal{H}^0_{\mathrm{gen}}\sqcup\mathcal{H}^0_{\mathrm{iso}} H 0 = H gen 0 ⊔ H iso 0 (34) , if x 1 ∈ V i x_1\in V_i x 1 ∈ V i x 2 ∈ V i x_2\in V_i x 2 ∈ V i H g e n 0 \mathcal{H}^0_{\mathrm{gen}} H gen 0 H i 0 = H g e n 0 ∩ ( V i × G Σ × V i ) \mathcal{H}^0_i=\mathcal{H}^0_{\mathrm{gen}}\cap(V_i\times\mathcal{G}_\Sigma\times V_i) H i 0 = H gen 0 ∩ ( V i × G Σ × V i )
Since ( s , t ) (s,t) ( s , t ) G Σ ∣ V i = V i × V i \mathcal{G}_\Sigma|_{V_i}=V_i\times V_i G Σ ∣ V i = V i × V i V i × V i V_i\times V_i V i × V i H i 0 = ( s , t ) − 1 ( ( N 1 ∩ V i ) × ( N 2 ∩ V i ) ) \mathcal{H}^0_i=(s,t)^{-1}((N_1\cap V_i)\times(N_2\cap V_i)) H i 0 = ( s , t ) − 1 (( N 1 ∩ V i ) × ( N 2 ∩ V i )) d ( s , t ) − 1 ( T x 1 N 1 × T x 2 N 2 ) d(s,t)^{-1}(T_{x_1}N_1\times T_{x_2}N_2) d ( s , t ) − 1 ( T x 1 N 1 × T x 2 N 2 )
( N 1 ∩ V i ) × ( N 2 ∩ V i ) ⟶ H i 0 , ( x 1 , x 2 ) ⟼ ( x 1 , ( x 2 , x 1 ) , x 2 ) , (43) (N_1\cap V_i)\times(N_2\cap V_i)\longrightarrow\mathcal{H}^0_i,\qquad(x_1,x_2)\longmapsto(x_1,(x_2,x_1),x_2), \tag*{(43)} ( N 1 ∩ V i ) × ( N 2 ∩ V i ) ⟶ H i 0 , ( x 1 , x 2 ) ⟼ ( x 1 , ( x 2 , x 1 ) , x 2 ) , (43) is a diffeomorphism. Therefore H \mathcal{H} H H g e n 0 \mathcal{H}^0_{\mathrm{gen}} H gen 0 Proposition 3.6 gives Ω Σ = t ∗ ω i − s ∗ ω i \Omega_\Sigma=t^*\omega_i-s^*\omega_i Ω Σ = t ∗ ω i − s ∗ ω i V i × V i V_i\times V_i V i × V i s ∘ pr 0 = pr 1 s\circ\operatorname{pr}_0=\operatorname{pr}_1 s ∘ pr 0 = pr 1 t ∘ pr 0 = pr 2 t\circ\operatorname{pr}_0=\operatorname{pr}_2 t ∘ pr 0 = pr 2
pr 0 ∗ Ω Σ = pr 2 ∗ ( ω i ∣ N 2 ∩ V i ) − pr 1 ∗ ( ω i ∣ N 1 ∩ V i ) = 0 \operatorname{pr}_0^*\Omega_\Sigma=\operatorname{pr}_2^*(\omega_i|_{N_2\cap V_i})-\operatorname{pr}_1^*(\omega_i|_{N_1\cap V_i})=0 pr 0 ∗ Ω Σ = pr 2 ∗ ( ω i ∣ N 2 ∩ V i ) − pr 1 ∗ ( ω i ∣ N 1 ∩ V i ) = 0 on H i 0 \mathcal{H}^0_i H i 0
The six local configurations
It remains to study H \mathcal{H} H H i s o 0 \mathcal{H}^0_{\mathrm{iso}} H iso 0 x ∈ D j ∩ N 1 ∩ N 2 x\in D_j\cap N_1\cap N_2 x ∈ D j ∩ N 1 ∩ N 2 t t t (26) , we take x = ( 0 , 0 ) x=(0,0) x = ( 0 , 0 ) G Σ , x ≅ Aff + ( R ) \mathcal{G}_{\Sigma,x}\cong\operatorname{Aff}^+(\mathbb{R}) G Σ , x ≅ Aff + ( R ) g = ( a , b , 0 , 0 ) g=(a,b,0,0) g = ( a , b , 0 , 0 ) α = log a \alpha=\log a α = log a § 3.3 . We first compute the expected tangent space, before comparing it with the local equations of H 0 \mathcal{H}^0 H 0
Lemma 5.1. For g = ( a , b , 0 , 0 ) ∈ G Σ , x g=(a,b,0,0)\in\mathcal{G}_{\Sigma,x} g = ( a , b , 0 , 0 ) ∈ G Σ , x M g = ( 1 b 0 a ) M_g=\begin{pmatrix}1&b\\0&a\end{pmatrix} M g = ( 1 0 b a ) ( ∂ ∂ t , ∂ ∂ r ) \left(\frac{\partial}{\partial t},\frac{\partial}{\partial r}\right) ( ∂ t ∂ , ∂ r ∂ ) T x Σ T_x\Sigma T x Σ d t g = M g ∘ d s g dt_g=M_g\circ ds_g d t g = M g ∘ d s g ker d s g = ker d t g \ker ds_g=\ker dt_g ker d s g = ker d t g 2 2 2 ( x , g , x ) (x,g,x) ( x , g , x )
dim T e x p = 2 + dim ( T x N 1 ∩ M g − 1 T x N 2 ) . (44) \dim T^{\mathrm{exp}}=2+\dim\left(T_xN_1\cap M_g^{-1}T_xN_2\right). \tag*{(44)} dim T exp = 2 + dim ( T x N 1 ∩ M g − 1 T x N 2 ) . (44) (1) If T x N 1 = T x N 2 = T x D j T_xN_1=T_xN_2=T_xD_j T x N 1 = T x N 2 = T x D j dim T e x p = 3 \dim T^{\mathrm{exp}}=3 dim T exp = 3 g g g
(2) If exactly one of T x N 1 T_xN_1 T x N 1 T x N 2 T_xN_2 T x N 2 T x D j T_xD_j T x D j dim T e x p = 2 \dim T^{\mathrm{exp}}=2 dim T exp = 2 g g g
(3) If both N 1 N_1 N 1 N 2 N_2 N 2 D D D x x x γ \gamma γ γ ~ \widetilde{\gamma} γ Lemma 4.2 , then dim T e x p = 2 \dim T^{\mathrm{exp}}=2 dim T exp = 2 { b = a γ ~ ′ ( 0 ) − γ ′ ( 0 ) } \{b=a\widetilde{\gamma}'(0)-\gamma'(0)\} { b = a γ ′ ( 0 ) − γ ′ ( 0 )} G Σ , x \mathcal{G}_{\Sigma,x} G Σ , x 3 3 3
Proof. Since t ( a , b , t , r ) = ( t + b r , a r ) t(a,b,t,r)=(t+br,ar) t ( a , b , t , r ) = ( t + b r , a r ) r = 0 r=0 r = 0 g g g w = ( a ˙ , b ˙ , t ˙ , r ˙ ) w=(\dot a,\dot b,\dot t,\dot r) w = ( a ˙ , b ˙ , t ˙ , r ˙ )
d s g ( w ) = ( t ˙ , r ˙ ) , d t g ( w ) = ( t ˙ + b r ˙ , a r ˙ ) = M g ( t ˙ , r ˙ ) . ds_g(w)=(\dot t,\dot r),\qquad dt_g(w)=(\dot t+b\dot r,a\dot r)=M_g(\dot t,\dot r). d s g ( w ) = ( t ˙ , r ˙ ) , d t g ( w ) = ( t ˙ + b r ˙ , a r ˙ ) = M g ( t ˙ , r ˙ ) . Consequently,
ker d s g = ker d t g = ⟨ ∂ ∂ a , ∂ ∂ b ⟩ . \ker ds_g=\ker dt_g=\left\langle\frac{\partial}{\partial a},\frac{\partial}{\partial b}\right\rangle. ker d s g = ker d t g = ⟨ ∂ a ∂ , ∂ b ∂ ⟩ . The surjection d s g ds_g d s g T e x p T^{\mathrm{exp}} T exp { u ∈ T x N 1 : M g u ∈ T x N 2 } \{u\in T_xN_1:M_gu\in T_xN_2\} { u ∈ T x N 1 : M g u ∈ T x N 2 } ker d s g \ker ds_g ker d s g (44) .
Since M g ∂ ∂ t = ∂ ∂ t M_g\frac{\partial}{\partial t}=\frac{\partial}{\partial t} M g ∂ t ∂ = ∂ t ∂ M g M_g M g T x D j = ⟨ ∂ ∂ t ⟩ T_xD_j=\left\langle\frac{\partial}{\partial t}\right\rangle T x D j = ⟨ ∂ t ∂ ⟩ T x N 1 = T x D j ≠ T x N 2 T_xN_1=T_xD_j\ne T_xN_2 T x N 1 = T x D j = T x N 2 M g − 1 T x N 2 ≠ T x D j M_g^{-1}T_xN_2\ne T_xD_j M g − 1 T x N 2 = T x D j (44) is zero. The other order is treated in the same way. In case (3), the tangent lines are
T x N 1 = ⟨ γ ′ ( 0 ) ∂ ∂ t + ∂ ∂ r ⟩ , T_xN_1=\left\langle\gamma'(0)\frac{\partial}{\partial t}+\frac{\partial}{\partial r}\right\rangle, T x N 1 = ⟨ γ ′ ( 0 ) ∂ t ∂ + ∂ r ∂ ⟩ , T x N 2 = ⟨ γ ~ ′ ( 0 ) ∂ ∂ t + ∂ ∂ r ⟩ . T_xN_2=\left\langle\widetilde{\gamma}'(0)\frac{\partial}{\partial t}+\frac{\partial}{\partial r}\right\rangle. T x N 2 = ⟨ γ ′ ( 0 ) ∂ t ∂ + ∂ r ∂ ⟩ . The vector
M g ( γ ′ ( 0 ) ∂ ∂ t + ∂ ∂ r ) = ( γ ′ ( 0 ) + b ) ∂ ∂ t + a ∂ ∂ r M_g\left(\gamma'(0)\frac{\partial}{\partial t}+\frac{\partial}{\partial r}\right)=(\gamma'(0)+b)\frac{\partial}{\partial t}+a\frac{\partial}{\partial r} M g ( γ ′ ( 0 ) ∂ t ∂ + ∂ r ∂ ) = ( γ ′ ( 0 ) + b ) ∂ t ∂ + a ∂ r ∂ lies in T x N 2 T_xN_2 T x N 2 γ ′ ( 0 ) + b = a γ ~ ′ ( 0 ) \gamma'(0)+b=a\widetilde{\gamma}'(0) γ ′ ( 0 ) + b = a γ ′ ( 0 )
We now compare these expected tangent spaces with the local equations of H 0 \mathcal{H}^0 H 0 Lemmas 4.2 and 4.3 , there are three possibilities for each curve near x x x D j D_j D j D D D t = γ ( r ) t=\gamma(r) t = γ ( r ) D D D r = h ( t ) r=h(t) r = h ( t ) N 2 N_2 N 2 γ ~ \widetilde{\gamma} γ h ~ \widetilde{h} h Figure 6 .
Figure 6. The six configurations of two curves at a common point of D D D
We fix one box W = J × I W=J\times I W = J × I x x x N 1 N_1 N 1 N 2 N_2 N 2 x x x I I I J J J D j D_j D j I I I J J J I = ( − ϵ , ϵ ) I=(-\epsilon,\epsilon) I = ( − ϵ , ϵ ) J = ( − C ϵ , C ϵ ) J=(-C\epsilon,C\epsilon) J = ( − C ϵ , C ϵ ) C > sup ∣ γ ′ ∣ C>\sup|\gamma'| C > sup ∣ γ ′ ∣ ϵ \epsilon ϵ ∣ γ ( r ) ∣ ≤ sup ∣ γ ′ ∣ ∣ r ∣ |\gamma(r)|\leq\sup|\gamma'|\,|r| ∣ γ ( r ) ∣ ≤ sup ∣ γ ′ ∣ ∣ r ∣ h ( t ) = O ( t 2 ) h(t)=O(t^2) h ( t ) = O ( t 2 )
We use the open set G j , W l o c \mathcal{G}^{\mathrm{loc}}_{j,W} G j , W loc Lemma 4.4 . It contains the isotropy group G j , x \mathcal{G}_{j,x} G j , x q j q_j q j G ‾ j ∣ W \overline{\mathcal{G}}_j|_W G j ∣ W G Σ \mathcal{G}_{\Sigma} G Σ
H 0 ∩ G j , W l o c = { ( a , b , t , r ) ∈ G j , W l o c : ( t , r ) ∈ N 1 , ( t + b r , a r ) ∈ N 2 } . \mathcal{H}^{0} \cap\mathcal{G}^{\mathrm{loc}}_{j,W} = \{(a,b,t,r) \in\mathcal{G}^{\mathrm{loc}}_{j,W} : (t,r) \in N_1,\ (t+br,ar) \in N_2\}. H 0 ∩ G j , W loc = {( a , b , t , r ) ∈ G j , W loc : ( t , r ) ∈ N 1 , ( t + b r , a r ) ∈ N 2 } . Proposition 5.2. Let x ∈ D j ∩ N 1 ∩ N 2 x \in D_j \cap N_1 \cap N_2 x ∈ D j ∩ N 1 ∩ N 2 g 0 ∈ G Σ , x g_0 \in\mathcal{G}_{\Sigma,x} g 0 ∈ G Σ , x g ^ 0 ∈ G j , W l o c \widehat{g}_0 \in\mathcal{G}^{\mathrm{loc}}_{j,W} g 0 ∈ G j , W loc O ⊂ G j , W l o c O \subset\mathcal{G}^{\mathrm{loc}}_{j,W} O ⊂ G j , W loc g ^ 0 \widehat{g}_0 g 0 H 0 \mathcal{H}^{0} H 0 Figure 6 .
(1) If D j ⊂ N 1 D_j \subset N_1 D j ⊂ N 1 D j ⊂ N 2 D_j \subset N_2 D j ⊂ N 2 H 0 ∩ O = O ∩ { r = 0 } \mathcal{H}^{0} \cap O = O \cap\{r=0\} H 0 ∩ O = O ∩ { r = 0 } 3 3 3 D j × Aff + ( R ) D_j \times\operatorname{Aff}^{+}(\mathbb{R}) D j × Aff + ( R ) H \mathcal{H} H
(2) If one of N 1 , N 2 N_1,N_2 N 1 , N 2 D j D_j D j H 0 ∩ O = O ∩ { t = r = 0 } \mathcal{H}^{0} \cap O = O \cap\{t=r=0\} H 0 ∩ O = O ∩ { t = r = 0 } H \mathcal{H} H
(3) If one of N 1 , N 2 N_1,N_2 N 1 , N 2 H 0 ∩ O = O ∩ { t = r = 0 } \mathcal{H}^{0} \cap O = O \cap\{t=r=0\} H 0 ∩ O = O ∩ { t = r = 0 } H \mathcal{H} H
(4) If both N 1 N_1 N 1 N 2 N_2 N 2 γ ( r ) = r γ 1 ( r ) \gamma(r)=r\gamma_1(r) γ ( r ) = r γ 1 ( r ) γ ~ ( r ) = r γ ~ 1 ( r ) \widetilde{\gamma}(r)=r\widetilde{\gamma}_1(r) γ ( r ) = r γ 1 ( r )
H 0 ∩ O = O ∩ ( { t = r = 0 } ∪ { t = γ ( r ) , b = a γ ~ 1 ( a r ) − γ 1 ( r ) } ) . \mathcal{H}^{0} \cap O = O \cap\left(\{t=r=0\} \cup\{t=\gamma(r),\ b=a\widetilde{\gamma}_1(ar)-\gamma_1(r)\}\right). H 0 ∩ O = O ∩ ( { t = r = 0 } ∪ { t = γ ( r ) , b = a γ 1 ( a r ) − γ 1 ( r )} ) . These two embedded surfaces meet along the curve { t = r = 0 , b = a γ ~ ′ ( 0 ) − γ ′ ( 0 ) } \{t=r=0,\ b=a\widetilde{\gamma}'(0)-\gamma'(0)\} { t = r = 0 , b = a γ ′ ( 0 ) − γ ′ ( 0 )} H 0 \mathcal{H}^{0} H 0
(5) If one of N 1 , N 2 N_1,N_2 N 1 , N 2 D j D_j D j H 0 ∩ O = O ∩ { t = r = 0 } \mathcal{H}^{0} \cap O = O \cap\{t=r=0\} H 0 ∩ O = O ∩ { t = r = 0 } dim T exp = 3 \dim T^{\exp}=3 dim T e x p = 3 H \mathcal{H} H
(6) If both N 1 N_1 N 1 N 2 N_2 N 2 H 0 ∩ O \mathcal{H}^{0} \cap O H 0 ∩ O O ∩ { t = r = 0 } O \cap\{t=r=0\} O ∩ { t = r = 0 } 3 3 3 H 0 ∩ O \mathcal{H}^{0} \cap O H 0 ∩ O 3 3 3 { r = h ( t ) } \{r=h(t)\} { r = h ( t )} H \mathcal{H} H h ( t ) = h ~ ( t ) = t 2 h(t)=\widetilde{h}(t)=t^2 h ( t ) = h ( t ) = t 2 H 0 ∩ O \mathcal{H}^{0} \cap O H 0 ∩ O { t = 0 } \{t=0\} { t = 0 } { a = ( 1 + b t ) 2 } \{a=(1+bt)^2\} { a = ( 1 + b t ) 2 } { r = t 2 } \{r=t^2\} { r = t 2 } { t = 0 , a = 1 } \{t=0,\ a=1\} { t = 0 , a = 1 }
Proof. The inversion ( x 1 , g , x 2 ) ↦ ( x 2 , g − 1 , x 1 ) (x_1,g,x_2) \mapsto(x_2,g^{-1},x_1) ( x 1 , g , x 2 ) ↦ ( x 2 , g − 1 , x 1 ) L 1 × Σ Z h L 2 \mathcal{L}_1 \mathbin{\times^h_{\Sigma_Z}} \mathcal{L}_2 L 1 × Σ Z h L 2 L 2 × Σ Z h L 1 \mathcal{L}_2 \mathbin{\times^h_{\Sigma_Z}} \mathcal{L}_1 L 2 × Σ Z h L 1 { t = r = 0 } \{t=r=0\} { t = r = 0 } O = G j , W loc O=G^{{\operatorname{loc}}}_{j,W} O = G j , W loc
In case (1), ( t , r ) ∈ N 1 (t,r)\in N_1 ( t , r ) ∈ N 1 r = 0 r=0 r = 0 ( t + b r , a r ) = ( t , 0 ) ∈ N 2 (t+br,ar)=(t,0)\in N_2 ( t + b r , a r ) = ( t , 0 ) ∈ N 2 T { r = 0 } ⊂ T exp T\{r=0\}\subset T^{\exp} T { r = 0 } ⊂ T e x p dim T exp = 3 \dim T^{\exp}=3 dim T e x p = 3 Lemma 5.1 (1).
In case (2), say N 1 ⊃ D j N_1\supset D_j N 1 ⊃ D j N 2 N_2 N 2 r = 0 r=0 r = 0 ( t , 0 ) ∈ N 2 (t,0)\in N_2 ( t , 0 ) ∈ N 2 t = γ ~ ( 0 ) = 0 t=\widetilde{\gamma}(0)=0 t = γ ( 0 ) = 0 T { t = r = 0 } = T exp T\{t=r=0\}=T^{\exp} T { t = r = 0 } = T e x p Lemma 5.1 (2).
In case (3), suppose that N 1 N_1 N 1 N 2 N_2 N 2 r = h ( t ) r=h(t) r = h ( t ) 0 = t + b h ( t ) − γ ~ ( a h ( t ) ) 0=t+bh(t)-\widetilde{\gamma}(ah(t)) 0 = t + bh ( t ) − γ ( ah ( t )) t = 0 t=0 t = 0 1 1 1 t t t h ′ ( 0 ) = 0 h'(0)=0 h ′ ( 0 ) = 0 O O O g ^ 0 \widehat{g}_0 g 0 t = 0 t=0 t = 0 r = 0 r=0 r = 0
In case (4), the source condition is t = γ ( r ) t=\gamma(r) t = γ ( r )
0 = γ ~ ( a r ) − γ ( r ) − b r = r ( a γ ~ 1 ( a r ) − γ 1 ( r ) − b ) . 0=\widetilde{\gamma}(ar)-\gamma(r)-br=r\left(a\widetilde{\gamma}_1(ar)-\gamma_1(r)-b\right). 0 = γ ( a r ) − γ ( r ) − b r = r ( a γ 1 ( a r ) − γ 1 ( r ) − b ) . The factor r r r { t = r = 0 } \{t=r=0\} { t = r = 0 } ( a , b ) (a,b) ( a , b ) ⟨ ∂ ∂ a , ∂ ∂ b ⟩ \left\langle\frac{\partial}{\partial a},\frac{\partial}{\partial b}\right\rangle ⟨ ∂ a ∂ , ∂ b ∂ ⟩
{ t = γ ( r ) , b = a γ ~ 1 ( a r ) − γ 1 ( r ) } , \{t=\gamma(r),\ b=a\widetilde{\gamma}_1(ar)-\gamma_1(r)\}, { t = γ ( r ) , b = a γ 1 ( a r ) − γ 1 ( r )} , parametrized by ( a , r ) (a,r) ( a , r ) ∂ ∂ r \frac{\partial}{\partial r} ∂ r ∂ 1 1 1 { t = r = 0 , b = a γ ~ ′ ( 0 ) − γ ′ ( 0 ) } \{t=r=0,\ b=a\widetilde{\gamma}'(0)-\gamma'(0)\} { t = r = 0 , b = a γ ′ ( 0 ) − γ ′ ( 0 )} γ 1 ( 0 ) = γ ′ ( 0 ) \gamma_1(0)=\gamma'(0) γ 1 ( 0 ) = γ ′ ( 0 ) γ ~ 1 ( 0 ) = γ ~ ′ ( 0 ) \widetilde{\gamma}_1(0)=\widetilde{\gamma}'(0) γ 1 ( 0 ) = γ ′ ( 0 ) H 0 \mathcal{H}^0 H 0
In case (5), say N 1 ⊃ D j N_1\supset D_j N 1 ⊃ D j N 2 N_2 N 2 r = 0 r=0 r = 0 h ~ ( t ) = 0 \widetilde{h}(t)=0 h ( t ) = 0 t = 0 t=0 t = 0 dim T exp = 3 \dim T^{\exp}=3 dim T e x p = 3 Lemma 5.1 (1).
In case (6), the equations are r = h ( t ) r=h(t) r = h ( t ) a h ( t ) = h ~ ( t + b h ( t ) ) ah(t)=\widetilde{h}(t+bh(t)) ah ( t ) = h ( t + bh ( t )) { t = r = 0 } \{t=r=0\} { t = r = 0 } dim T exp = 3 \dim T^{\exp}=3 dim T e x p = 3 Lemma 5.1 (1). On the 3-manifold { r = h ( t ) } \{r=h(t)\} { r = h ( t )} ( a , b , t ) (a,b,t) ( a , b , t ) F ( a , b , t ) = a h ( t ) − h ~ ( t + b h ( t ) ) F(a,b,t)=ah(t)-\widetilde{h}(t+bh(t)) F ( a , b , t ) = ah ( t ) − h ( t + bh ( t )) ∂ F / ∂ a = h ( t ) ≠ 0 \partial F/\partial a=h(t)\ne0 ∂ F / ∂ a = h ( t ) = 0 t ≠ 0 t\ne0 t = 0 t = 0 t=0 t = 0 { t = 0 } \{t=0\} { t = 0 } H 0 ∩ O \mathcal{H}^0\cap O H 0 ∩ O { r = h ( t ) } \{r=h(t)\} { r = h ( t )} H \mathcal{H} H ( x , g 0 , x ) (x,g_0,x) ( x , g 0 , x ) H 0 \mathcal{H}^0 H 0 g ^ 0 \widehat{g}_0 g 0 g ^ 0 \widehat{g}_0 g 0 h = h ~ = t 2 h=\widetilde{h}=t^2 h = h = t 2 a t 2 − ( t + b t 2 ) 2 = t 2 ( a − ( 1 + b t ) 2 ) at^2-(t+bt^2)^2=t^2\left(a-(1+bt)^2\right) a t 2 − ( t + b t 2 ) 2 = t 2 ( a − ( 1 + b t ) 2 )
We can now decide when H \mathcal{H} H
Proposition 5.3. The homotopy fibre product H = L 1 × Σ Z h L 2 \mathcal{H}=\mathcal{L}_1\mathbin{\times^h_{\Sigma_Z}}\mathcal{L}_2 H = L 1 × Σ Z h L 2 x ∈ D ∩ N 1 ∩ N 2 x\in D\cap N_1\cap N_2 x ∈ D ∩ N 1 ∩ N 2 D j D_j D j D D D x x x
either D j ⊂ N 1 D_j\subset N_1 D j ⊂ N 1 D j ⊂ N 2 D_j\subset N_2 D j ⊂ N 2
or exactly one of the lines T x N 1 T_xN_1 T x N 1 T x N 2 T_xN_2 T x N 2 T x D j T_xD_j T x D j
In that case, Γ pr 0 ∗ Ω Z \Gamma_{\operatorname{pr}_0^*\Omega_Z} Γ pr 0 ∗ Ω Z H \mathcal{H} H
pr 0 ∗ Ω Z = { 0 on H g e n 0 and at the points where (2) holds , d t ∧ d α on the components D j × Aff + ( R ) where (1) holds . \operatorname{pr}_0^*\Omega_Z=
\begin{cases}
0 & \text{on }\mathcal{H}^0_{\mathrm{gen}}\text{ and at the points where (2) holds},\\
\mathrm{d}t\wedge\mathrm{d}\alpha& \text{on the components }D_j\times\operatorname{Aff}^{+}(\mathbb{R})\text{ where (1) holds}.
\end{cases} pr 0 ∗ Ω Z = { 0 d t ∧ d α on H gen 0 and at the points where (2) holds , on the components D j × Aff + ( R ) where (1) holds . Proof. Cleanness is local, and H \mathcal{H} H H g e n 0 \mathcal{H}^0_{\mathrm{gen}} H gen 0 § 5.1 . At x ∈ D ∩ N 1 ∩ N 2 x\in D\cap N_1\cap N_2 x ∈ D ∩ N 1 ∩ N 2 H \mathcal{H} H { x } × Σ Z × { x } \{x\}\times\Sigma_Z\times\{x\} { x } × Σ Z × { x } { b = a γ ~ ′ ( 0 ) − γ ′ ( 0 ) } \{b=a\widetilde{\gamma}'(0)-\gamma'(0)\} { b = a γ ′ ( 0 ) − γ ′ ( 0 )} T x N 1 = T x D j T_xN_1=T_xD_j T x N 1 = T x D j N 1 N_1 N 1 D j D_j D j D D D x x x N 2 N_2 N 2
The form pr 0 ∗ Ω Z \operatorname{pr}_0^*\Omega_Z pr 0 ∗ Ω Z H g e n 0 \mathcal{H}^0_{\mathrm{gen}} H gen 0 § 5.1 . In cases (1)–(3) of Proposition 5.2, H 0 \mathcal{H}^0 H 0 { r = 0 } \{r=0\} { r = 0 } { x } × Σ Z × { x } \{x\}\times\Sigma_Z\times\{x\} { x } × Σ Z × { x } { t = r = 0 } \{t=r=0\} { t = r = 0 } (31) , Ω ∣ { r = 0 } = d t ∧ d α \left.\Omega\right|_{\{r=0\}}=\mathrm{d}t\wedge\mathrm{d}\alpha Ω ∣ { r = 0 } = d t ∧ d α Ω ∣ { t = r = 0 } = 0 \left.\Omega\right|_{\{t=r=0\}}=0 Ω ∣ { t = r = 0 } = 0
If H \mathcal{H} H Γ pr 0 ∗ Ω Z \Gamma_{\operatorname{pr}_0^*\Omega_Z} Γ pr 0 ∗ Ω Z [21 Corollary 4.12(1)] .
:::
For the shifted self-intersection of a single curve, the criterion takes a simple form.
Corollary 5.4. Let N N N L \mathcal{L} L L × Σ Z h L \mathcal{L}\mathbin{\times^h_{\Sigma_Z}}\mathcal{L} L × Σ Z h L N N N D D D D D D
Proof. At x ∈ D ∩ N x \in D \cap N x ∈ D ∩ N Proposition 5.3 fails since the two lines coincide. So L × G h L L \times_{\mathcal{G}}^{h} L L × G h L D j ⊂ N D_j \subset N D j ⊂ N D j D_j D j D D D x ∈ D ∩ N x \in D \cap N x ∈ D ∩ N D j D_j D j N N N D j ⊂ N D_j \subset N D j ⊂ N D j D_j D j N N N x x x
Morita equivalence and modular periods
We now describe the clean homotopy fibre products up to Morita equivalence and identify the function that recovers the modular period. On a component D j × Aff + ( R ) D_j \times\operatorname{Aff}^{+}(\mathbb{R}) D j × Aff + ( R ) H 0 \mathcal{H}^{0} H 0 ( t , a , b ) (t,a,b) ( t , a , b ) ( a , b , t , 0 ) (a,b,t,0) ( a , b , t , 0 ) α = log a \alpha=\log a α = log a a a a M g M_g M g Lemma 5.1 , characterized by d t g = M g ∘ d s g dt_g=M_g\circ ds_g d t g = M g ∘ d s g a a a T x Σ / T x D j T_x\Sigma/T_xD_j T x Σ/ T x D j
Proposition 5.5. Suppose that H \mathcal{H} H Then H \mathcal{H} H D j D_j D j D D D N 1 ∩ N 2 N_1 \cap N_2 N 1 ∩ N 2 ( S ℓ j × R ) × R (S^{\ell_j} \times\mathbb{R}) \times\mathbb{R} ( S ℓ j × R ) × R S ℓ j × R S^{\ell_j} \times\mathbb{R} S ℓ j × R Proposition 5.3 restricts to the graph of the zero form on the points and of the symplectic form d α ∧ d t d\alpha\wedge dt d α ∧ d t R \mathbb{R} R
Proof. We decompose H 0 \mathcal{H}^{0} H 0
the sets
H K 1 , K 2 0 = ( s , t ) − 1 ( K 1 × K 2 ) ≅ K 1 × K 2 , \mathcal{H}^{0}_{K_1,K_2}=(s,t)^{-1}(K_1\times K_2)\cong K_1\times K_2, H K 1 , K 2 0 = ( s , t ) − 1 ( K 1 × K 2 ) ≅ K 1 × K 2 , where K 1 K_1 K 1 K 2 K_2 K 2 N 1 ∖ D N_1\setminus D N 1 ∖ D N 2 ∖ D N_2\setminus D N 2 ∖ D Σ ∖ D \Sigma\setminus D Σ ∖ D
the sets { x } × G Σ , x × { x } ≅ Aff + ( R ) \{x\}\times G_{\Sigma,x}\times\{x\}\cong\operatorname{Aff}^{+}(\mathbb{R}) { x } × G Σ , x × { x } ≅ Aff + ( R ) x x x D ∩ N 1 ∩ N 2 D\cap N_1\cap N_2 D ∩ N 1 ∩ N 2
the sets D j × Aff + ( R ) D_j\times\operatorname{Aff}^{+}(\mathbb{R}) D j × Aff + ( R ) D j ⊂ N 1 ∩ N 2 D_j\subset N_1\cap N_2 D j ⊂ N 1 ∩ N 2 D D D
An arrow ( l , l ~ ) (l,\widetilde l) ( l , l ) H \mathcal{H} H x 1 x_1 x 1 t ( l ) t(l) t ( l ) x 2 x_2 x 2 t ( l ~ ) t(\widetilde l) t ( l ) Proposition 4.12 and (33) , it preserves the component of N 1 ∖ D N_1\setminus D N 1 ∖ D N 1 ∩ D N_1\cap D N 1 ∩ D x 1 x_1 x 1 x 2 x_2 x 2 x 1 ∈ D x_1\in D x 1 ∈ D x 1 = x 2 x_1=x_2 x 1 = x 2 (33) , and condition (1) or (2) of Proposition 5.3 holds at x 1 x_1 x 1 D D D x 1 x_1 x 1 x 1 x_1 x 1 D ∩ N 1 ∩ N 2 D\cap N_1\cap N_2 D ∩ N 1 ∩ N 2 H 0 \mathcal{H}^{0} H 0 ( pr 1 , pr 2 ) (\operatorname{pr}_1,\operatorname{pr}_2) ( pr 1 , pr 2 ) Proposition 5.3 .
On a set of type (1), L 1 \mathcal{L}_1 L 1 L 2 \mathcal{L}_2 L 2 K 1 K_1 K 1 K 2 K_2 K 2 Proposition 4.12 ), so, by (43) , H \mathcal{H} H K 1 × K 2 K_1\times K_2 K 1 × K 2 Proposition 5.3 .
On a set of type (2), H \mathcal{H} H L 1 , x × L 2 , x \mathcal{L}_{1,x}\times\mathcal{L}_{2,x} L 1 , x × L 2 , x Aff + ( R ) \operatorname{Aff}^{+}(\mathbb{R}) Aff + ( R ) ( l , l ~ ) ⋅ g = l ~ g l − 1 (l,\widetilde l)\cdot g=\widetilde l g l^{-1} ( l , l ) ⋅ g = l g l − 1 ( a , b ) ↦ a (a,b)\mapsto a ( a , b ) ↦ a R > 0 \mathbb{R}_{>0} R > 0
L 1 , x ∩ g − 1 L 2 , x g = { 1 } , L 2 , x ( g L 1 , x g − 1 ) = Aff + ( R ) . \mathcal{L}_{1,x}\cap g^{-1}\mathcal{L}_{2,x}g=\{1\},\qquad\mathcal{L}_{2,x}(g\mathcal{L}_{1,x}g^{-1})=\operatorname{Aff}^{+}(\mathbb{R}). L 1 , x ∩ g − 1 L 2 , x g = { 1 } , L 2 , x ( g L 1 , x g − 1 ) = Aff + ( R ) . The first identity says that the stabilizer of g g g
L 2 , x g L 1 , x = L 2 , x ( g L 1 , x g − 1 ) g = Aff + ( R ) . \mathcal{L}_{2,x}g\mathcal{L}_{1,x}=\mathcal{L}_{2,x}(g\mathcal{L}_{1,x}g^{-1})g=\operatorname{Aff}^{+}(\mathbb{R}). L 2 , x g L 1 , x = L 2 , x ( g L 1 , x g − 1 ) g = Aff + ( R ) . The action is therefore free and transitive, so its action groupoid is Morita equivalent to a point. The form vanishes by Proposition 5.3 .
On a set of type (3), L 1 \mathcal{L}_1 L 1 L 2 \mathcal{L}_2 L 2 D j D_j D j l = ( 1 , u , t , 0 ) l=(1,u,t,0) l = ( 1 , u , t , 0 ) l ~ = ( 1 , u ~ , t , 0 ) \widetilde l=(1,\widetilde u,t,0) l = ( 1 , u , t , 0 ) (29) gives
( 1 , u ~ ) ( a , b ) ( 1 , u ) − 1 = ( a , b − u + a u ~ ) . (1,\widetilde u)(a,b)(1,u)^{-1}=(a,b-u+a\widetilde u). ( 1 , u ) ( a , b ) ( 1 , u ) − 1 = ( a , b − u + a u ) . The orbits are the lines { ( t , a ) } × R \{(t,a)\}\times\mathbb{R} {( t , a )} × R { u = a u ~ } ≅ R \{u=a\widetilde u\}\cong\mathbb{R} { u = a u } ≅ R { b = 0 } \{b=0\} { b = 0 } R \mathbb{R} R S ℓ j × R S^{\ell_j}\times\mathbb{R} S ℓ j × R ( t , α ) (t,\alpha) ( t , α ) ( t , a , u ~ , u ) ⟼ ( t , a , a u ~ − u ) (t,a,\widetilde u,u)\longmapsto(t,a,a\widetilde u-u) ( t , a , u , u ) ⟼ ( t , a , a u − u ) D j × Aff + ( R ) D_j\times\operatorname{Aff}^{+}(\mathbb{R}) D j × Aff + ( R ) Proposition 5.3 , − pr 0 ∗ Ω = − d t ∧ d α = d α ∧ d t -\operatorname{pr}_0^{*}\Omega=-dt\wedge d\alpha=d\alpha\wedge dt − pr 0 ∗ Ω = − d t ∧ d α = d α ∧ d t b b b d α ∧ d t d\alpha\wedge dt d α ∧ d t H 0 \mathcal{H}^{0} H 0 H \mathcal{H} H
The canonical function α \alpha α
ι − ∂ ∂ t ( d α ∧ d t ) = d α , X α = − ∂ ∂ t . \iota_{-\frac{\partial}{\partial t}}(d\alpha\wedge dt)=d\alpha,\qquad X_\alpha=-\frac{\partial}{\partial t}. ι − ∂ t ∂ ( d α ∧ d t ) = d α , X α = − ∂ t ∂ . By (27) , X α X_\alpha X α D j D_j D j S ℓ j 1 S^1_{\ell_j} S ℓ j 1 ℓ j \ell_j ℓ j ℓ > 0 \ell>0 ℓ > 0
( t , α ) ⟼ ( ℓ ℓ j t , ℓ j ℓ α ) (t,\alpha)\longmapsto\left(\frac{\ell}{\ell_j}t,\frac{\ell_j}{\ell}\alpha\right) ( t , α ) ⟼ ( ℓ j ℓ t , ℓ ℓ j α ) is a symplectomorphism onto S ℓ 1 × R S^1_\ell\times\mathbb{R} S ℓ 1 × R α \alpha α
For a clean self-intersection L × Σ h L L\times^h_{\Sigma}L L × Σ h L D ∩ N D\cap N D ∩ N N ∖ D N\setminus D N ∖ D Σ ∖ D \Sigma\setminus D Σ ∖ D D D D N N N
Example 5.6. Let S 2 ⊂ R 3 S^2\subset\mathbb{R}^3 S 2 ⊂ R 3 ( θ , z ) (\theta,z) ( θ , z ) θ ∈ R / 2 π Z \theta\in\mathbb{R}/2\pi\mathbb{Z} θ ∈ R /2 π Z d θ ∧ d z d\theta\wedge dz d θ ∧ d z ∂ ∂ θ ∧ ∂ ∂ z \frac{\partial}{\partial\theta}\wedge\frac{\partial}{\partial z} ∂ θ ∂ ∧ ∂ z ∂ S 2 S^2 S 2 σ = z ∂ ∂ z ∧ ∂ ∂ θ \sigma=z\frac{\partial}{\partial z}\wedge\frac{\partial}{\partial\theta} σ = z ∂ z ∂ ∧ ∂ θ ∂ D D D ( t , r ) = ( θ , z ) (t,r)=(\theta,z) ( t , r ) = ( θ , z ) (26) . So the modular period of D D D ℓ 1 = 2 π \ell_1=2\pi ℓ 1 = 2 π N 1 N_1 N 1 θ ∈ { θ 0 , θ 0 + π } \theta\in\{\theta_0,\theta_0+\pi\} θ ∈ { θ 0 , θ 0 + π } D D D p 1 , p 2 p_1,p_2 p 1 , p 2 N 2 ≠ D N_2\ne D N 2 = D p 1 p_1 p 1 p 2 p_2 p 2 N 2 = N 1 N_2=N_1 N 2 = N 1 p 1 p_1 p 1 L 1 × Σ h L 2 L_1\times^h_{\Sigma}L_2 L 1 × Σ h L 2 N 1 N_1 N 1 N 2 N_2 N 2 N 2 ≠ N 1 N_2\ne N_1 N 2 = N 1 D D D ( S 2 π 1 × R ) × R (S^1_{2\pi}\times\mathbb{R})\times\mathbb{R} ( S 2 π 1 × R ) × R d α ∧ d θ d\alpha\wedge d\theta d α ∧ d θ α \alpha α 2 π 2\pi 2 π N 1 N_1 N 1 D D D
Proof of Theorem C. This is Proposition 5.3. □
The homotopy Poisson algebra of a curve
In this section, we study the homotopy Poisson algebras of curves. Let N N N ( Σ , σ ) (\Sigma,\sigma) ( Σ , σ ) D D D ψ \psi ψ N N N ( a , λ ψ ) (\mathfrak a,\lambda^\psi) ( a , λ ψ ) N N N D D D L ∞ L_\infty L ∞ L ∞ L_\infty L ∞
The brackets in adapted coordinates
We first reduce to a single component of N N N
The normal bundle of N N N ∧ k ν N = 0 \wedge^k\nu_N=0 ∧ k ν N = 0 k ≥ 2 k\ge2 k ≥ 2 a \mathfrak a a 0 0 0 1 1 1 N N N N c N_c N c N N N a = ⨁ c a c \mathfrak a=\bigoplus_c\mathfrak a_c a = ⨁ c a c a c = C ∞ ( ∧ ∙ ν N c ) = a c 0 ⊕ a c 1 \mathfrak a_c=C^\infty(\wedge^\bullet\nu_{N_c})=\mathfrak a_c^0\oplus\mathfrak a_c^1 a c = C ∞ ( ∧ ∙ ν N c ) = a c 0 ⊕ a c 1 C ∞ ( ν N c ) C^\infty(\nu_{N_c}) C ∞ ( ν N c ) N N N N c N_c N c N N N Q ψ Q_\psi Q ψ a c \mathfrak a_c a c ( a , λ ψ ) (\mathfrak a,\lambda^\psi) ( a , λ ψ ) ( a c , λ ψ ) (\mathfrak a_c,\lambda^\psi) ( a c , λ ψ ) N c N_c N c λ n = λ n ψ \lambda_n=\lambda_n^\psi λ n = λ n ψ (21) on a c \mathfrak a_c a c
Each N c N_c N c ν N c \nu_{N_c} ν N c adapted chart near a point of N c N_c N c ( x , y ) (x,y) ( x , y ) Σ \Sigma Σ U U U N c ∩ U = { y = 0 } N_c\cap U=\{y=0\} N c ∩ U = { y = 0 } ( x , y e ( x ) ) ↦ ( x , y ) (x,ye(x))\mapsto(x,y) ( x , y e ( x )) ↦ ( x , y ) e e e ν N c \nu_{N_c} ν N c ∂ ∂ y \frac{\partial}{\partial y} ∂ y ∂ ψ \psi ψ x x x N c N_c N c e e e y y y e e e N N N
In the adapted chart determined by ψ \psi ψ
σ ψ = v ( x , y ) ∂ ∂ x ∧ ∂ ∂ y , v ( x , y ) = ∑ k ≥ 0 v k ( x ) y k , v k ( x ) = 1 k ! ∂ k v ∂ y k ( x , 0 ) . \sigma_{\psi}=v(x,y)\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y},\qquad v(x,y)=\sum_{k\geq0}v_k(x)y^k,
\qquad
v_k(x)=\frac{1}{k!}\frac{\partial^k v}{\partial y^k}(x,0). σ ψ = v ( x , y ) ∂ x ∂ ∧ ∂ y ∂ , v ( x , y ) = k ≥ 0 ∑ v k ( x ) y k , v k ( x ) = k ! 1 ∂ y k ∂ k v ( x , 0 ) . The term v k y k ∂ ∂ x ∧ ∂ ∂ y v_k y^k\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y} v k y k ∂ x ∂ ∧ ∂ y ∂ k − 1 k-1 k − 1 σ ψ \sigma_{\psi} σ ψ Definition 2.2 ). We identify functions on N c N_c N c ν N c \nu N_c ν N c y y y g e ge g e ν N c \nu N_c ν N c g ( x ) ∂ ∂ y g(x)\frac{\partial}{\partial y} g ( x ) ∂ y ∂
Proposition 6.1. Up to graded antisymmetry, the only possibly nonzero brackets of ( a c , λ ψ ) (\mathfrak a_c,\lambda^\psi) ( a c , λ ψ ) a c 0 \mathfrak a_c^0 a c 0 n ≥ 1 n\geq1 n ≥ 1 f , g 1 , … , g n − 1 ∈ C ∞ ( N c ) f,g_1,\ldots,g_{n-1}\in C^\infty(N_c) f , g 1 , … , g n − 1 ∈ C ∞ ( N c )
λ n ( f , g 1 e , … , g n − 1 e ) = ( n − 1 ) ! v n − 1 ∂ f ∂ x g 1 ⋯ g n − 1 e . (45) \lambda_n(f,g_1e,\ldots,g_{n-1}e)=(n-1)!\,v_{n-1}\frac{\partial f}{\partial x}g_1\cdots g_{n-1}e. \tag*{(45)} λ n ( f , g 1 e , … , g n − 1 e ) = ( n − 1 )! v n − 1 ∂ x ∂ f g 1 ⋯ g n − 1 e . (45) Proof. We use (22) in an adapted chart. A bracket λ n \lambda_n λ n j j j a c 0 \mathfrak a_c^0 a c 0 n − j n-j n − j a c 1 \mathfrak a_c^1 a c 1 ( n − j ) + ( 2 − n ) = 2 − j (n-j)+(2-n)=2-j ( n − j ) + ( 2 − n ) = 2 − j j = 0 j=0 j = 0 j ≥ 3 j\geq3 j ≥ 3 λ 0 ∈ a c 2 = 0 \lambda_0\in\mathfrak a_c^2=0 λ 0 ∈ a c 2 = 0
Let j ∈ { 1 , 2 } j\in\{1,2\} j ∈ { 1 , 2 } (4) for ( Q ψ ) n (Q_\psi)_n ( Q ψ ) n (21) , replacing an argument e e e g e ge g e g g g j + 1 j+1 j + 1 a c 0 \mathfrak a_c^0 a c 0 j = 2 j=2 j = 2 j = 3 j=3 j = 3 a c 1 \mathfrak a_c^1 a c 1 e e e f f f g g g y y y
{ f , g } = v ( ∂ f ∂ x ∂ g ∂ y − ∂ f ∂ y ∂ g ∂ x ) = 0. \begin{aligned}
\{f,g\}=v\left(\frac{\partial f}{\partial x}\frac{\partial g}{\partial y}-\frac{\partial f}{\partial y}\frac{\partial g}{\partial x}\right) \\
&=0.
\end{aligned} { f , g } = v ( ∂ x ∂ f ∂ y ∂ g − ∂ y ∂ f ∂ x ∂ g ) = 0. The first formula of (22) therefore gives λ n ( e , … , e , f , g ) = 0 \lambda_n(e,\ldots,e,f,g)=0 λ n ( e , … , e , f , g ) = 0 j = 1 j=1 j = 1 j = 2 j=2 j = 2 g i = 1 g_i=1 g i = 1 { f , y } = v ∂ f ∂ x \{f,y\}=v\frac{\partial f}{\partial x} { f , y } = v ∂ x ∂ f (22) gives
λ n ( e , … , e , f ) = ( − 1 ) n − 1 ∂ n − 1 ∂ y n − 1 ( v ∂ f ∂ x ) ∣ y = 0 e = ( − 1 ) n − 1 ( n − 1 ) ! v n − 1 ∂ f ∂ x e . \begin{aligned}
\lambda_n(e,\ldots,e,f)=(-1)^{n-1}\left.\frac{\partial^{n-1}}{\partial y^{n-1}}\left(v\frac{\partial f}{\partial x}\right)\right|_{y=0}e \\
&=(-1)^{n-1}(n-1)!\,v_{n-1}\frac{\partial f}{\partial x}e.
\end{aligned} λ n ( e , … , e , f ) = ( − 1 ) n − 1 ∂ y n − 1 ∂ n − 1 ( v ∂ x ∂ f ) y = 0 e = ( − 1 ) n − 1 ( n − 1 )! v n − 1 ∂ x ∂ f e . By graded antisymmetry, moving f f f n − 1 n-1 n − 1 e e e ( − 1 ) n − 1 (-1)^{n-1} ( − 1 ) n − 1 (45) .
Since f , g 1 , … , g n − 1 f,g_1,\ldots,g_{n-1} f , g 1 , … , g n − 1 (45) , λ n = 0 \lambda_n=0 λ n = 0 v n − 1 = 0 v_{n-1}=0 v n − 1 = 0 n ≥ 1 n\geq1 n ≥ 1 λ n = 0 \lambda_n=0 λ n = 0 n ≥ 2 n\geq2 n ≥ 2 v k = 0 v_k=0 v k = 0 k ≥ 1 k\geq1 k ≥ 1 ( a c , λ 1 ) (\mathfrak a_c,\lambda_1) ( a c , λ 1 ) λ 1 : C ∞ ( N c ) → C ∞ ( ν N c ) \lambda_1:C^\infty(N_c)\to C^\infty(\nu N_c) λ 1 : C ∞ ( N c ) → C ∞ ( ν N c )
λ 1 ( f ) = v 0 ∂ f ∂ x e . (46) \lambda_1(f)=v_0\frac{\partial f}{\partial x}e. \tag*{(46)} λ 1 ( f ) = v 0 ∂ x ∂ f e . (46) Corollary 6.2. One has λ 2 ∣ a c 0 ⊗ a c 0 = 0 \lambda_2|_{\mathfrak a_c^0\otimes\mathfrak a_c^0}=0 λ 2 ∣ a c 0 ⊗ a c 0 = 0 X X X N c N_c N c λ 2 ( f , v ) = X ( f ) v \lambda_2(f,v)=X(f)v λ 2 ( f , v ) = X ( f ) v f ∈ a c 0 f\in\mathfrak a_c^0 f ∈ a c 0 v ∈ a c 1 v\in\mathfrak a_c^1 v ∈ a c 1 X = v 1 ∂ ∂ x X=v_1\frac{\partial}{\partial x} X = v 1 ∂ x ∂
Proof. The first claim is part of Proposition 6.1. By (45) , λ 2 ( f , g e ) = v 1 ∂ f ∂ x g e \lambda_2(f,ge)=v_1\frac{\partial f}{\partial x}ge λ 2 ( f , g e ) = v 1 ∂ x ∂ f g e λ 2 ( f , v ) = X ( f ) v \lambda_2(f,v)=X(f)v λ 2 ( f , v ) = X ( f ) v X X X ν N c \nu N_c ν N c
Contact multiplicities
The unary bracket depends on the restriction of σ \sigma σ N c N_c N c λ 1 \lambda_1 λ 1
Keep the tubular neighbourhood, replace the coordinate x x x θ \theta θ x = x ( θ ) x=x(\theta) x = x ( θ ) e e e e ˉ = μ e \bar e=\mu e e ˉ = μ e μ \mu μ y = y ˉ y=\bar y y = y ˉ μ \mu μ x ( θ ) x(\theta) x ( θ ) ( θ , y ˉ ) (\theta,\bar y) ( θ , y ˉ )
∂ ∂ θ = x ′ ( θ ) ∂ ∂ x + y ˉ d ( μ ∘ x ) d θ ∂ ∂ y , ∂ ∂ y ˉ = μ ∂ ∂ y , ∂ ∂ θ ∧ ∂ ∂ y ˉ = x ′ ( θ ) μ ∂ ∂ x ∧ ∂ ∂ y . \frac{\partial}{\partial\theta}=x'(\theta)\frac{\partial}{\partial x}+\bar y\frac{d(\mu\circ x)}{d\theta}\frac{\partial}{\partial y},
\qquad
\frac{\partial}{\partial\bar y}=\mu\frac{\partial}{\partial y},
\qquad
\frac{\partial}{\partial\theta}\wedge\frac{\partial}{\partial\bar y}
=x'(\theta)\mu\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}. ∂ θ ∂ = x ′ ( θ ) ∂ x ∂ + y ˉ d θ d ( μ ∘ x ) ∂ y ∂ , ∂ y ˉ ∂ = μ ∂ y ∂ , ∂ θ ∂ ∧ ∂ y ˉ ∂ = x ′ ( θ ) μ ∂ x ∂ ∧ ∂ y ∂ . Hence v ˉ ( θ , y ˉ ) = v ( x ( θ ) , μ y ˉ ) / ( x ′ ( θ ) μ ) \bar{v}(\theta,\bar{y})=v(x(\theta),\mu\bar{y})/(x'(\theta)\mu) v ˉ ( θ , y ˉ ) = v ( x ( θ ) , μ y ˉ ) / ( x ′ ( θ ) μ ) y ˉ 0 \bar{y}^{0} y ˉ 0 y ˉ 1 \bar{y}^{1} y ˉ 1
v ˉ 0 ( θ ) = v 0 ( x ( θ ) ) x ′ ( θ ) μ ( x ( θ ) ) , (47) \bar{v}_{0}(\theta)=\frac{v_{0}(x(\theta))}{x'(\theta)\mu(x(\theta))}, \tag*{(47)} v ˉ 0 ( θ ) = x ′ ( θ ) μ ( x ( θ )) v 0 ( x ( θ )) , (47) v ˉ 1 ( θ ) = v 1 ( x ( θ ) ) x ′ ( θ ) , so that v ˉ 1 ∂ ∂ θ = v 1 ∂ ∂ x on N c . (48) \bar{v}_{1}(\theta)=\frac{v_{1}(x(\theta))}{x'(\theta)},\quad\text{so that}\quad\bar{v}_{1}\frac{\partial}{\partial\theta}=v_{1}\frac{\partial}{\partial x}\quad\text{on }N_{c}. \tag*{(48)} v ˉ 1 ( θ ) = x ′ ( θ ) v 1 ( x ( θ )) , so that v ˉ 1 ∂ θ ∂ = v 1 ∂ x ∂ on N c . (48) More intrinsically, there is a canonical identification
T N c ⊗ ν N c ⟶ ⋀ 2 T Σ ∣ N c , u ⊗ [ v ] ⟼ u ∧ v . TN_{c}\otimes\nu_{N_{c}}\longrightarrow\bigwedge^{2}T\Sigma|_{N_{c}},\quad u\otimes[v]\longmapsto u\wedge v. T N c ⊗ ν N c ⟶ ⋀ 2 T Σ ∣ N c , u ⊗ [ v ] ⟼ u ∧ v . It is well defined because T N c TN_{c} T N c T N c TN_{c} T N c ν N c \nu_{N_{c}} ν N c
v 0 ∂ ∂ x ⊗ e = σ ∣ N c . (49) v_{0}\frac{\partial}{\partial x}\otimes e=\sigma|_{N_{c}}. \tag*{(49)} v 0 ∂ x ∂ ⊗ e = σ ∣ N c . (49) So a change of tubular neighbourhood, coordinate or frame multiplies v 0 v_{0} v 0 (47) , and the zeros of v 0 v_{0} v 0 N c N_{c} N c ν N c \nu_{N_{c}} ν N c x ∈ R / Z x\in\mathbb{R}/\mathbb{Z} x ∈ R / Z N c N_{c} N c e e e v k v_{k} v k N c N_{c} N c
By (15) , two tubular neighbourhoods satisfy σ ψ ′ = φ ∗ σ ψ \sigma_{\psi'}=\varphi^{*}\sigma_{\psi} σ ψ ′ = φ ∗ σ ψ φ = φ ψ ψ ′ ∈ Diff 1 ( ν N c ) \varphi=\varphi_{\psi\psi'}\in\operatorname{Diff}_{1}(\nu_{N_{c}}) φ = φ ψ ψ ′ ∈ Diff 1 ( ν N c ) Lemma 2.4 , φ = e Z \varphi=e^{Z} φ = e Z Z ∈ X ≥ 1 ( ν N c ) Z\in\mathfrak{X}_{\geq1}(\nu_{N_{c}}) Z ∈ X ≥ 1 ( ν N c ) φ ∗ = e ad Z \varphi^{*}=e^{\operatorname{ad}Z} φ ∗ = e ad Z (13) . In an adapted chart, the component of weight m m m
Z ( m ) = y m κ ( x ) ∂ ∂ x + y m + 1 ρ ( x ) ∂ ∂ y , with κ , ρ ∈ C ∞ ( N c ) . Z_{(m)}=y^{m}\kappa(x)\frac{\partial}{\partial x}+y^{m+1}\rho(x)\frac{\partial}{\partial y},\quad\text{with }\kappa,\rho\in C^{\infty}(N_{c}). Z ( m ) = y m κ ( x ) ∂ x ∂ + y m + 1 ρ ( x ) ∂ y ∂ , with κ , ρ ∈ C ∞ ( N c ) . Lemma 6.3. Let m ≥ 1 m\geq1 m ≥ 1 π = v ∂ ∂ x ∧ ∂ ∂ y \pi=v\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y} π = v ∂ x ∂ ∧ ∂ y ∂ ≥ − 1 \geq-1 ≥ − 1 Z ∈ X ≥ m ( ν N c ) Z\in\mathfrak{X}_{\geq m}(\nu_{N_{c}}) Z ∈ X ≥ m ( ν N c ) Z ( m ) Z_{(m)} Z ( m ) v ~ \tilde{v} v ~ e ad Z π = v ~ ∂ ∂ x ∧ ∂ ∂ y e^{\operatorname{ad}Z}\pi=\tilde{v}\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y} e ad Z π = v ~ ∂ x ∂ ∧ ∂ y ∂
v ~ k = v k ( 0 ≤ k ≤ m − 1 ) , \tilde{v}_{k}=v_{k}\quad(0\leq k\leq m-1), v ~ k = v k ( 0 ≤ k ≤ m − 1 ) , v ~ m = v m + κ v 0 ′ − ( κ ′ + ( m + 1 ) ρ ) v 0 , (50) \tilde{v}_{m}=v_{m}+\kappa v_{0}'-\bigl(\kappa'+(m+1)\rho\bigr)v_{0}, \tag*{(50)} v ~ m = v m + κ v 0 ′ − ( κ ′ + ( m + 1 ) ρ ) v 0 , (50) where the prime denotes d d x \frac{d}{dx} d x d
Proof. Since π \pi π ≥ − 1 \geq-1 ≥ − 1 v 0 ∂ ∂ x ∧ ∂ ∂ y v_{0}\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y} v 0 ∂ x ∂ ∧ ∂ y ∂ − 1 -1 − 1 ad Z \operatorname{ad}Z ad Z m m m e ad Z π = π + [ Z , π ] + 1 2 [ Z , [ Z , π ] ] + ⋯ e^{\operatorname{ad}Z}\pi=\pi+[Z,\pi]+\frac{1}{2}[Z,[Z,\pi]]+\cdots e ad Z π = π + [ Z , π ] + 2 1 [ Z , [ Z , π ]] + ⋯ ≤ m − 2 \leq m-2 ≤ m − 2 π \pi π m − 1 m-1 m − 1 [ Z ( m ) , v 0 ∂ ∂ x ∧ ∂ ∂ y ] \left[Z_{(m)},v_{0}\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}\right] [ Z ( m ) , v 0 ∂ x ∂ ∧ ∂ y ∂ ] (10) ,
Z ( m ) ( v 0 ) = y m κ v 0 ′ , Z_{(m)}(v_{0})=y^{m}\kappa v_{0}', Z ( m ) ( v 0 ) = y m κ v 0 ′ , [ Z ( m ) , ∂ ∂ x ] = − y m κ ′ ∂ ∂ x − y m + 1 ρ ′ ∂ ∂ y , \left[Z_{(m)},\frac{\partial}{\partial x}\right]=-y^{m}\kappa'\frac{\partial}{\partial x}-y^{m+1}\rho'\frac{\partial}{\partial y}, [ Z ( m ) , ∂ x ∂ ] = − y m κ ′ ∂ x ∂ − y m + 1 ρ ′ ∂ y ∂ , [ Z ( m ) , ∂ ∂ y ] = − m y m − 1 κ ∂ ∂ x − ( m + 1 ) y m ρ ∂ ∂ y , \left[Z_{(m)},\frac{\partial}{\partial y}\right]=-my^{m-1}\kappa\frac{\partial}{\partial x}-(m+1)y^{m}\rho\frac{\partial}{\partial y}, [ Z ( m ) , ∂ y ∂ ] = − m y m − 1 κ ∂ x ∂ − ( m + 1 ) y m ρ ∂ y ∂ , so, by the Leibniz rule,
[ Z ( m ) , v 0 ∂ ∂ x ∧ ∂ ∂ y ] = Z ( m ) ( v 0 ) ∂ ∂ x ∧ ∂ ∂ y + v 0 [ Z ( m ) , ∂ ∂ x ] ∧ ∂ ∂ y + v 0 ∂ ∂ x ∧ [ Z ( m ) , ∂ ∂ y ] = y m ( κ v 0 ′ − κ ′ v 0 − ( m + 1 ) ρ v 0 ) ∂ ∂ x ∧ ∂ ∂ y . \begin{aligned}
\left[Z_{(m)},v_{0}\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}\right]
&=Z_{(m)}(v_{0})\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}
+v_{0}\left[Z_{(m)},\frac{\partial}{\partial x}\right]\wedge\frac{\partial}{\partial y}
+v_{0}\frac{\partial}{\partial x}\wedge\left[Z_{(m)},\frac{\partial}{\partial y}\right]\\
&=y^{m}\bigl(\kappa v_{0}'-\kappa'v_{0}-(m+1)\rho v_{0}\bigr)
\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}.
\end{aligned} [ Z ( m ) , v 0 ∂ x ∂ ∧ ∂ y ∂ ] = Z ( m ) ( v 0 ) ∂ x ∂ ∧ ∂ y ∂ + v 0 [ Z ( m ) , ∂ x ∂ ] ∧ ∂ y ∂ + v 0 ∂ x ∂ ∧ [ Z ( m ) , ∂ y ∂ ] = y m ( κ v 0 ′ − κ ′ v 0 − ( m + 1 ) ρ v 0 ) ∂ x ∂ ∧ ∂ y ∂ . This gives (50) . □
To interpret v 0 v_{0} v 0 v 1 v_{1} v 1
Lemma 6.4. (1) At a point q ∈ N c q\in N_{c} q ∈ N c v 0 ( q ) = 0 v_{0}(q)=0 v 0 ( q ) = 0 q ∈ D q\in D q ∈ D N c ⊂ Σ ∖ D N_{c}\subset\Sigma\setminus D N c ⊂ Σ ∖ D v 0 v_{0} v 0
(2) If N c = D j N_{c}=D_{j} N c = D j ( x , y ) = ( t , r ) (x,y)=(t,r) ( x , y ) = ( t , r ) (26) , v = − y v=-y v = − y v 0 ≡ 0 v_{0}\equiv0 v 0 ≡ 0 v 1 ≡ − 1 v_{1}\equiv-1 v 1 ≡ − 1 v k = 0 v_{k}=0 v k = 0 k ≥ 2 k\geq2 k ≥ 2
(3) At a transverse contact p ∈ N c ∩ D j p\in N_{c}\cap D_{j} p ∈ N c ∩ D j ( x , y ) = ( r , s ) (x,y)=(r,s) ( x , y ) = ( r , s ) Lemma 4.2 , v = x v=x v = x v 0 ( x ) = x v_{0}(x)=x v 0 ( x ) = x v k = 0 v_{k}=0 v k = 0 k ≥ 1 k\geq1 k ≥ 1
(4) At a tangential contact p ∈ N c ∩ D j p \in N_c \cap D_j p ∈ N c ∩ D j x = t x=t x = t y = r − h ( t ) y=r-h(t) y = r − h ( t ) Lemma 4.3 , υ = − ( y + h ( x ) ) \upsilon=-(y+h(x)) υ = − ( y + h ( x )) υ 0 = − h \upsilon_0=-h υ 0 = − h υ 1 ≡ − 1 \upsilon_1\equiv-1 υ 1 ≡ − 1 υ k = 0 \upsilon_k=0 υ k = 0 k ≥ 2 k\ge2 k ≥ 2
Proof. (1) For q ∈ N c q\in N_c q ∈ N c σ q = υ 0 ( q ) ∂ ∂ x ∧ ∂ ∂ y \sigma_q=\upsilon_0(q)\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y} σ q = υ 0 ( q ) ∂ x ∂ ∧ ∂ y ∂ σ q = 0 \sigma_q=0 σ q = 0 q ∈ D q\in D q ∈ D σ = r ∂ ∂ r ∧ ∂ ∂ t = − r ∂ ∂ t ∧ ∂ ∂ r \sigma=r\frac{\partial}{\partial r}\wedge\frac{\partial}{\partial t}=-r\frac{\partial}{\partial t}\wedge\frac{\partial}{\partial r} σ = r ∂ r ∂ ∧ ∂ t ∂ = − r ∂ t ∂ ∧ ∂ r ∂ σ = r ∂ ∂ r ∧ ∂ ∂ s \sigma=r\frac{\partial}{\partial r}\wedge\frac{\partial}{\partial s} σ = r ∂ r ∂ ∧ ∂ s ∂ N c ∩ W = { s = 0 } N_c\cap W=\{s=0\} N c ∩ W = { s = 0 } Lemma 4.2 . (4) D j ∩ W = { r = 0 } D_j\cap W=\{r=0\} D j ∩ W = { r = 0 } N c ∩ W = { r = h ( t ) } N_c\cap W=\{r=h(t)\} N c ∩ W = { r = h ( t )} Lemma 4.3 . With x = t x=t x = t y = r − h ( t ) y=r-h(t) y = r − h ( t )
∂ ∂ x = ∂ ∂ t , ∂ ∂ y = ∂ ∂ r − h ′ ( t ) ∂ ∂ t , ∂ ∂ x ∧ ∂ ∂ y = ∂ ∂ t ∧ ∂ ∂ r , σ = − r ∂ ∂ t ∧ ∂ ∂ r = − ( y + h ( x ) ) ∂ ∂ x ∧ ∂ ∂ y . \frac{\partial}{\partial x}=\frac{\partial}{\partial t},\qquad\frac{\partial}{\partial y}=\frac{\partial}{\partial r}-h'(t)\frac{\partial}{\partial t},\qquad\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}=\frac{\partial}{\partial t}\wedge\frac{\partial}{\partial r},\qquad\sigma=-r\frac{\partial}{\partial t}\wedge\frac{\partial}{\partial r}=-(y+h(x))\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}. ∂ x ∂ = ∂ t ∂ , ∂ y ∂ = ∂ r ∂ − h ′ ( t ) ∂ t ∂ , ∂ x ∂ ∧ ∂ y ∂ = ∂ t ∂ ∧ ∂ r ∂ , σ = − r ∂ t ∂ ∧ ∂ r ∂ = − ( y + h ( x )) ∂ x ∂ ∧ ∂ y ∂ . Definition 6.5. Let N c ⊄ D N_c\not\subset D N c ⊂ D N N N p ∈ N c ∩ D p\in N_c\cap D p ∈ N c ∩ D m p ∈ { 1 , 2 , … } ∪ { ∞ } m_p\in\{1,2,\ldots\}\cup\{\infty\} m p ∈ { 1 , 2 , … } ∪ { ∞ } N c N_c N c p p p p p p r ∣ N c r|_{N_c} r ∣ N c r r r D D D p p p m p = ∞ m_p=\infty m p = ∞ r ∣ N c r|_{N_c} r ∣ N c p p p
The multiplicity m p m_p m p r r r
Proposition 6.6. Let N c N_c N c N N N N c ⊄ D N_c\not\subset D N c ⊂ D
(1) For every tubular neighbourhood, coordinate and frame, υ 0 \upsilon_0 υ 0 p ∈ N c ∩ D p\in N_c\cap D p ∈ N c ∩ D m p m_p m p m p = 1 m_p=1 m p = 1 p p p
(2) H 0 ( a c , λ 1 ) = R H^0(\mathfrak{a}_c,\lambda_1)=\mathbb{R} H 0 ( a c , λ 1 ) = R
im λ 1 = { υ 0 u e : u ∈ C ∞ ( N c ) , ∫ N c u d x = 0 } . \operatorname{im}\lambda_1=\left\{\upsilon_0u\,e:u\in C^\infty(N_c),\ \int_{N_c}u\,dx=0\right\}. im λ 1 = { υ 0 u e : u ∈ C ∞ ( N c ) , ∫ N c u d x = 0 } . (3) If every m p m_p m p ∑ p m p \sum_p m_p ∑ p m p
dim H 1 ( a c , λ 1 ) = 1 + ∑ p ∈ N c ∩ D m p . \dim H^1(\mathfrak{a}_c,\lambda_1)=1+\sum_{p\in N_c\cap D}m_p. dim H 1 ( a c , λ 1 ) = 1 + p ∈ N c ∩ D ∑ m p . (4) If some m p m_p m p H 1 ( a c , λ 1 ) H^1(\mathfrak{a}_c,\lambda_1) H 1 ( a c , λ 1 )
Proof. (1) Near p ∈ D j p\in D_j p ∈ D j σ = r ∂ ∂ r ∧ ∂ ∂ t \sigma=r\frac{\partial}{\partial r}\wedge\frac{\partial}{\partial t} σ = r ∂ r ∂ ∧ ∂ t ∂ (49) , υ 0 \upsilon_0 υ 0 r ∣ N c r|_{N_c} r ∣ N c r ∣ N c r|_{N_c} r ∣ N c p p p d r ∣ T p N c ≠ 0 dr|_{T_pN_c}\ne0 d r ∣ T p N c = 0 T p N c ≠ T p D j T_pN_c\ne T_pD_j T p N c = T p D j
(2) By (46) , λ 1 ( f ) = υ 0 f ′ e \lambda_1(f)=\upsilon_0f'e λ 1 ( f ) = υ 0 f ′ e υ 0 \upsilon_0 υ 0 N c ∩ D N_c\cap D N c ∩ D λ 1 ( f ) = 0 \lambda_1(f)=0 λ 1 ( f ) = 0 f ′ = 0 f'=0 f ′ = 0 f f f u = f ′ u=f' u = f ′ f ∈ C ∞ ( N c ) f\in C^\infty(N_c) f ∈ C ∞ ( N c ) ∫ u d x = 0 \int u\,dx=0 ∫ u d x = 0
(3) As in (2), u ↦ υ 0 u u\mapsto\upsilon_0u u ↦ υ 0 u C ∞ ( N c ) C^\infty(N_c) C ∞ ( N c ) I m ⊂ C ∞ ( N c ) I_{\mathbf m}\subset C^\infty(N_c) I m ⊂ C ∞ ( N c ) m = ( m p ) p \mathbf m=(m_p)_p m = ( m p ) p m p − 1 m_p-1 m p − 1 p ∈ N c ∩ D p\in N_c\cap D p ∈ N c ∩ D υ 0 C ∞ ( N c ) ⊂ I m \upsilon_0C^\infty(N_c)\subset I_{\mathbf m} υ 0 C ∞ ( N c ) ⊂ I m g ∈ I m g\in I_{\mathbf m} g ∈ I m υ 0 = ( x − x ( p ) ) m p w p \upsilon_0=(x-x(p))^{m_p}w_p υ 0 = ( x − x ( p ) ) m p w p g = ( x − x ( p ) ) m p g p g=(x-x(p))^{m_p}g_p g = ( x − x ( p ) ) m p g p p p p w p w_p w p w p ( p ) ≠ 0 w_p(p)\ne0 w p ( p ) = 0 g / υ 0 g/\upsilon_0 g / υ 0 N c ∩ D N_c\cap D N c ∩ D g p / w p g_p/w_p g p / w p p p p g ∈ υ 0 C ∞ ( N c ) g\in\upsilon_0C^\infty(N_c) g ∈ υ 0 C ∞ ( N c ) x − x ( p ) x-x(p) x − x ( p ) C ∞ ( N c ) C^\infty(N_c) C ∞ ( N c ) m p − 1 m_p-1 m p − 1 p p p I m I_{\mathbf m} I m ∑ p m p \sum_p m_p ∑ p m p im λ 1 \operatorname{im}\lambda_1 im λ 1 { ∫ u d x = 0 } \{\int u\,dx=0\} { ∫ u d x = 0 } u ↦ υ 0 u e u\mapsto\upsilon_0u\,e u ↦ υ 0 u e 1 1 1 I m e I_{\mathbf m}e I m e 1 + ∑ p m p 1+\sum_p m_p 1 + ∑ p m p a c \mathfrak{a}_c a c υ 0 \upsilon_0 υ 0 p p p m p m_p m p N c N_c N c ∑ p m p \sum_p m_p ∑ p m p
(4) If m p = ∞ m_p=\infty m p = ∞ im λ 1 \operatorname{im}\lambda_1 im λ 1 p p p H 1 ( a c , λ 1 ) H^1(\mathfrak{a}_c,\lambda_1) H 1 ( a c , λ 1 ) p p p
For transverse contacts, the dimension of H 1 ( a c , λ 1 ) H^1(\mathfrak{a}_c,\lambda_1) H 1 ( a c , λ 1 ) b b b [15 Theorem 27] (see also [16 Proposition 3.19] ). For contacts of equal order, it agrees with the cohomology of b k b^k b k [31 Proposition 4.3] .
Example 6.7. On the sphere of Example 5.6, let g : R / 2 π Z → [ 0 , 1 ] g:\mathbb{R}/2\pi\mathbb{Z}\to[0,1] g : R /2 π Z → [ 0 , 1 ] g ( θ ) = exp ( − 1 / θ 2 ) g(\theta)=\exp(-1/\theta^2) g ( θ ) = exp ( − 1/ θ 2 ) 0 < ∣ θ ∣ ≤ 1 2 0<|\theta|\le\frac{1}{2} 0 < ∣ θ ∣ ≤ 2 1 g ( 0 ) = 0 g(0)=0 g ( 0 ) = 0 g > 0 g>0 g > 0 N = { z = g ( θ ) } N=\{z=g(\theta)\} N = { z = g ( θ )} D D D p = ( 0 , 0 ) p=(0,0) p = ( 0 , 0 ) m p = ∞ m_p=\infty m p = ∞ H 1 ( a c , λ 1 ) H^1(\mathfrak{a}_c,\lambda_1) H 1 ( a c , λ 1 ) Proposition 6.6 (4).
Remark 6.8. By [9 Theorem 4] , for a Hausdorff source-connected Lie groupoid whose source fibres are homologically n n n ≤ n \le n ≤ n n + 1 n+1 n + 1 L \mathcal{L} L Proposition 4.12 . Every arrow of L \mathcal{L} L N N N L ∣ N c = s − 1 ( N c ) \mathcal{L}|_{N_c}=s^{-1}(N_c) L ∣ N c = s − 1 ( N c ) L \mathcal{L} L L \mathcal{L} L G Σ \mathcal{G}_{\Sigma} G Σ T N c ∘ TN_c^\circ T N c ∘ ( a c , λ 1 ) (\mathfrak{a}_c,\lambda_1) ( a c , λ 1 ) N c N_c N c D D D Proposition 4.12 they are arcs of N c ∖ D N_c\setminus D N c ∖ D R > 0 \mathbb{R}_{>0} R > 0 R \mathbb{R} R N c ⊄ D N_c\not\subset D N c ⊂ D R \mathbb{R} R N c = D j N_c=D_j N c = D j H d i f f ∙ ( L ∣ N c ) ≅ H ∙ ( a c , λ 1 ) H^\bullet_{\mathrm{diff}}(\mathcal{L}|_{N_c})\cong H^\bullet(\mathfrak{a}_c,\lambda_1) H diff ∙ ( L ∣ N c ) ≅ H ∙ ( a c , λ 1 ) N c ⊄ D N_c\not\subset D N c ⊂ D Proposition 6.6 , dim H d i f f 1 ( L ∣ N c ) = 1 + ∑ p m p \dim H^1_{\mathrm{diff}}(\mathcal{L}|_{N_c})=1+\sum_p m_p dim H diff 1 ( L ∣ N c ) = 1 + ∑ p m p m p m_p m p H d i f f 1 ( L ∣ N c ) H^1_{\mathrm{diff}}(\mathcal{L}|_{N_c}) H diff 1 ( L ∣ N c ) N c ⊂ Σ ∖ D N_c\subset\Sigma\setminus D N c ⊂ Σ ∖ D L ∣ N c = N c × N c \mathcal{L}|_{N_c}=N_c\times N_c L ∣ N c = N c × N c H d i f f 1 ( L ∣ N c ) = 0 H^1_{\mathrm{diff}}(\mathcal{L}|_{N_c})=0 H diff 1 ( L ∣ N c ) = 0 [9 Proposition 1] , while H 1 ( a c , λ 1 ) = R H^1(\mathfrak{a}_c,\lambda_1)=\mathbb{R} H 1 ( a c , λ 1 ) = R Proposition 6.6 (3). This is consistent with [9 Theorem 4] , which only gives injectivity in degree 1 here, since the source fibres N c N_c N c
A normal form
We now construct a tubular neighbourhood in which the formal expansion σ ψ \sigma_\psi σ ψ (45) , this eliminates all brackets of arity at least three. When all contacts are transverse, the same construction also eliminates the binary bracket. The normal form concerns the formal expansion along N c N_c N c
Throughout this subsection, we fix a tubular neighbourhood ψ 0 \psi_0 ψ 0 N c N_c N c x ∈ R / Z x\in\mathbb{R}/\mathbb{Z} x ∈ R / Z e e e ν N c \nu_{N_c} ν N c N c × R N_c\times\mathbb{R} N c × R ψ 0 \psi_0 ψ 0 N c × ( − ε , ε ) N_c\times(-\varepsilon,\varepsilon) N c × ( − ε , ε ) ( x , y ) (x,y) ( x , y ) N c N_c N c ∑ k f k ( x ) y k \sum_k f_k(x)y^k ∑ k f k ( x ) y k f k ∈ C ∞ ( N c ) f_k\in C^\infty(N_c) f k ∈ C ∞ ( N c ) x = x ( θ ) x=x(\theta) x = x ( θ ) e ˉ = μ e \bar e=\mu e e ˉ = μ e v ( x ( θ ) , μ y ˉ ) / ( x ′ ( θ ) μ ) v(x(\theta),\mu\bar y)/(x'(\theta)\mu) v ( x ( θ ) , μ y ˉ ) / ( x ′ ( θ ) μ ) v v v y y y
Lemma 6.9. (1) For every φ ∈ Diff 1 ( ν N c ) \varphi\in\operatorname{Diff}_1(\nu_{N_c}) φ ∈ Diff 1 ( ν N c ) ψ \psi ψ N c N_c N c φ ψ 0 ψ = φ \varphi\psi_0\psi=\varphi φ ψ 0 ψ = φ
(2) Let p 1 , … , p s ∈ N c p_1,\ldots,p_s\in N_c p 1 , … , p s ∈ N c ψ i \psi_i ψ i J i ∋ p i J_i\ni p_i J i ∋ p i N c N_c N c ψ \psi ψ N c N_c N c ψ i \psi_i ψ i p i p_i p i i i i
Proof. Let a , b a,b a , b N c × ( − ε , ε ) N_c\times(-\varepsilon,\varepsilon) N c × ( − ε , ε ) a ( x , 0 ) = b ( x , 0 ) = ∂ b ∂ y ( x , 0 ) = 0 a(x,0)=b(x,0)=\frac{\partial b}{\partial y}(x,0)=0 a ( x , 0 ) = b ( x , 0 ) = ∂ y ∂ b ( x , 0 ) = 0 γ ( x , y ) = ( x + a ( x , y ) , y + b ( x , y ) ) \gamma(x,y)=(x+a(x,y),y+b(x,y)) γ ( x , y ) = ( x + a ( x , y ) , y + b ( x , y )) x + a x+a x + a Z \mathbb{Z} Z γ \gamma γ ( x ˙ , y ˙ ) ↦ ( x ˙ + ∂ a ∂ y ( x , 0 ) y ˙ , y ˙ ) (\dot x,\dot y)\mapsto(\dot x+\frac{\partial a}{\partial y}(x,0)\dot y,\dot y) ( x ˙ , y ˙ ) ↦ ( x ˙ + ∂ y ∂ a ( x , 0 ) y ˙ , y ˙ ) γ \gamma γ ν N c \nu_{N_c} ν N c u k ≠ u k ′ u_k\ne u'_k u k = u k ′ γ ( u k ) = γ ( u k ′ ) \gamma(u_k)=\gamma(u'_k) γ ( u k ) = γ ( u k ′ ) u u u u ′ ∈ N c u'\in N_c u ′ ∈ N c u = γ ( u ) = γ ( u ′ ) = u ′ u=\gamma(u)=\gamma(u')=u' u = γ ( u ) = γ ( u ′ ) = u ′ u u u ψ 0 ∘ γ \psi_0\circ\gamma ψ 0 ∘ γ φ ψ 0 , ψ 0 ∘ γ \varphi_{\psi_0,\psi_0\circ\gamma} φ ψ 0 , ψ 0 ∘ γ
(1) Since φ ∈ Diff 1 ( ν N c ) \varphi\in\operatorname{Diff}_1(\nu_{N_c}) φ ∈ Diff 1 ( ν N c ) φ ∗ ( x ) = x + ∑ k ≥ 1 a k ( x ) y k \varphi^*(x)=x+\sum_{k\geq1}a_k(x)y^k φ ∗ ( x ) = x + ∑ k ≥ 1 a k ( x ) y k φ ∗ ( y ) = y + ∑ k ≥ 2 b k ( x ) y k \varphi^*(y)=y+\sum_{k\geq2}b_k(x)y^k φ ∗ ( y ) = y + ∑ k ≥ 2 b k ( x ) y k φ ∗ ( x ) \varphi^*(x) φ ∗ ( x ) R / Z \mathbb{R}/\mathbb{Z} R / Z N c N_c N c a a a b b b y y y φ γ ∈ Diff 1 ( ν N c ) \varphi_\gamma\in\operatorname{Diff}_1(\nu_{N_c}) φ γ ∈ Diff 1 ( ν N c ) γ \gamma γ φ γ − 1 = φ γ ∘ φ γ − 1 \varphi_\gamma^{-1}=\varphi_\gamma\circ\varphi_\gamma^{-1} φ γ − 1 = φ γ ∘ φ γ − 1 φ γ − 1 \varphi_\gamma^{-1} φ γ − 1 x x x y y y φ γ ∗ − i d \varphi_\gamma^*-id φ γ ∗ − i d log φ γ ∗ = 0 \log\varphi_\gamma^*=0 log φ γ ∗ = 0 Lemma 2.4 (2), φ γ = φ \varphi_\gamma=\varphi φ γ = φ ψ = ψ 0 ∘ γ \psi=\psi_0\circ\gamma ψ = ψ 0 ∘ γ
(2) Since ψ i \psi_i ψ i ψ 0 \psi_0 ψ 0 N c N_c N c ν N c \nu_{N_c} ν N c J i ′ ∋ p i J_i'\ni p_i J i ′ ∋ p i J i J_i J i ψ 0 − 1 ∘ ψ i ( x , y ) = ( x + a i ( x , y ) , y + b i ( x , y ) ) \psi_0^{-1}\circ\psi_i(x,y)=(x+a_i(x,y),y+b_i(x,y)) ψ 0 − 1 ∘ ψ i ( x , y ) = ( x + a i ( x , y ) , y + b i ( x , y )) J i ′ × { 0 } J_i'\times\{0\} J i ′ × { 0 } a i ( x , 0 ) = b i ( x , 0 ) = ∂ b i ∂ y ( x , 0 ) = 0 a_i(x,0)=b_i(x,0)=\frac{\partial b_i}{\partial y}(x,0)=0 a i ( x , 0 ) = b i ( x , 0 ) = ∂ y ∂ b i ( x , 0 ) = 0 β i ∈ C ∞ ( N c ) \beta_i\in C^\infty(N_c) β i ∈ C ∞ ( N c ) J i ′ J_i' J i ′ p i p_i p i a = ∑ i β i a i a=\sum_i\beta_i a_i a = ∑ i β i a i b = ∑ i β i b i b=\sum_i\beta_i b_i b = ∑ i β i b i ε \varepsilon ε ψ = ψ 0 ∘ γ \psi=\psi_0\circ\gamma ψ = ψ 0 ∘ γ ψ i \psi_i ψ i p i p_i p i
The construction has two steps. First, we choose a tubular neighbourhood that is already affine near the tangential contacts. We then remove the remaining coefficients one at a time without changing the neighbourhood formally near those contacts. The following lemma provides the second step.
Lemma 6.10. Let N c ⊄ D N_c\not\subset D N c ⊂ D A ⊂ N c A\subset N_c A ⊂ N c N c N_c N c π = v ∂ ∂ x ∧ ∂ ∂ y \pi=v\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y} π = v ∂ x ∂ ∧ ∂ y ∂ y 0 y^0 y 0 v 0 v_0 v 0 σ ψ 0 \sigma_{\psi_0} σ ψ 0 m ≥ 1 m\geq1 m ≥ 1 g ∈ C ∞ ( N c ) g\in C^\infty(N_c) g ∈ C ∞ ( N c ) A A A Z ∈ Ξ ≥ m ( ν N c ) Z\in\Xi_{\geq m}(\nu_{N_c}) Z ∈ Ξ ≥ m ( ν N c ) m m m A A A v ~ \tilde v v ~ e ad Z π e^{\operatorname{ad}_Z}\pi e ad Z π v ~ k = v k \tilde v_k=v_k v ~ k = v k k < m k<m k < m v ~ m = v m − g \tilde v_m=v_m-g v ~ m = v m − g v ~ = v \tilde v=v v ~ = v A A A
Proof. By Proposition 6.6 (1), v 0 ′ ( q ) ≠ 0 v_0'(q)\ne0 v 0 ′ ( q ) = 0 q q q B B B v 0 ′ ≠ 0 v_0'\ne0 v 0 ′ = 0 A A A O = N c ∖ D O=N_c\setminus D O = N c ∖ D A A A B B B O O O N c N_c N c χ A + χ B + χ O = 1 \chi_A+\chi_B+\chi_O=1 χ A + χ B + χ O = 1
Z = y m κ ∂ ∂ x + y m + 1 ρ ∂ ∂ y , κ = − χ B g / v 0 ′ , ρ = 1 m + 1 ( χ O g / v 0 − κ ′ ) . Z=y^m\kappa\frac{\partial}{\partial x}+y^{m+1}\rho\frac{\partial}{\partial y},\qquad\kappa=-\chi_Bg/v_0',\qquad\rho=\frac{1}{m+1}(\chi_Og/v_0-\kappa'). Z = y m κ ∂ x ∂ + y m + 1 ρ ∂ y ∂ , κ = − χ B g / v 0 ′ , ρ = m + 1 1 ( χ O g / v 0 − κ ′ ) . These functions are smooth, since v 0 ′ ≠ 0 v'_0 \ne0 v 0 ′ = 0 B B B v 0 ≠ 0 v_0 \ne0 v 0 = 0 O O O A A A χ B = g = 0 \chi_B=g=0 χ B = g = 0 κ \kappa κ κ ′ \kappa' κ ′ ρ \rho ρ χ A g = 0 \chi_Ag=0 χ A g = 0
κ v 0 ′ − ( κ ′ + ( m + 1 ) ρ ) v 0 = − χ B g − χ O g = − g , \kappa v'_0-(\kappa'+(m+1)\rho)v_0=-\chi_Bg-\chi_Og=-g, κ v 0 ′ − ( κ ′ + ( m + 1 ) ρ ) v 0 = − χ B g − χ O g = − g , and Lemma 6.3 gives v ~ k = v k \widetilde{v}_k=v_k v k = v k k < m k<m k < m v ~ m = v m − g \widetilde{v}_m=v_m-g v m = v m − g ad Z \operatorname{ad}_Z ad Z Z Z Z A A A e ad Z π = π e^{\operatorname{ad}_Z}\pi=\pi e ad Z π = π A A A
Proposition 6.11
Proposition 6.11. Let x x x N c N_c N c e e e ν N c \nu_{N_c} ν N c
There is a tubular neighbourhood ψ \psi ψ N c N_c N c σ ψ = ( v 0 + v 1 y ) ∂ ∂ x ∧ ∂ ∂ y \sigma_\psi=(v_0+v_1y)\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y} σ ψ = ( v 0 + v 1 y ) ∂ x ∂ ∧ ∂ y ∂ ψ \psi ψ λ n ψ = 0 \lambda^\psi_n=0 λ n ψ = 0 n ≥ 3 n\geq3 n ≥ 3 ( a c , λ ψ ) (a_c,\lambda^\psi) ( a c , λ ψ )
If N c ⊄ D N_c\not\subset D N c ⊂ D U U U N c N_c N c ψ \psi ψ v 1 = 0 v_1=0 v 1 = 0 U U U N c N_c N c D D D σ ψ = v 0 ∂ ∂ x ∧ ∂ ∂ y \sigma_\psi=v_0\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y} σ ψ = v 0 ∂ x ∂ ∧ ∂ y ∂ λ n ψ = 0 \lambda^\psi_n=0 λ n ψ = 0 n ≥ 2 n\geq2 n ≥ 2
Proof. It suffices to construct the stated formal expansion, since the claims about the brackets then follow from (45) . If N c = D j N_c=D_j N c = D j ( t , r ) (t,r) ( t , r ) (26) gives a tubular neighbourhood with v = − y v=-y v = − y t t t ∂ ∂ r \frac{\partial}{\partial r} ∂ r ∂ x x x e e e
Suppose henceforth that N c ⊄ D N_c\not\subset D N c ⊂ D p 1 , … , p s p_1,\ldots,p_s p 1 , … , p s
We first choose an affine model near the tangential contacts. By the beginning of this subsection, we may use x x x e e e Lemma 6.9 (2), we may assume that ψ 0 \psi_0 ψ 0 p i p_i p i ψ i \psi_i ψ i v v v y y y U U U A ⊂ U A\subset U A ⊂ U p i p_i p i ψ 0 = ψ i \psi_0=\psi_i ψ 0 = ψ i β ∈ C ∞ ( N c ) \beta\in C^\infty(N_c) β ∈ C ∞ ( N c ) U U U β ∣ A = 1 \beta|_A=1 β ∣ A = 1 v k = 0 v_k=0 v k = 0 A A A k ≥ 2 k\geq2 k ≥ 2 ψ 0 = ψ i \psi_0=\psi_i ψ 0 = ψ i s = 0 s=0 s = 0 A = ∅ A=\varnothing A = ∅ β = 0 \beta=0 β = 0
We then remove the remaining coefficients successively. Let π ( 0 ) = σ ψ 0 \pi^{(0)}=\sigma_{\psi_0} π ( 0 ) = σ ψ 0 Lemma 6.10 with m = 1 m=1 m = 1 g = ( 1 − β ) v 1 g=(1-\beta)v_1 g = ( 1 − β ) v 1 Z 1 Z_1 Z 1 A A A π ( 1 ) = e ad Z 1 π ( 0 ) \pi^{(1)}=e^{\operatorname{ad}_{Z_1}}\pi^{(0)} π ( 1 ) = e ad Z 1 π ( 0 ) v 0 v_0 v 0 β v 1 \beta v_1 β v 1 y 0 y^0 y 0 y 1 y^1 y 1 π ( 0 ) \pi^{(0)} π ( 0 ) A A A m ≥ 2 m\geq2 m ≥ 2 Lemma 6.10 with g g g y m y^m y m π ( m − 1 ) \pi^{(m-1)} π ( m − 1 ) A A A Z m Z_m Z m m m m A A A π ( m ) = e ad Z m π ( m − 1 ) \pi^{(m)}=e^{\operatorname{ad}_{Z_m}}\pi^{(m-1)} π ( m ) = e ad Z m π ( m − 1 ) π ( m − 1 ) \pi^{(m-1)} π ( m − 1 ) y 0 , … , y m − 1 y^0,\ldots,y^{m-1} y 0 , … , y m − 1 y m y^m y m π ( m − 1 ) \pi^{(m-1)} π ( m − 1 ) A A A
Finally, we take the formal limit and realize it smoothly. Set φ M = e Z 1 ∘ ⋯ ∘ e Z M \varphi_M=e^{Z_1}\circ\cdots\circ e^{Z_M} φ M = e Z 1 ∘ ⋯ ∘ e Z M φ M ∗ σ ψ 0 = π ( M ) \varphi_M^*\sigma_{\psi_0}=\pi^{(M)} φ M ∗ σ ψ 0 = π ( M ) (13) . Since Z M + 1 Z_{M+1} Z M + 1 O ^ \widehat{O} O I M + 1 I^{M+1} I M + 1 φ M + 1 − φ M = ( e Z M + 1 − id ) ∘ φ M \varphi_{M+1}-\varphi_M=(e^{Z_{M+1}}-\operatorname{id})\circ\varphi_M φ M + 1 − φ M = ( e Z M + 1 − id ) ∘ φ M φ M \varphi_M φ M φ ∗ \varphi^* φ ∗ O ^ \widehat{O} O φ ∗ − id \varphi^*-\operatorname{id} φ ∗ − id φ M ∗ − id \varphi_M^*-\operatorname{id} φ M ∗ − id I w I^w I w I w + 1 I^{w+1} I w + 1 φ ∗ \varphi^* φ ∗ ∑ k ≥ 0 ( id − φ ∗ ) k \sum_{k\geq0}(\operatorname{id}-\varphi^*)^k ∑ k ≥ 0 ( id − φ ∗ ) k φ ∈ Diff 1 ( ν N c ) \varphi\in\operatorname{Diff}_1(\nu_{N_c}) φ ∈ Diff 1 ( ν N c ) e Z M ′ + 1 ∘ ⋯ ∘ e Z M ′ e^{Z_{M'+1}}\circ\cdots\circ e^{Z_{M'}} e Z M ′ + 1 ∘ ⋯ ∘ e Z M ′ M ′ → ∞ M'\to\infty M ′ → ∞ φ M ∈ Diff 1 ( ν N c ) \varphi_M\in\operatorname{Diff}_1(\nu_{N_c}) φ M ∈ Diff 1 ( ν N c ) φ = φ M ∘ φ M ′ \varphi=\varphi_M\circ\varphi'_{M} φ = φ M ∘ φ M ′ φ M ′ ∗ − id \varphi'^*_M-\operatorname{id} φ M ′ ∗ − id log φ M ′ \log\varphi'_M log φ M ′ I w I^w I w I w + 1 I^{w+1} I w + 1 log φ M ′ ∈ X M + 1 ( ν N c ) \log\varphi'_M\in\mathfrak{X}_{M+1}(\nu_{N_c}) log φ M ′ ∈ X M + 1 ( ν N c ) f ∂ ∂ x f\frac{\partial}{\partial x} f ∂ x ∂ f y k + 1 ∂ ∂ y f y^{k+1}\frac{\partial}{\partial y} f y k + 1 ∂ y ∂ k k k (13) , φ ∗ σ ψ 0 = e ad log φ M ′ π ( M ) \varphi^*\sigma_{\psi_0}=e^{\operatorname{ad}_{\log\varphi'_M}}\pi^{(M)} φ ∗ σ ψ 0 = e ad l o g φ M ′ π ( M ) π ( M ) \pi^{(M)} π ( M ) ≥ M \geq M ≥ M y k y^k y k k ≥ M + 1 k\geq M+1 k ≥ M + 1 M M M φ ∗ σ ψ 0 = ( v 0 + β v 1 y ) ∂ ∂ x ∧ ∂ ∂ y \varphi^*\sigma_{\psi_0}=(v_0+\beta v_1y)\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y} φ ∗ σ ψ 0 = ( v 0 + β v 1 y ) ∂ x ∂ ∧ ∂ y ∂ Lemma 6.9 (1), φ = ψ ψ 0 − 1 \varphi=\psi\psi_0^{-1} φ = ψ ψ 0 − 1 ψ \psi ψ σ ψ = φ ∗ σ ψ 0 \sigma_\psi=\varphi^*\sigma_{\psi_0} σ ψ = φ ∗ σ ψ 0 (15) . This proves (1) and (2), with U = ∅ U=\varnothing U = ∅
For transverse contacts, the last claim also follows from the b b b [12 Theorem 2.13 and Proposition 2.15] . Indeed, N c N_c N c b b b [12 Definition 2.2] , whose model is the b b b ( N c , N c ∩ D ) (N_c,N_c\cap D) ( N c , N c ∩ D ) [12 Example 1.17] , up to a fibrewise linear change of coordinates which makes the neighbourhood induce the identity on ν N c \nu_{N_c} ν N c
Proposition 6.12
Proposition 6.12. Let N c ⊄ D N_c\not\subset D N c ⊂ D D D D ∣ N c ∩ D ∣ = 2 k |N_c\cap D|=2k ∣ N c ∩ D ∣ = 2 k k ≥ 0 k\geq0 k ≥ 0 N c N_c N c θ ∈ R / 2 π Z \theta\in\mathbb{R}/2\pi\mathbb{Z} θ ∈ R /2 π Z N c N_c N c e e e ν N c \nu_{N_c} ν N c λ n = 0 \lambda_n=0 λ n = 0 n ≥ 2 n\geq2 n ≥ 2
λ 1 ( f ) = { ∂ f ∂ θ e , k = 0 , sin ( k θ ) ∂ f ∂ θ e , k ≥ 1. (51) \lambda_1(f)=
\begin{cases}
\frac{\partial f}{\partial\theta}e, & k=0,\\
\sin(k\theta)\frac{\partial f}{\partial\theta}e, & k\geq1.
\end{cases} \tag*{(51)} λ 1 ( f ) = { ∂ θ ∂ f e , sin ( k θ ) ∂ θ ∂ f e , k = 0 , k ≥ 1. (51) Proof. By Proposition 6.6 (1), m p = 1 m_p=1 m p = 1 p ∈ N c ∩ D p\in N_c\cap D p ∈ N c ∩ D ∣ N c ∩ D ∣ = ∑ p m p |N_c\cap D|=\sum_p m_p ∣ N c ∩ D ∣ = ∑ p m p Proposition 6.6 (3). By (49) , v 0 v_0 v 0 k = 0 k=0 k = 0 v 0 v_0 v 0
Lemma 6.4(1). Choose a diffeomorphism θ ↦ x ( θ ) \theta\mapsto x(\theta) θ ↦ x ( θ ) R / 2 π Z \mathbb{R}/2\pi\mathbb{Z} R /2 π Z N c N_c N c μ ( x ( θ ) ) = υ 0 ( x ( θ ) ) x ′ ( θ ) \mu(x(\theta)) = \frac{\upsilon_0(x(\theta))}{x'(\theta)} μ ( x ( θ )) = x ′ ( θ ) υ 0 ( x ( θ )) υ ‾ 0 ≡ 1 \overline{\upsilon}_0 \equiv1 υ 0 ≡ 1 (47) .
If k ≥ 1 k \ge1 k ≥ 1 N c ∩ D = { p 1 , … , p 2 k } N_c \cap D = \{p_1,\ldots,p_{2k}\} N c ∩ D = { p 1 , … , p 2 k } x ( i π / k ) = x ( p i ) x(i\pi/k) = x(p_i) x ( iπ / k ) = x ( p i ) Proposition 6.6 (1), υ 0 ∘ x \upsilon_0 \circ x υ 0 ∘ x sin ( k θ ) \sin(k\theta) sin ( k θ ) i π / k i\pi/k iπ / k μ ( x ( θ ) ) = υ 0 ( x ( θ ) ) x ′ ( θ ) sin ( k θ ) \mu(x(\theta)) = \frac{\upsilon_0(x(\theta))}{x'(\theta)\sin(k\theta)} μ ( x ( θ )) = x ′ ( θ ) s i n ( k θ ) υ 0 ( x ( θ )) υ ‾ 0 ( θ ) = sin ( k θ ) \overline{\upsilon}_0(\theta) = \sin(k\theta) υ 0 ( θ ) = sin ( k θ ) (47) .
By Proposition 6.11(2) for θ \theta θ e ‾ \overline{e} e σ ψ = υ ‾ 0 ∂ ∂ θ ∧ ∂ ∂ y ~ \sigma_\psi= \overline{\upsilon}_0 \frac{\partial}{\partial\theta} \wedge\frac{\partial}{\partial\widetilde{y}} σ ψ = υ 0 ∂ θ ∂ ∧ ∂ y ∂ ψ \psi ψ (45) gives the brackets.
Example 6.13. On the sphere of Example 5.6, let N = { z = z 0 } N = \{z = z_0\} N = { z = z 0 } 0 < ∣ z 0 ∣ < 1 0 < |z_0| < 1 0 < ∣ z 0 ∣ < 1 ( x , y ) = ( θ , z − z 0 ) (x,y) = (\theta,z-z_0) ( x , y ) = ( θ , z − z 0 ) ( x , y ~ ) (x,\widetilde{y}) ( x , y ) y ~ = z 0 log ( 1 + y / z 0 ) \widetilde{y} = z_0 \log(1+y/z_0) y = z 0 log ( 1 + y / z 0 ) ∣ y ∣ < ∣ z 0 ∣ |y| < |z_0| ∣ y ∣ < ∣ z 0 ∣ ∂ y ~ ∂ y = z 0 z 0 + y \frac{\partial\widetilde{y}}{\partial y} = \frac{z_0}{z_0+y} ∂ y ∂ y = z 0 + y z 0 1 1 1 N N N N N N
σ = − ( z 0 + y ) ∂ ∂ x ∧ ∂ ∂ y = − z 0 ∂ ∂ x ∧ ∂ ∂ y ~ . \sigma= -(z_0+y)\frac{\partial}{\partial x} \wedge\frac{\partial}{\partial y} = -z_0\frac{\partial}{\partial x} \wedge\frac{\partial}{\partial\widetilde{y}}. σ = − ( z 0 + y ) ∂ x ∂ ∧ ∂ y ∂ = − z 0 ∂ x ∂ ∧ ∂ y ∂ . In the first chart υ 1 = − 1 \upsilon_1=-1 υ 1 = − 1 λ 2 ≠ 0 \lambda_2 \ne0 λ 2 = 0 λ n \lambda_n λ n n ≥ 2 n \ge2 n ≥ 2
Tangential contacts and the modular vector field
The preceding construction removes the binary bracket when all contacts are transverse. We now identify the obstruction at a tangential contact. The vector field X X X Corollary 6.2 agrees there with the modular vector field. Higher contact multiplicity forces agreement of higher jets, and along a component of D D D
Proposition 6.14. Suppose that N c N_c N c p ∈ D j p \in D_j p ∈ D j N c = D j N_c = D_j N c = D j p ∈ D j p \in D_j p ∈ D j υ 1 ( p ) ∂ ∂ x ∣ p \upsilon_1(p)\frac{\partial}{\partial x}\big|_p υ 1 ( p ) ∂ x ∂ p X v o l X_{\mathrm{vol}} X vol (27) ,
υ 1 ( p ) ∂ ∂ x ∣ p = X v o l ( p ) ≠ 0. (52) \upsilon_1(p)\frac{\partial}{\partial x}\bigg|_p = X_{\mathrm{vol}}(p) \ne0. \tag*{(52)} υ 1 ( p ) ∂ x ∂ p = X vol ( p ) = 0. (52) In particular, if N c = D j N_c = D_j N c = D j X X X Corollary 6.2 is X v o l ∣ D j X_{\mathrm{vol}}|_{D_j} X vol ∣ D j λ 2 ≠ 0 \lambda_2 \ne0 λ 2 = 0 N c N_c N c
Proof. By Lemma 6.4(2) and (4) and (49) , υ 0 ( p ) = υ 0 ′ ( p ) = 0 \upsilon_0(p) = \upsilon'_0(p) = 0 υ 0 ( p ) = υ 0 ′ ( p ) = 0 υ 1 ∂ ∂ x \upsilon_1\frac{\partial}{\partial x} υ 1 ∂ x ∂ (48) . Under a change of tubular neighbourhood, possibly only defined near p p p Lemma 6.3 with m = 1 m=1 m = 1 N c N_c N c
υ ~ 1 ( p ) = υ 1 ( p ) + κ ( p ) υ 0 ′ ( p ) − ( κ ′ ( p ) + 2 ρ ( p ) ) υ 0 ( p ) = υ 1 ( p ) . \begin{aligned}
\widetilde{\upsilon}_1(p) = \upsilon_1(p) + \kappa(p)\upsilon'_0(p) - (\kappa'(p)+2\rho(p))\upsilon_0(p) \\
&= \upsilon_1(p).
\end{aligned} υ 1 ( p ) = υ 1 ( p ) + κ ( p ) υ 0 ′ ( p ) − ( κ ′ ( p ) + 2 ρ ( p )) υ 0 ( p ) = υ 1 ( p ) . Since every change of data is a composite of the two, υ 1 ( p ) ∂ ∂ x ∣ p \upsilon_1(p)\frac{\partial}{\partial x}\big|_p υ 1 ( p ) ∂ x ∂ p υ 1 ( p ) = − 1 \upsilon_1(p)=-1 υ 1 ( p ) = − 1 ∂ ∂ x ∣ p = ∂ ∂ t ∣ p \frac{\partial}{\partial x}\big|_p = \frac{\partial}{\partial t}\big|_p ∂ x ∂ p = ∂ t ∂ p ∂ ∂ x = ∂ ∂ t \frac{\partial}{\partial x} = \frac{\partial}{\partial t} ∂ x ∂ = ∂ t ∂ N c = D j N_c=D_j N c = D j ∂ ∂ x = ∂ ∂ t + h ′ ( t ) ∂ ∂ r \frac{\partial}{\partial x} = \frac{\partial}{\partial t}+h'(t)\frac{\partial}{\partial r} ∂ x ∂ = ∂ t ∂ + h ′ ( t ) ∂ r ∂ h ′ ( 0 ) = 0 h'(0)=0 h ′ ( 0 ) = 0 υ 1 ( p ) ∂ ∂ x ∣ p = − ∂ ∂ t ∣ p = X v o l ( p ) ≠ 0 \upsilon_1(p)\frac{\partial}{\partial x}\big|_p = -\frac{\partial}{\partial t}\big|_p = X_{\mathrm{vol}}(p) \ne0 υ 1 ( p ) ∂ x ∂ p = − ∂ t ∂ p = X vol ( p ) = 0 (27) . If N c = D j N_c=D_j N c = D j p ∈ D j p \in D_j p ∈ D j X = υ 1 ∂ ∂ x = X v o l ∣ D j X=\upsilon_1\frac{\partial}{\partial x}=X_{\mathrm{vol}}|_{D_j} X = υ 1 ∂ x ∂ = X vol ∣ D j λ 2 ( f , v ) = X ( f ) v \lambda_2(f,v)=X(f)v λ 2 ( f , v ) = X ( f ) v X = υ 1 ∂ ∂ x X=\upsilon_1\frac{\partial}{\partial x} X = υ 1 ∂ x ∂ Corollary 6.2 , and X ( p ) ≠ 0 X(p)\ne0 X ( p ) = 0
A section of T Σ ∣ N c T\Sigma|_{N_c} T Σ ∣ N c k k k p p p x x x N c N_c N c p p p T Σ T\Sigma T Σ O ( x k ) O(x^k) O ( x k ) k k k
Proposition 6.15. Let p p p N c N_c N c D j D_j D j m p ∈ { 2 , 3 , … } ∪ { ∞ } m_p \in\{2,3,\ldots\}\cup\{\infty\} m p ∈ { 2 , 3 , … } ∪ { ∞ } vol \operatorname{vol} vol p p p X X X Corollary 6.2 .
If m p < ∞ m_p<\infty m p < ∞ N c N_c N c X − X v o l ∣ N c X-X_{\mathrm{vol}}|_{N_c} X − X vol ∣ N c T Σ ∣ N c T\Sigma|_{N_c} T Σ ∣ N c m p − 1 m_p-1 m p − 1 p p p m p − 2 m_p-2 m p − 2 X X X p p p m p = ∞ m_p=\infty m p = ∞ p p p X X X p p p
If m p < ∞ m_p<\infty m p < ∞ m p − 1 m_p-1 m p − 1 X X X p p p
Proof. (1) In the chart ( t , r ) (t,r) ( t , r ) (26) near p = ( 0 , 0 ) p=(0,0) p = ( 0 , 0 ) Lemma 4.3 gives N c = { r = h ( t ) } N_c=\{r=h(t)\} N c = { r = h ( t )} m p m_p m p h h h 0 0 0 m p m_p m p m p = ∞ m_p=\infty m p = ∞ h h h 0 0 0 v o l 0 = d r ∧ d t \mathrm{vol}_0=dr\wedge dt vol 0 = d r ∧ d t X v o l 0 = − ∂ ∂ t X_{\mathrm{vol}_0}=-\frac{\partial}{\partial t} X vol 0 = − ∂ t ∂ p p p (27) . Changing the volume form adds to the modular vector field a Hamiltonian vector field σ ( d u ) = r ( ∂ u ∂ r ∂ ∂ t − ∂ u ∂ t ∂ ∂ r ) \sigma(du)=r\left(\frac{\partial u}{\partial r}\frac{\partial}{\partial t}-\frac{\partial u}{\partial t}\frac{\partial}{\partial r}\right) σ ( d u ) = r ( ∂ r ∂ u ∂ t ∂ − ∂ t ∂ u ∂ r ∂ ) X v o l = − ∂ ∂ t + r Y X_{\mathrm{vol}}=-\frac{\partial}{\partial t}+rY X vol = − ∂ t ∂ + r Y Y Y Y
Let ψ p \psi_p ψ p x = t x=t x = t y = r − h ( t ) y=r-h(t) y = r − h ( t ) ∂ ∂ y \frac{\partial}{\partial y} ∂ y ∂ υ 0 = − h \upsilon_0=-h υ 0 = − h υ 1 ≡ − 1 \upsilon_1\equiv-1 υ 1 ≡ − 1 ∂ ∂ x = ∂ ∂ t + h ′ ( t ) ∂ ∂ r \frac{\partial}{\partial x}=\frac{\partial}{\partial t}+h'(t)\frac{\partial}{\partial r} ∂ x ∂ = ∂ t ∂ + h ′ ( t ) ∂ r ∂ N c N_c N c (48) , we may compute X X X ψ \psi ψ p p p ψ p \psi_p ψ p ψ \psi ψ e Z e^Z e Z Z ∈ X ≥ 1 Z\in\mathfrak{X}_{\geq1} Z ∈ X ≥ 1 Lemma 2.4 , and Lemma 6.3 with m = 1 m=1 m = 1
υ 1 ψ = − 1 + κ υ 0 ′ − ( κ ′ + 2 ρ ) υ 0 = − 1 − κ h ′ + ( κ ′ + 2 ρ ) h . \begin{aligned}
\upsilon_1^\psi=-1+\kappa\upsilon_0'-\left(\kappa'+2\rho\right)\upsilon_0 \\
&=-1-\kappa h'+\left(\kappa'+2\rho\right)h.
\end{aligned} υ 1 ψ = − 1 + κ υ 0 ′ − ( κ ′ + 2 ρ ) υ 0 = − 1 − κ h ′ + ( κ ′ + 2 ρ ) h . Suppose first that m p < ∞ m_p<\infty m p < ∞ υ 1 ψ = − 1 + O ( t m p − 1 ) \upsilon_1^\psi=-1+O(t^{m_p-1}) υ 1 ψ = − 1 + O ( t m p − 1 ) N c N_c N c
X − X v o l = υ 1 ψ ∂ ∂ x + ∂ ∂ t − h Y = − h ′ ( t ) ∂ ∂ r + O ( t m p − 1 ) − h ( t ) Y = O ( t m p − 1 ) . X-X_{\mathrm{vol}}=\upsilon_1^\psi\frac{\partial}{\partial x}+\frac{\partial}{\partial t}-hY=-h'(t)\frac{\partial}{\partial r}+O(t^{m_p-1})-h(t)Y=O(t^{m_p-1}). X − X vol = υ 1 ψ ∂ x ∂ + ∂ t ∂ − hY = − h ′ ( t ) ∂ r ∂ + O ( t m p − 1 ) − h ( t ) Y = O ( t m p − 1 ) . Here we used h ′ ( t ) = O ( t m p − 1 ) h'(t)=O(t^{m_p-1}) h ′ ( t ) = O ( t m p − 1 ) h ( t ) = O ( t m p ) h(t)=O(t^{m_p}) h ( t ) = O ( t m p ) X v o l ∣ N c X_{\mathrm{vol}}|_{N_c} X vol ∣ N c ψ \psi ψ X X X N c N_c N c X v o l X_{\mathrm{vol}} X vol ∂ ∂ r \frac{\partial}{\partial r} ∂ r ∂ ( ∂ ∂ x , ∂ ∂ r ) \left(\frac{\partial}{\partial x},\frac{\partial}{\partial r}\right) ( ∂ x ∂ , ∂ r ∂ ) h ′ + O ( t m p ) h'+O(t^{m_p}) h ′ + O ( t m p ) m p − 1 m_p-1 m p − 1
If m p = ∞ m_p=\infty m p = ∞ h h h h ′ h' h ′ υ 1 ψ \upsilon_1^\psi υ 1 ψ υ 1 ψ + 1 \upsilon_1^\psi+1 υ 1 ψ + 1 ∂ ∂ x = ∂ ∂ t + h ′ ∂ ∂ r \frac{\partial}{\partial x}=\frac{\partial}{\partial t}+h'\frac{\partial}{\partial r} ∂ x ∂ = ∂ t ∂ + h ′ ∂ r ∂ N c N_c N c X v o l = − ∂ ∂ t + h Y X_{\mathrm{vol}}=-\frac{\partial}{\partial t}+hY X vol = − ∂ t ∂ + hY X − X v o l ∣ N c X-X_{\mathrm{vol}}|_{N_c} X − X vol ∣ N c
(2) Fix ψ \psi ψ x x x N c N_c N c x ( p ) = 0 x(p)=0 x ( p ) = 0 ν N c \nu_{N_c} ν N c N c × ( − ε , ε ) N_c\times(-\varepsilon,\varepsilon) N c × ( − ε , ε ) κ ∈ C ∞ ( N c ) \kappa\in C^\infty(N_c) κ ∈ C ∞ ( N c ) κ ( p ) ≠ 0 \kappa(p)\ne0 κ ( p ) = 0 γ ( x , y ) = ( x + y κ ( x ) , y ) \gamma(x,y)=(x+y\kappa(x),y) γ ( x , y ) = ( x + y κ ( x ) , y ) Lemma 6.9 , ψ ′ = ψ ∘ γ \psi'=\psi\circ\gamma ψ ′ = ψ ∘ γ φ ψ ψ ′ \varphi_{\psi\psi'} φ ψ ψ ′ γ \gamma γ Z = log φ ψ ψ ′ Z=\log\varphi_{\psi\psi'} Z = log φ ψ ψ ′ φ ψ ψ ′ ∗ ( y ) = y \varphi_{\psi\psi'}^*(y)=y φ ψ ψ ′ ∗ ( y ) = y ( φ ψ ψ ′ − id ) k (\varphi_{\psi\psi'}-\operatorname{id})^k ( φ ψ ψ ′ − id ) k O ^ \widehat{\mathcal{O}} O I k \mathcal{I}^k I k Z ( y ) = 0 Z(y)=0 Z ( y ) = 0 Z ( x ) = y κ Z(x)=y\kappa Z ( x ) = y κ I 2 \mathcal{I}^2 I 2
Z ( 1 ) = y κ ∂ ∂ x , Z_{(1)}=y\kappa\frac{\partial}{\partial x}, Z ( 1 ) = y κ ∂ x ∂ , and, by Lemma 6.3 , the coefficients of σ ψ \sigma_\psi σ ψ σ ψ ′ \sigma_{\psi'} σ ψ ′
υ ~ 1 − υ 1 = κ υ 0 ′ − κ ′ υ 0 . \widetilde{\upsilon}_1-\upsilon_1=\kappa\upsilon_0'-\kappa'\upsilon_0. υ 1 − υ 1 = κ υ 0 ′ − κ ′ υ 0 . By Proposition 6.6 (1), υ 0 = x m p w \upsilon_0=x^{m_p}w υ 0 = x m p w p p p w ( 0 ) ≠ 0 w(0)\ne0 w ( 0 ) = 0
υ ~ 1 − υ 1 = m p κ ( 0 ) w ( 0 ) x m p − 1 + O ( x m p ) , m p κ ( 0 ) w ( 0 ) ≠ 0. \widetilde{\upsilon}_1-\upsilon_1=m_p\kappa(0)w(0)x^{m_p-1}+O(x^{m_p}),\qquad m_p\kappa(0)w(0)\ne0. υ 1 − υ 1 = m p κ ( 0 ) w ( 0 ) x m p − 1 + O ( x m p ) , m p κ ( 0 ) w ( 0 ) = 0. Thus the jets of order m p − 1 m_p-1 m p − 1 υ 1 ∂ ∂ x \upsilon_1\frac{\partial}{\partial x} υ 1 ∂ x ∂ υ ~ 1 ∂ ∂ x \widetilde{\upsilon}_1\frac{\partial}{\partial x} υ 1 ∂ x ∂ p p p
L ∞ L_\infty L ∞ We now classify the underlying L ∞ L_\infty L ∞ ( a c , λ ψ ) (\mathfrak{a}_c,\lambda^\psi) ( a c , λ ψ ) D D D
M ( N c ) = ∑ p ∈ N c ∩ D m p ∈ { 0 , 1 , 2 , … } ∪ { ∞ } , M(N_c)=\sum_{p\in N_c\cap D}m_p\in\{0,1,2,\ldots\}\cup\{\infty\}, M ( N c ) = p ∈ N c ∩ D ∑ m p ∈ { 0 , 1 , 2 , … } ∪ { ∞ } , where the empty sum is 0 0 0 ∞ \infty ∞ m p m_p m p Theorem A to determine which isomorphisms are induced by changes of tubular neighbourhood.
Throughout this classification, L ∞ L_\infty L ∞ L ∞ L_\infty L ∞ L ∞ L_\infty L ∞ Q Q Q W W W minimal if Q 1 = 0 Q_1=0 Q 1 = 0 ( C , d C ) (C,d_C) ( C , d C ) L ∞ L_\infty L ∞ C C C d C d_C d C ( a c , λ 1 ) (\mathfrak{a}_c,\lambda_1) ( a c , λ 1 ) P ∞ P_\infty P ∞ λ 1 \lambda_1 λ 1 Q l i n Q^{\mathrm{lin}} Q lin L ∞ L_\infty L ∞ a c [ 1 ] \mathfrak{a}_c[1] a c [ 1 ] ( Q a c ) 1 (Q_{\mathfrak{a}_c})_1 ( Q a c ) 1
By the decomposition theorem of Kontsevich [17 §4.5.1, Lemma] , every flat L ∞ L_\infty L ∞ ( V , Q ) (V,Q) ( V , Q ) L ∞ L_\infty L ∞ ( W , Q min ) ⊕ ( C , d C ) (W,Q^{\min})\oplus(C,d_C) ( W , Q m i n ) ⊕ ( C , d C ) ( W , Q min ) (W,Q^{\min}) ( W , Q m i n ) ( C , d C ) (C,d_C) ( C , d C )
Proposition 6.16. Let ψ \psi ψ N c N_c N c Q = Q ψ Q=Q_\psi Q = Q ψ V = a c [ 1 ] V=\mathfrak{a}_c[1] V = a c [ 1 ] N c = D j N_c=D_j N c = D j Q 2 ′ ≠ 0 Q'_2\ne0 Q 2 ′ = 0 L ∞ L_\infty L ∞ ( V ′ , Q ′ ) (V',Q') ( V ′ , Q ′ ) L ∞ L_\infty L ∞ ( V , Q ) (V,Q) ( V , Q ) N c ⊄ D N_c\not\subset D N c ⊂ D L ∞ L_\infty L ∞ F : ( a c , λ ψ ) → ( a c , λ 1 ) F:(\mathfrak{a}_c,\lambda^\psi)\to(\mathfrak{a}_c,\lambda_1) F : ( a c , λ ψ ) → ( a c , λ 1 )
(3) Suppose that N c ⊄ D N_c \not\subset D N c ⊂ D N c ′ ′ ⊄ D ′ N'_{c'} \not\subset D' N c ′ ′ ⊂ D ′ ( Σ ′ , σ ′ ) (\Sigma',\sigma') ( Σ ′ , σ ′ ) ψ ′ \psi' ψ ′ a c ′ ′ = C ∞ ( ∧ ∙ ν N c ′ ′ ) \mathfrak{a}'_{c'}=C^\infty(\wedge^\bullet\nu_{N'_{c'}}) a c ′ ′ = C ∞ ( ∧ ∙ ν N c ′ ′ ) ( a c , λ ψ ) (\mathfrak{a}_c,\lambda^\psi) ( a c , λ ψ ) ( a c ′ ′ , λ ψ ′ ) (\mathfrak{a}'_{c'},\lambda^{\psi'}) ( a c ′ ′ , λ ψ ′ ) L ∞ L_\infty L ∞ ∑ p m p = ∑ p ′ m p ′ \sum_p m_p=\sum_{p'}m_{p'} ∑ p m p = ∑ p ′ m p ′ { 0 , 1 , 2 , … } ∪ { ∞ } \{0,1,2,\ldots\}\cup\{\infty\} { 0 , 1 , 2 , … } ∪ { ∞ }
(4) If N c = D j N_c=D_j N c = D j N c ′ ′ = D j ′ ′ N'_{c'}=D'_{j'} N c ′ ′ = D j ′ ′ ψ \psi ψ ψ ′ \psi' ψ ′ ( a c , λ ψ ) (\mathfrak{a}_c,\lambda^\psi) ( a c , λ ψ ) ( a c ′ ′ , λ ψ ′ ) (\mathfrak{a}'_{c'},\lambda^{\psi'}) ( a c ′ ′ , λ ψ ′ ) L ∞ L_\infty L ∞
Proof. (1) By Lemma 6.4(2) and (49) , ν 0 ≡ 0 \nu_0\equiv0 ν 0 ≡ 0 Q 1 = 0 Q_1=0 Q 1 = 0 Q 2 ≠ 0 Q_2\ne0 Q 2 = 0 Proposition 6.14 . For an L ∞ L_\infty L ∞ F : ( V , Q ) → ( V ′ , Q ′ ) F:(V,Q)\to(V',Q') F : ( V , Q ) → ( V ′ , Q ′ ) F F F F − 1 F^{-1} F − 1 ( F − 1 ) 1 = F 1 − 1 (F^{-1})_1=F_1^{-1} ( F − 1 ) 1 = F 1 − 1 (1) and (2) , the components of length one of F ∘ Q = Q ′ ∘ F F\circ Q=Q'\circ F F ∘ Q = Q ′ ∘ F V V V S 2 V S^2V S 2 V
F 1 ∘ Q 1 = Q 1 ′ ∘ F 1 , F 1 ∘ Q 2 + F 2 ∘ Q 1 ( 2 ) = Q 1 ′ ∘ F 2 + Q 2 ′ ∘ ( F 1 ⋅ F 1 ) . F_1\circ Q_1=Q'_1\circ F_1,\qquad F_1\circ Q_2+F_2\circ Q_1^{(2)}=Q'_1\circ F_2+Q'_2\circ(F_1\cdot F_1). F 1 ∘ Q 1 = Q 1 ′ ∘ F 1 , F 1 ∘ Q 2 + F 2 ∘ Q 1 ( 2 ) = Q 1 ′ ∘ F 2 + Q 2 ′ ∘ ( F 1 ⋅ F 1 ) . where Q 1 ( 2 ) ( a 1 a 2 ) = Q 1 ( a 1 ) a 2 + ( − 1 ) ∣ a 1 ∣ a 1 Q 1 ( a 2 ) Q_1^{(2)}(a_1a_2)=Q_1(a_1)a_2+(-1)^{|a_1|}a_1Q_1(a_2) Q 1 ( 2 ) ( a 1 a 2 ) = Q 1 ( a 1 ) a 2 + ( − 1 ) ∣ a 1 ∣ a 1 Q 1 ( a 2 ) ( F 1 ⋅ F 1 ) ( a 1 a 2 ) = F 1 ( a 1 ) F 1 ( a 2 ) (F_1\cdot F_1)(a_1a_2)=F_1(a_1)F_1(a_2) ( F 1 ⋅ F 1 ) ( a 1 a 2 ) = F 1 ( a 1 ) F 1 ( a 2 ) Q 1 ′ = 0 Q'_1=0 Q 1 ′ = 0 Q 2 ′ = F 1 ∘ Q 2 ∘ ( F 1 ⋅ F 1 ) − 1 ≠ 0 Q'_2=F_1\circ Q_2\circ(F_1\cdot F_1)^{-1}\ne0 Q 2 ′ = F 1 ∘ Q 2 ∘ ( F 1 ⋅ F 1 ) − 1 = 0
(2) We first show that the minimal summand in the decomposition theorem has zero brackets. We then compose with a linear isomorphism to obtain a linearization with identity linear part.
By Proposition 6.6 (2), H 0 ( a c , λ 1 ) = R ⋅ 1 H^0(\mathfrak{a}_c,\lambda_1)=\mathbb{R}\cdot1 H 0 ( a c , λ 1 ) = R ⋅ 1 V = V H ⊕ V B ⊕ V ′ V=V_H\oplus V_B\oplus V' V = V H ⊕ V B ⊕ V ′ V B = im Q 1 V_B=\operatorname{im}Q_1 V B = im Q 1 ker Q 1 = V H ⊕ V B \ker Q_1=V_H\oplus V_B ker Q 1 = V H ⊕ V B 1 ∈ V H 1\in V_H 1 ∈ V H V H ⊂ R ⋅ 1 ⊕ a c V_H\subset\mathbb{R}\cdot1\oplus\mathfrak{a}_c V H ⊂ R ⋅ 1 ⊕ a c Proposition 6.1 , Q n Q_n Q n n ≥ 2 n\ge2 n ≥ 2 1 ⋅ 1 = 0 1\cdot1=0 1 ⋅ 1 = 0 S 2 V S^2V S 2 V 1 1 1 Q Q Q S ( V H ) S(V_H) S ( V H ) i : ( V H , 0 ) → ( V , Q ) i:(V_H,0)\to(V,Q) i : ( V H , 0 ) → ( V , Q ) G : ( V , Q ) → ( W , Q min ) ⊕ ( C , d C ) G:(V,Q)\to(W,Q^{\min})\oplus(C,d_C) G : ( V , Q ) → ( W , Q m i n ) ⊕ ( C , d C ) q q q W W W G ′ = q ∘ G ∘ i : ( V H , 0 ) → ( W , Q min ) G'=q\circ G\circ i:(V_H,0)\to(W,Q^{\min}) G ′ = q ∘ G ∘ i : ( V H , 0 ) → ( W , Q m i n ) G 1 ′ G'_1 G 1 ′ G 1 ′ G'_1 G 1 ′ Q min = G 1 ′ ∘ 0 ∘ G 1 ′ − 1 = 0 Q^{\min}=G'_1\circ0\circ G_1^{'-1}=0 Q m i n = G 1 ′ ∘ 0 ∘ G 1 ′ − 1 = 0 G 1 : ( V , Q 1 ) → ( W ⊕ C , 0 ⊕ d C ) G_1:(V,Q_1)\to(W\oplus C,0\oplus d_C) G 1 : ( V , Q 1 ) → ( W ⊕ C , 0 ⊕ d C )
F=S(G_1^{-1})\circ G:(V,Q)\longrightarrow(V,Q^{\lin})
is an L ∞ L_\infty L ∞ F 1 = id F_1=\operatorname{id} F 1 = id S ( G 1 − 1 ) S(G_1^{-1}) S ( G 1 − 1 ) G 1 − 1 G_1^{-1} G 1 − 1
(3) As in (1), the linear part of an L ∞ L_\infty L ∞ Proposition 6.6 (3) and (4). Conversely, if they agree, the spaces H 1 H^1 H 1 Proposition 6.6 (3) in the finite case. Otherwise both dimensions are 2 ℵ 0 2^{\aleph_0} 2 ℵ 0 a c 0 \mathfrak{a}_c^0 a c 0 2 ℵ 0 2^{\aleph_0} 2 ℵ 0 m p = ∞ m_p=\infty m p = ∞ im λ 1 \operatorname{im}\lambda_1 im λ 1 p p p Proposition 6.6 (4)), so by Borel’s lemma, H 1 ( a c , λ 1 ) H^1(\mathfrak{a}_c,\lambda_1) H 1 ( a c , λ 1 ) R [ [ x ] ] \mathbb{R}[[x]] R [[ x ]] x x x p p p ∑ k s k x k \sum_k s_kx^k ∑ k s k x k s ∈ R s\in\mathbb{R} s ∈ R ( a c , λ 1 ) ≅ ( a c ′ ′ , λ 1 ′ ) (\mathfrak{a}_c,\lambda_1)\cong(\mathfrak{a}'_{c'},\lambda'_1) ( a c , λ 1 ) ≅ ( a c ′ ′ , λ 1 ′ ) E 0 E^0 E 0 R \mathbb{R} R a c 0 \mathfrak{a}_c^0 a c 0 E 1 E^1 E 1 im λ 1 \operatorname{im}\lambda_1 im λ 1 a c 1 \mathfrak{a}_c^1 a c 1 E ′ 0 , E ′ 1 E'^0,E'^1 E ′0 , E ′1 E 0 E^0 E 0 E ′ 0 E'^0 E ′0 C ∞ ( N c ) C^\infty(N_c) C ∞ ( N c ) C ∞ ( N c ′ ′ ) C^\infty(N'_{c'}) C ∞ ( N c ′ ′ ) N c → N c ′ ′ N_c\to N'_{c'} N c → N c ′ ′ dim E 1 = dim E ′ 1 \dim E^1=\dim E'^1 dim E 1 = dim E ′1 L 0 : E 0 → E ′ 0 L^0:E^0\to E'^0 L 0 : E 0 → E ′0 L 1 : E 1 → E ′ 1 L^1:E^1\to E'^1 L 1 : E 1 → E ′1 λ 1 \lambda_1 λ 1 λ 1 ′ \lambda'_1 λ 1 ′ E 0 E^0 E 0 E ′ 0 E'^0 E ′0 1 ↦ 1 1\mapsto1 1 ↦ 1 f ↦ L 0 ( f ) f\mapsto L^0(f) f ↦ L 0 ( f ) λ 1 ( f ) ↦ λ 1 ′ ( L 0 ( f ) ) \lambda_1(f)\mapsto\lambda'_1(L^0(f)) λ 1 ( f ) ↦ λ 1 ′ ( L 0 ( f )) v ↦ L 1 ( v ) v\mapsto L^1(v) v ↦ L 1 ( v ) f ∈ E 0 f\in E^0 f ∈ E 0 v ∈ E 1 v\in E^1 v ∈ E 1
(4) Let ψ j \psi_j ψ j ( t , r ) (t,r) ( t , r ) (26) , with t ∈ S ℓ j 1 t\in S^1_{\ell_j} t ∈ S ℓ j 1 Φ ( ψ ψ j ) : ( a c , λ ψ ) → ( a c , λ ψ j ) \Phi(\psi_{\psi_j}):(\mathfrak{a}_c,\lambda^\psi)\to(\mathfrak{a}_c,\lambda^{\psi_j}) Φ ( ψ ψ j ) : ( a c , λ ψ ) → ( a c , λ ψ j ) L ∞ L_\infty L ∞ Proposition 6.1 and Lemma 6.4(2), the only nonzero brackets of λ ψ j \lambda^{\psi_j} λ ψ j e e e ∂ ∂ r \frac{\partial}{\partial r} ∂ r ∂ λ 2 ( f , g e ) = − ∂ f ∂ θ g e \lambda_2(f,ge)=-\frac{\partial f}{\partial\theta}ge λ 2 ( f , g e ) = − ∂ θ ∂ f g e u j : R / Z → D j u_j:\mathbb{R}/\mathbb{Z}\to D_j u j : R / Z → D j θ ↦ ℓ j θ \theta\mapsto\ell_j\theta θ ↦ ℓ j θ b = C ∞ ( R / Z ) ⊕ C ∞ ( R / Z ) e 0 \mathfrak{b}=C^\infty(\mathbb{R}/\mathbb{Z})\oplus C^\infty(\mathbb{R}/\mathbb{Z})e_0 b = C ∞ ( R / Z ) ⊕ C ∞ ( R / Z ) e 0 λ 2 ( f , g e 0 ) = − ∂ f ∂ θ g e 0 \lambda_2(f,ge_0)=-\frac{\partial f}{\partial\theta}ge_0 λ 2 ( f , g e 0 ) = − ∂ θ ∂ f g e 0 Θ ( f ) = ℓ j − 1 f ∘ u j \Theta(f)=\ell_j^{-1}f\circ u_j Θ ( f ) = ℓ j − 1 f ∘ u j Θ ( g e ) = ( g ∘ u j ) e 0 \Theta(ge)=(g\circ u_j)e_0 Θ ( g e ) = ( g ∘ u j ) e 0
λ 2 ( Θ ( f ) , Θ ( g e ) ) = − ℓ j − 1 ∂ ( f ∘ u j ) ∂ θ ( g ∘ u j ) e 0 = − ( ∂ f ∂ t g ) ∘ u j e 0 = Θ ( λ 2 ( f , g e ) ) . \begin{aligned}
\lambda_2(\Theta(f),\Theta(ge))
&=-\ell_j^{-1}\frac{\partial(f\circ u_j)}{\partial\theta}(g\circ u_j)e_0\\
&=-\left(\frac{\partial f}{\partial t}g\right)\circ u_j e_0\\
&=\Theta(\lambda_2(f,ge)).
\end{aligned} λ 2 ( Θ ( f ) , Θ ( g e )) = − ℓ j − 1 ∂ θ ∂ ( f ∘ u j ) ( g ∘ u j ) e 0 = − ( ∂ t ∂ f g ) ∘ u j e 0 = Θ ( λ 2 ( f , g e )) . So Θ \Theta Θ L ∞ L_\infty L ∞ b \mathfrak{b} b ℓ j \ell_j ℓ j D j ′ ′ D'_{j'} D j ′ ′
By Proposition 6.16(4), components of D D D L ∞ L_\infty L ∞
Proposition 6.17. Let N c = D j N_c = D_j N c = D j N c ′ = D j ′ ′ N'_c = D'_{j'} N c ′ = D j ′ ′ F : ( a c , λ ψ ) → ( a c ′ , λ ψ ′ ) F : (\mathfrak{a}_c,\lambda^\psi) \to(\mathfrak{a}'_c,\lambda^{\psi'}) F : ( a c , λ ψ ) → ( a c ′ , λ ψ ′ ) L ∞ L_\infty L ∞ F 1 ( f v ) = F 1 ( f ) F 1 ( v ) F_1(fv)=F_1(f)F_1(v) F 1 ( f v ) = F 1 ( f ) F 1 ( v ) f ∈ a c 0 f\in\mathfrak{a}^0_c f ∈ a c 0 v ∈ a c 1 v\in\mathfrak{a}^1_c v ∈ a c 1 ℓ j = ℓ j ′ ′ \ell_j=\ell'_{j'} ℓ j = ℓ j ′ ′
Proof. By the proof of Proposition 6.16(1), F 1 F_1 F 1 Q 2 ′ ∘ ( F 1 ⋅ F 1 ) = F 1 ∘ Q 2 Q'_2\circ(F_1\cdot F_1)=F_1\circ Q_2 Q 2 ′ ∘ ( F 1 ⋅ F 1 ) = F 1 ∘ Q 2 λ 2 ′ ( F 1 a , F 1 b ) = F 1 ( λ 2 ( a , b ) ) \lambda'_2(F_1a,F_1b)=F_1(\lambda_2(a,b)) λ 2 ′ ( F 1 a , F 1 b ) = F 1 ( λ 2 ( a , b )) λ 2 ( f , v ) = X ( f ) v \lambda_2(f,v)=X(f)v λ 2 ( f , v ) = X ( f ) v λ 2 ′ ( f ′ , v ′ ) = X ′ ( f ′ ) v ′ \lambda'_2(f',v')=X'(f')v' λ 2 ′ ( f ′ , v ′ ) = X ′ ( f ′ ) v ′ X = X v o l ∣ D j X=X_{\mathrm{vol}|D_j} X = X vol ∣ D j X ′ = X v o l ∣ D j ′ ′ ′ X'=X'_{\mathrm{vol}|D'_{j'}} X ′ = X vol ∣ D j ′ ′ ′ f ∈ C ∞ ( D j ) f\in C^\infty(D_j) f ∈ C ∞ ( D j ) v ∈ a c 1 v\in\mathfrak{a}^1_c v ∈ a c 1
X ′ ( F 1 ( f ) ) F 1 ( v ) = λ 2 ′ ( F 1 f , F 1 v ) = F 1 ( X ( f ) v ) = F 1 ( X ( f ) ) F 1 ( v ) . X'(F_1(f))F_1(v)=\lambda'_2(F_1f,F_1v)=F_1(X(f)v)=F_1(X(f))F_1(v). X ′ ( F 1 ( f )) F 1 ( v ) = λ 2 ′ ( F 1 f , F 1 v ) = F 1 ( X ( f ) v ) = F 1 ( X ( f )) F 1 ( v ) . Choosing v v v F 1 ( v ) F_1(v) F 1 ( v ) X ′ ∘ F 1 = F 1 ∘ X X'\circ F_1=F_1\circ X X ′ ∘ F 1 = F 1 ∘ X C ∞ ( D j ) C^\infty(D_j) C ∞ ( D j ) X X X X ′ X' X ′ t ∈ S ℓ j 1 t\in S^1_{\ell_j} t ∈ S ℓ j 1 (26) , X = − ∂ ∂ t X=-\frac{\partial}{\partial t} X = − ∂ t ∂ (27) , and − d f d t = μ f -\frac{df}{dt}=\mu f − d t df = μ f f = e − μ t f=e^{-\mu t} f = e − μ t S ℓ j 1 S^1_{\ell_j} S ℓ j 1 μ ∈ 2 π i ℓ j Z \mu\in\frac{2\pi i}{\ell_j}\mathbb{Z} μ ∈ ℓ j 2 π i Z 2 π i ℓ j Z = 2 π i ℓ j ′ ′ Z \frac{2\pi i}{\ell_j}\mathbb{Z}=\frac{2\pi i}{\ell'_{j'}}\mathbb{Z} ℓ j 2 π i Z = ℓ j ′ ′ 2 π i Z ℓ j = ℓ j ′ ′ \ell_j=\ell'_{j'} ℓ j = ℓ j ′ ′
It remains to identify the isomorphisms induced by changes of tubular neighbourhood. By Theorem A, an isomorphism F F F F 1 = id F_1=\operatorname{id} F 1 = id log F \log F log F e e e ν N c \nu_{N_c} ν N c y y y Y Y Y N c N_c N c ν N c ≅ N c × R \nu_{N_c}\cong N_c\times\mathbb{R} ν N c ≅ N c × R Y Y Y Y = Y x ( x ) ∂ ∂ x Y=Y^x(x)\frac{\partial}{\partial x} Y = Y x ( x ) ∂ x ∂
Proposition 6.18. Let n ≥ 2 n\ge2 n ≥ 2 l : S n ( a c [ 1 ] ) → a c [ 1 ] l:S^n(\mathfrak{a}_c[1])\to\mathfrak{a}_c[1] l : S n ( a c [ 1 ]) → a c [ 1 ] 0 0 0 a c \mathfrak{a}_c a c X n X_n X n N c N_c N c ρ n ∈ C ∞ ( N c ) \rho_n\in C^\infty(N_c) ρ n ∈ C ∞ ( N c ) f , g 1 , … , g n ∈ C ∞ ( N c ) f,g_1,\ldots,g_n\in C^\infty(N_c) f , g 1 , … , g n ∈ C ∞ ( N c )
l ( f , g 1 e , … , g n − 1 e ) = g 1 ⋯ g n − 1 X n ( f ) , (53) l(f,g_1e,\ldots,g_{n-1}e)=g_1\cdots g_{n-1}X_n(f), \tag*{(53)} l ( f , g 1 e , … , g n − 1 e ) = g 1 ⋯ g n − 1 X n ( f ) , (53) l ( g 1 e , … , g n e ) = ( ∑ i = 1 n ( ∏ k ≠ i g k ) X n ( g i ) + ρ n g 1 ⋯ g n ) e . (54) l(g_1e,\ldots,g_ne)=\left(\sum_{i=1}^{n}\left(\prod_{k\ne i}g_k\right)X_n(g_i)+\rho_n g_1\cdots g_n\right)e. \tag*{(54)} l ( g 1 e , … , g n e ) = i = 1 ∑ n k = i ∏ g k X n ( g i ) + ρ n g 1 ⋯ g n e . (54) The pair ( X n , ρ n ) (X_n,\rho_n) ( X n , ρ n ) l = ( D Z ( n − 1 ) ) n l=(D_{Z_{(n-1)}})_n l = ( D Z ( n − 1 ) ) n n − 1 n-1 n − 1
Z ( n − 1 ) = ( − 1 ) n − 1 ( n − 1 ) ! y n − 1 X n + ( − 1 ) n n ! ρ n y n ∂ ∂ y . (55) Z_{(n-1)}=\frac{(-1)^{n-1}}{(n-1)!}y^{n-1}X_n+\frac{(-1)^n}{n!}\rho_n y^n\frac{\partial}{\partial y}. \tag*{(55)} Z ( n − 1 ) = ( n − 1 )! ( − 1 ) n − 1 y n − 1 X n + n ! ( − 1 ) n ρ n y n ∂ y ∂ . (55) Proof. In a c [ 1 ] \mathfrak{a}_c[1] a c [ 1 ] − 1 -1 − 1 ν N c \nu_{N_c} ν N c 0 0 0 l l l a c 0 \mathfrak{a}^0_c a c 0
Let l l l g , h ∈ C ∞ ( N c ) g,h\in C^\infty(N_c) g , h ∈ C ∞ ( N c ) a 2 , … , a n ∈ a c [ 1 ] a_2,\ldots,a_n\in\mathfrak{a}_c[1] a 2 , … , a n ∈ a c [ 1 ] d = 0 d=0 d = 0
l ( g h , a 2 , … , a n ) = l ( g , a 2 , … , a n ) h + g l ( h , a 2 , … , a n ) , l(gh,a_2,\ldots,a_n)=l(g,a_2,\ldots,a_n)h+g\,l(h,a_2,\ldots,a_n), l ( g h , a 2 , … , a n ) = l ( g , a 2 , … , a n ) h + g l ( h , a 2 , … , a n ) , l ( g e , a 2 , … , a n ) = ± l ( g , a 2 , … , a n ) e + g l ( e , a 2 , … , a n ) , l(ge,a_2,\ldots,a_n)=\pm l(g,a_2,\ldots,a_n)e+g\,l(e,a_2,\ldots,a_n), l ( g e , a 2 , … , a n ) = ± l ( g , a 2 , … , a n ) e + g l ( e , a 2 , … , a n ) , with the sign + + + a k a_k a k a k a_k a k f ↦ l ( f , e , … , e ) f\mapsto l(f,e,\ldots,e) f ↦ l ( f , e , … , e ) C ∞ ( N c ) C^\infty(N_c) C ∞ ( N c ) X n X_n X n ρ n \rho_n ρ n l ( e , … , e ) = ρ n e l(e,\ldots,e)=\rho_n e l ( e , … , e ) = ρ n e (53) and (54) .
Conversely, since a c 0 = C ∞ ( N c ) e \mathfrak{a}^0_c=C^\infty(N_c)e a c 0 = C ∞ ( N c ) e 0 0 0 (53) and (54) is determined by ( X n , ρ n ) (X_n,\rho_n) ( X n , ρ n ) g i = 1 g_i=1 g i = 1 ( X n , ρ n ) (X_n,\rho_n) ( X n , ρ n ) Z ( n − 1 ) Z_{(n-1)} Z ( n − 1 ) (55) , (20) and (19) give
( D Z ( n − 1 ) ) n ( e , … , e , x ) = ( − 1 ) n − 1 ∂ n − 1 ∂ y n − 1 ( ( − 1 ) n − 1 ( n − 1 ) ! y n − 1 X n x ) = X n x , (D_{Z_{(n-1)}})_n(e,\ldots,e,x)=(-1)^{n-1}\frac{\partial^{n-1}}{\partial y^{n-1}}\left(\frac{(-1)^{n-1}}{(n-1)!}y^{n-1}X_n^x\right)=X_n^x, ( D Z ( n − 1 ) ) n ( e , … , e , x ) = ( − 1 ) n − 1 ∂ y n − 1 ∂ n − 1 ( ( n − 1 )! ( − 1 ) n − 1 y n − 1 X n x ) = X n x , ( D Z ( n − 1 ) ) n ( e , … , e ) = ( − 1 ) n ∂ n ∂ y n ( ( − 1 ) n n ! ρ n y n ) e = ρ n e . (D_{Z_{(n-1)}})_n(e,\ldots,e)=(-1)^n\frac{\partial^n}{\partial y^n}\left(\frac{(-1)^n}{n!}\rho_n y^n\right)e=\rho_n e. ( D Z ( n − 1 ) ) n ( e , … , e ) = ( − 1 ) n ∂ y n ∂ n ( n ! ( − 1 ) n ρ n y n ) e = ρ n e . By §2.2, ( D Z ( n − 1 ) ) n (D_{Z_{(n-1)}})_n ( D Z ( n − 1 ) ) n (53) and (54) for the pair ( X n , ρ n ) (X_n,\rho_n) ( X n , ρ n ) (53) and (54) equals ( D Z ( n − 1 ) ) n (D_{Z_{(n-1)}})_n ( D Z ( n − 1 ) ) n
Let ψ , ψ ′ \psi,\psi' ψ , ψ ′ N c N_c N c F : ( a c , λ ψ ) ⟶ ( a c , λ ψ ′ ) F:(\mathfrak a_c,\lambda^\psi)\longrightarrow(\mathfrak a_c,\lambda^{\psi'}) F : ( a c , λ ψ ) ⟶ ( a c , λ ψ ′ ) L ∞ L_\infty L ∞ F 1 = id F_1=\operatorname{id} F 1 = id log F ∈ Coder ≥ 2 ( a c ) \log F\in\operatorname{Coder}_{\ge2}(\mathfrak a_c) log F ∈ Coder ≥ 2 ( a c ) Proposition 6.18 , F F F F 1 = id F_1=\operatorname{id} F 1 = id ( log F ) n (\log F)_n ( log F ) n n ≥ 2 n\ge2 n ≥ 2 ( X n , ρ n ) (X_n,\rho_n) ( X n , ρ n ) log F = D Z \log F=D_Z log F = D Z Lemma 2.6 (1), and F = e D Z = Φ ( e Z ) F=e^{D_Z}=\Phi(e^Z) F = e D Z = Φ ( e Z ) Z = ∑ n ≥ 2 Z ( n − 1 ) Z=\sum_{n\ge2}Z^{(n-1)} Z = ∑ n ≥ 2 Z ( n − 1 ) n = 2 n=2 n = 2 Z ( 1 ) = − y X 2 + 1 2 ρ 2 y 2 ∂ ∂ y Z^{(1)}=-yX_2+\frac{1}{2}\rho_2y^2\frac{\partial}{\partial y} Z ( 1 ) = − y X 2 + 2 1 ρ 2 y 2 ∂ y ∂ κ = − X 2 x \kappa=-X_2^x κ = − X 2 x ρ = 1 2 ρ 2 \rho=\frac{1}{2}\rho_2 ρ = 2 1 ρ 2 Lemma 6.3 , so, by (6.6), the coefficients υ 1 \upsilon_1 υ 1 σ ψ \sigma_\psi σ ψ υ ~ 1 \widetilde{\upsilon}_1 υ 1 σ ψ ′ \sigma_{\psi'} σ ψ ′
υ ~ 1 = υ 1 − X 2 ( υ 0 ) + ( ∂ X 2 x ∂ x − ρ 2 ) υ 0 . (56) \widetilde{\upsilon}_1=\upsilon_1-X_2(\upsilon_0)+\left(\frac{\partial X_2^x}{\partial x}-\rho_2\right)\upsilon_0. \tag*{(56)} υ 1 = υ 1 − X 2 ( υ 0 ) + ( ∂ x ∂ X 2 x − ρ 2 ) υ 0 . (56) To apply the criterion to linearization, we also need to specify when the target complex is realized geometrically. We say that an L ∞ L_\infty L ∞
F : ( a c , λ ψ ) ⟶ ( a c , λ 1 ) F:(\mathfrak a_c,\lambda^\psi)\longrightarrow(\mathfrak a_c,\lambda_1) F : ( a c , λ ψ ) ⟶ ( a c , λ 1 ) is induced by a change of tubular neighbourhood if there is a tubular neighbourhood ψ ′ ′ \psi'' ψ ′′ λ n ψ ′ ′ = 0 \lambda_n^{\psi''}=0 λ n ψ ′′ = 0 n ≥ 2 n\ge2 n ≥ 2 F = Φ ( φ ) F=\Phi(\varphi) F = Φ ( φ ) φ ∈ Diff 1 ( ψ , ψ ′ ′ ) \varphi\in\operatorname{Diff}_1(\psi,\psi'') φ ∈ Diff 1 ( ψ , ψ ′′ ) λ 1 \lambda_1 λ 1
Corollary 6.19. Let N c ⊄ D N_c\not\subset D N c ⊂ D ψ \psi ψ N c N_c N c
If N c N_c N c D D D L ∞ L_\infty L ∞ ( a c , λ ψ ) ⟶ ( a c , λ 1 ) (\mathfrak a_c,\lambda^\psi)\longrightarrow(\mathfrak a_c,\lambda_1) ( a c , λ ψ ) ⟶ ( a c , λ 1 ) Φ ( φ ) ∘ Q ψ ∘ Φ ( φ ) − 1 ≠ Q l i n \Phi(\varphi)\circ Q_\psi\circ\Phi(\varphi)^{-1}\ne Q^{\mathrm{lin}} Φ ( φ ) ∘ Q ψ ∘ Φ ( φ ) − 1 = Q lin φ ∈ Diff 1 ( ν N c ) \varphi\in\operatorname{Diff}_1(\nu_{N_c}) φ ∈ Diff 1 ( ν N c )
If every contact of N c N_c N c D D D Q l i n = Q ψ ′ ′ Q^{\mathrm{lin}}=Q_{\psi''} Q lin = Q ψ ′′ ψ ′ ′ \psi'' ψ ′′ Φ ( φ ψ ψ ′ ′ ) \Phi(\varphi_{\psi\psi''}) Φ ( φ ψ ψ ′′ ) L ∞ L_\infty L ∞ ( a c , λ ψ ) ⟶ ( a c , λ 1 ) (\mathfrak a_c,\lambda^\psi)\longrightarrow(\mathfrak a_c,\lambda_1) ( a c , λ ψ ) ⟶ ( a c , λ 1 )
Proof. (1) By Proposition 6.14 , the binary bracket of Q ψ ′ Q_{\psi'} Q ψ ′ ψ ′ \psi' ψ ′ Q l i n ≠ Q ψ ′ Q^{\mathrm{lin}}\ne Q_{\psi'} Q lin = Q ψ ′ φ ∈ Diff 1 ( ν N c ) \varphi\in\operatorname{Diff}_1(\nu_{N_c}) φ ∈ Diff 1 ( ν N c ) Proposition 2.8 (1) gives Φ ( φ ) ∘ Q ψ ∘ Φ ( φ ) − 1 = Q φ ∗ σ ψ \Phi(\varphi)\circ Q_\psi\circ\Phi(\varphi)^{-1}=Q_{\varphi^*\sigma_\psi} Φ ( φ ) ∘ Q ψ ∘ Φ ( φ ) − 1 = Q φ ∗ σ ψ φ ∗ σ ψ = υ ~ ∂ ∂ x ∧ ∂ ∂ y \varphi^*\sigma_\psi=\widetilde{\upsilon}\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y} φ ∗ σ ψ = υ ∂ x ∂ ∧ ∂ y ∂ p p p Lemma 6.3 with m = 1 m=1 m = 1 Z = log φ Z=\log\varphi Z = log φ υ ~ 1 ( p ) = υ 1 ( p ) ≠ 0 \widetilde{\upsilon}_1(p)=\upsilon_1(p)\ne0 υ 1 ( p ) = υ 1 ( p ) = 0 Proposition 6.14 . By the computation of Proposition 6.1 for the formal bivector φ ∗ σ ψ \varphi^*\sigma_\psi φ ∗ σ ψ Q φ ∗ σ ψ Q_{\varphi^*\sigma_\psi} Q φ ∗ σ ψ ( f , g e ) ↦ υ ~ 1 ∂ f ∂ x g e (f,g e)\mapsto\widetilde{\upsilon}_1\frac{\partial f}{\partial x}g e ( f , g e ) ↦ υ 1 ∂ x ∂ f g e p p p
(2) By Proposition 6.12, λ n ψ ′ ′ = 0 \lambda_n^{\psi''}=0 λ n ψ ′′ = 0 n ≥ 2 n\ge2 n ≥ 2 ψ ′ ′ \psi'' ψ ′′ ( Q ψ ′ ′ ) 1 = ( Q ψ ) 1 (Q_{\psi''})_1=(Q_\psi)_1 ( Q ψ ′′ ) 1 = ( Q ψ ) 1 Q ψ ′ ′ = Q l i n Q_{\psi''}=Q^{\mathrm{lin}} Q ψ ′′ = Q lin
Example 6.20. Let M = ( R / Z ) × R M=(\mathbb R/\mathbb Z)\times\mathbb R M = ( R / Z ) × R ( x , y ) (x,y) ( x , y ) σ = ∂ ∂ x ∧ ∂ ∂ y \sigma=\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y} σ = ∂ x ∂ ∧ ∂ y ∂ N = { y = 0 } N=\{y=0\} N = { y = 0 } Σ ∖ D \Sigma\setminus D Σ ∖ D ν N = N × R \nu_N=N\times\mathbb R ν N = N × R e = ∂ ∂ y e=\frac{\partial}{\partial y} e = ∂ y ∂ ψ ( x , y ) = ( x , y ) \psi(x,y)=(x,y) ψ ( x , y ) = ( x , y ) ψ ′ ( x , y ) = ( x , y + y 2 ) \psi'(x,y)=(x,y+y^2) ψ ′ ( x , y ) = ( x , y + y 2 ) ∣ y ∣ < 1 2 |y|<\frac{1}{2} ∣ y ∣ < 2 1 φ = φ ψ ψ ′ \varphi=\varphi_{\psi\psi'} φ = φ ψ ψ ′ ( x , y ) ↦ ( x , y + y 2 ) (x,y)\mapsto(x,y+y^2) ( x , y ) ↦ ( x , y + y 2 ) y ~ = y + y 2 \widetilde y=y+y^2 y = y + y 2 ∂ ∂ y ~ = ( 1 + 2 y ) − 1 ∂ ∂ y \frac{\partial}{\partial\widetilde y}=(1+2y)^{-1}\frac{\partial}{\partial y} ∂ y ∂ = ( 1 + 2 y ) − 1 ∂ y ∂
σ ψ = ∂ ∂ x ∧ ∂ ∂ y , \sigma_\psi=\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}, σ ψ = ∂ x ∂ ∧ ∂ y ∂ , σ ψ ′ = 1 1 + 2 y ∂ ∂ x ∧ ∂ ∂ y = ∑ k ≥ 0 ( − 2 ) k y k ∂ ∂ x ∧ ∂ ∂ y . \sigma_{\psi'}=\frac{1}{1+2y}\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}
=\sum_{k\ge0}(-2)^ky^k\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}. σ ψ ′ = 1 + 2 y 1 ∂ x ∂ ∧ ∂ y ∂ = k ≥ 0 ∑ ( − 2 ) k y k ∂ x ∂ ∧ ∂ y ∂ . By (6.1), λ n ψ = 0 \lambda_n^\psi=0 λ n ψ = 0 n ≥ 2 n\ge2 n ≥ 2
λ n ψ ′ ( f , g 1 e , … , g n − 1 e ) = ( n − 1 ) ! ( − 2 ) n − 1 ∂ f ∂ x g 1 ⋯ g n − 1 e . \lambda_n^{\psi'}(f,g_1e,\ldots,g_{n-1}e)=(n-1)!(-2)^{n-1}\frac{\partial f}{\partial x}g_1\cdots g_{n-1}e. λ n ψ ′ ( f , g 1 e , … , g n − 1 e ) = ( n − 1 )! ( − 2 ) n − 1 ∂ x ∂ f g 1 ⋯ g n − 1 e . So the two P ∞ P_\infty P ∞ λ 1 ψ = λ 1 ψ ′ \lambda_1^\psi=\lambda_1^{\psi'} λ 1 ψ = λ 1 ψ ′
Let Z = log φ = z ( y ) ∂ ∂ y Z=\log\varphi=z(y)\frac{\partial}{\partial y} Z = log φ = z ( y ) ∂ y ∂ e Z ( y ) = y + y 2 e^{Z}(y)=y+y^2 e Z ( y ) = y + y 2 y + z + 1 2 z z ′ + ⋯ = y + y 2 y+z+\frac{1}{2}zz'+\cdots=y+y^2 y + z + 2 1 z z ′ + ⋯ = y + y 2 z = ∑ k ≥ 2 z k y k z=\sum_{k\ge2}z_ky^k z = ∑ k ≥ 2 z k y k z ( y ) = y 2 − y 3 + 3 2 y 4 − ⋯ z(y)=y^2-y^3+\frac{3}{2}y^4-\cdots z ( y ) = y 2 − y 3 + 2 3 y 4 − ⋯ Proposition 6.18 , X n = 0 X_n=0 X n = 0 ρ n = ( − 1 ) n n ! z n \rho_n=(-1)^nn!z_n ρ n = ( − 1 ) n n ! z n ( log Φ ( φ ) ) n ( e , … , e ) = ρ n e (\log\Phi(\varphi))_n(e,\ldots,e)=\rho_ne ( log Φ ( φ ) ) n ( e , … , e ) = ρ n e ρ 2 = 2 \rho_2=2 ρ 2 = 2 ρ 3 = 6 \rho_3=6 ρ 3 = 6 ρ 4 = 36 \rho_4=36 ρ 4 = 36
By Proposition 6.18 with X n = 0 X_n=0 X n = 0 D Z D_Z D Z C ∞ ( N ) C^\infty(N) C ∞ ( N ) C ∞ ( N ) C^\infty(N) C ∞ ( N ) ν N \nu_N ν N D Z D_Z D Z Φ ( φ ) n \Phi(\varphi)_n Φ ( φ ) n n ≥ 2 n\ge2 n ≥ 2 X n = 0 X_n=0 X n = 0
Example 6.21. On the sphere of Example 5.6, let N ′ N' N ′ π / 4 \pi/4 π /4 π / 4 \pi/4 π /4 θ = 0 \theta= 0 θ = 0 D D D p = ( 0 , 0 ) p = (0, 0) p = ( 0 , 0 )
N ′ = { z = h ( θ ) , ∣ θ ∣ < π 2 } , N' = \left\{ z = h(\theta),\ |\theta| < \frac{\pi}{2} \right\}, N ′ = { z = h ( θ ) , ∣ θ ∣ < 2 π } , h ( θ ) = sin 2 θ 1 + cos 2 θ = 1 2 θ 2 + O ( θ 4 ) , h(\theta) = \frac{\sin^2\theta}{1+\cos^2\theta} = \frac{1}{2}\theta^2 + O(\theta^4), h ( θ ) = 1 + cos 2 θ sin 2 θ = 2 1 θ 2 + O ( θ 4 ) , so m p = 2 m_p = 2 m p = 2 N 1 N_1 N 1 D D D N ′ N' N ′ N 1 N_1 N 1 L ∞ L_\infty L ∞ N 1 N_1 N 1 λ n = 0 \lambda_n = 0 λ n = 0 n ≥ 2 n \ge2 n ≥ 2 Proposition 6.14 and (27) , N ′ N' N ′ λ 2 ( f , v ) = X ( f ) v \lambda_2(f,v) = X(f)\,v λ 2 ( f , v ) = X ( f ) v X ( p ) = − ∂ ∂ θ X(p) = -\frac{\partial}{\partial\theta} X ( p ) = − ∂ θ ∂
Proof of Theorem D. Part (1) is Proposition 6.16(2) and (3). In part (2), an induced L ∞ L_\infty L ∞ ( a c , λ ψ ) → ( a c , λ 1 ) (\mathfrak{a}_c,\lambda^\psi) \to(\mathfrak{a}_c,\lambda_1) ( a c , λ ψ ) → ( a c , λ 1 ) Corollary 6.19 (2), and none exists otherwise, by Corollary 6.19 (1). At a tangential contact p p p λ 2 ψ ( f , v ) = X ( f ) v \lambda^\psi_2(f,v) = X(f)\,v λ 2 ψ ( f , v ) = X ( f ) v X X X Corollary 6.2 , and X ( p ) = X v o l ( p ) X(p) = X_{\mathrm{vol}}(p) X ( p ) = X vol ( p ) (52) . □
References
Département de mathématiques, Université de Sherbrooke, 2500 Bd de l’Université, Sherbrooke, QC, J1K 2R1, Canada
Email address: yassine.ait.mohamed@usherbrooke.ca
Département de mathématiques, Université de Sherbrooke, 2500 Bd de l’Université, Sherbrooke, QC, J1K 2R1, Canada
Email address: maxence.mayrand@usherbrooke.ca
