Introduction
Background
The idea behind this article was to let AI explore geometry and topology freely and see what conclusions it could reach by focusing on “heuristic” mathematics.
Using the chat interface of GPT 6 Astra Pro, the prompter started fifteen separate conversations and asked the AI to propose new mathematical objects worth studying. The prompter then merged related topics, discarded problems that did not seem worth pursuing, and collected the directions he himself found interesting into this research guide. These questions are relatively new overall, but they are closely tied to existing human mathematical research.
According to the prompter, further work has shown clear mathematical structure in some of these theories and has even solved open problems in other areas. This gives the prompter reason to believe that at least some of these directions are promising. The prompter will discuss further work on these directions in later papers.
The prompter claims no credit for any of these theories and welcomes human-led mathematical discussion with anyone interested in them. The hope is that AI will create more interesting mathematical problems, rather than just kill existing ones.
References may be incomplete, since the prompter has limited knowledge of each of these areas and AI may have missed relevant works. The prompter apologizes for any omissions of works that should have been cited as sources of the ideas presented here and will correct them as soon as they are reported.
Critical kernels
Section 2 studies symmetric kernels whose Morse slice functions define finite critical equivalence classes. The associated coverings give strong topological restrictions when the Euler characteristic is nonzero. First-level Morse–Bott attachments yield quadratic cohomology extensions, and additional generation hypotheses lead to recognition by reflection groups. This connects the construction with invariant theory and rational homotopy theory. Further questions concern geometric realizability and what remains true without the generation hypotheses.
Uniformly Morse linear systems
Section 3 studies finite-dimensional spaces of functions modulo constants whose nonzero elements are Morse. On a closed connected n n n n + 1 n+1 n + 1 S 3 S^{3} S 3
Presymplectic dependency algebras
Section 4 studies positive systems of closed two-forms of constant rank zero or two and their finite wedge algebras. Restriction to characteristic foliations realizes matroid contraction. The algebras distinguish systems on a fixed torus with identical cohomology classes, matroids, and volume polynomials. These constructions relate multilinear matroid representations to symplectic pairs and polysymplectic geometry. Further questions concern local moduli and the extent to which the algebra captures the geometry and deformation behavior of the chosen system.
Diffusion transport defects
Section 5 studies rank defects of connection heat kernels and coverings by their invertibility domains. Characteristic classes give lower bounds for the covering number. Flat torus constructions exhibit arbitrarily many distinct covering levels as time varies, keeping the metric and connection fixed in each example while allowing dimension to grow. The framework relates vector diffusion methods, characteristic-class obstructions, and theta-function geometry. Further questions concern transversality and the possible time profiles of the covering number.
Tangent shapes and knots
Section 6 studies families of compact shapes embedded in punctured tangent spaces and oriented knots in spheres of tangent directions. Euler-class and holonomy arguments yield obstructions and rigidity results for shape fields. For a prescribed positive nontrivial torus-knot type on a closed, connected, oriented smooth four-manifold, existence is characterized by a weighted circle reduction of the tangent bundle and explicit characteristic-class conditions. These results connect embedding spaces, classical G G G
Finite-scale bundle comparisons
Section 7 organizes smooth fiber identifications by their distance from the diagonal, using an extension radius and a system of restriction maps. We establish metric stability and line-bundle criteria, including an exact half-systole formula for nontrivial real line bundles and computations on flat tori. For closed negatively curved oriented surfaces, the complete family of labelled radii on finite covers recovers the marked length spectrum. This links direct connections with systolic geometry. Further questions concern higher-rank obstructions, critical radii, and information retained by the full comparison system.
Harmonic closure of differential forms
Section 8 considers the algebra generated by harmonic forms under wedge product, exterior differentiation, and Hodge star on a closed oriented Riemannian manifold. Finite closure produces a finite de Rham model and a canonical Lie algebra. For compact nilmanifolds with descended left-invariant metrics, the construction recovers the metric Lie algebra. Other results include conformal rigidity in the presence of intermediate cohomology and a three-dimensional classification. Connections with geometric formality motivate further study of higher-dimensional finite-closure metrics and their associated Lie algebra actions.
Critical kernels
A critical kernel is a symmetric two-point function whose slice critical points form finite equivalence classes. The resulting covering structure imposes restrictions on the topology of the underlying manifold.
Throughout this section, manifolds are smooth, closed (compact without boundary), connected, and of positive dimension. Cohomology is singular cohomology. Put P M ( t ) = ∑ j dim H j ( M ; Q ) t j P_M(t)=\sum_j \dim H^j(M;\mathbb{Q})t^j P M ( t ) = ∑ j dim H j ( M ; Q ) t j ≃ Q \simeq_{\mathbb{Q}} ≃ Q M M M g ⋅ a = ( g − 1 ) ∗ a g\mathbin{\cdot}a=(g^{-1})^*a g ⋅ a = ( g − 1 ) ∗ a g ∗ g^* g ∗
Definition and covering geometry
Definition 2.1 (Critical kernel). A critical kernel on M M M F : M × M → [ 0 , ∞ ) F:M\times M\to[0,\infty) F : M × M → [ 0 , ∞ )
(i) F ( x , y ) = F ( y , x ) F(x,y)=F(y,x) F ( x , y ) = F ( y , x ) F ( x , y ) = 0 F(x,y)=0 F ( x , y ) = 0 x = y x=y x = y
(ii) for every x ∈ M x\in M x ∈ M F x ( y ) = F ( x , y ) F_x(y)=F(x,y) F x ( y ) = F ( x , y )
(iii) the relation x ∼ F y ⟺ d 2 F ( x , y ) = 0 x \sim_F y \Longleftrightarrow d_2F(x,y)=0 x ∼ F y ⟺ d 2 F ( x , y ) = 0
Write C F = { ( x , y ) : d 2 F ( x , y ) = 0 } C_F=\{(x,y):d_2F(x,y)=0\} C F = {( x , y ) : d 2 F ( x , y ) = 0 } B F = M / ∼ F B_F=M/\sim_F B F = M / ∼ F ϕ : M → N \phi:M\to N ϕ : M → N F N ( ϕ x , ϕ y ) = F M ( x , y ) F_N(\phi x,\phi y)=F_M(x,y) F N ( ϕ x , ϕ y ) = F M ( x , y )
Proposition 2.2 (Critical covering). Let n = dim M n=\dim M n = dim M C F C_F C F n n n C F → M C_F\to M C F → M r ≥ 2 r\ge2 r ≥ 2 B F B_F B F
q : M ⟶ B F , C F = M × B F M , q:M\longrightarrow B_F,\qquad C_F=M\times_{B_F}M, q : M ⟶ B F , C F = M × B F M , is an r r r r r r r r r
Proof. The section d 2 F d_2F d 2 F π 2 ∗ T ∗ M \pi_2^*T^*M π 2 ∗ T ∗ M F x F_x F x C F C_F C F y = ϕ ( x ) y=\phi(x) y = ϕ ( x ) ϕ \phi ϕ
For a class [ x ] = { x 1 , … , x r } [x]=\{x_1,\ldots,x_r\} [ x ] = { x 1 , … , x r } U U U x = x 1 x=x_1 x = x 1 ϕ i : U → U i \phi_i:U\to U_i ϕ i : U → U i ϕ 1 = id \phi_1=\operatorname{id} ϕ 1 = id U i U_i U i [ ϕ i ( u ) ] = { ϕ 1 ( u ) , … , ϕ r ( u ) } [\phi_i(u)]=\{\phi_1(u),\ldots,\phi_r(u)\} [ ϕ i ( u )] = { ϕ 1 ( u ) , … , ϕ r ( u )} ⋃ i U i \bigcup_i U_i ⋃ i U i q ∣ U i q|_{U_i} q ∣ U i B F B_F B F q q q r ≥ 2 r\ge2 r ≥ 2
Proposition 2.3 (Morse–Bott structure and metric). The full function F F F C F C_F C F ( x , y ) (x,y) ( x , y ) y y y F x F_x F x
g x ( u , v ) = Hess y = x F x ( u , v ) g_x(u,v)=\operatorname{Hess}_{y=x}F_x(u,v) g x ( u , v ) = Hess y = x F x ( u , v ) defines a Riemannian metric, natural under kernel isomorphisms.
Proof. If d 2 F ( x , y ) = 0 d_2F(x,y)=0 d 2 F ( x , y ) = 0 d 1 F ( x , y ) = 0 d_1F(x,y)=0 d 1 F ( x , y ) = 0 Crit ( F ) = C F \operatorname{Crit}(F)=C_F Crit ( F ) = C F y = ϕ ( x ) y=\phi(x) y = ϕ ( x ) A = D ϕ x A=D\phi_x A = D ϕ x H = D y y 2 F ( x , ϕ ( x ) ) H=D^2_{yy}F(x,\phi(x)) H = D y y 2 F ( x , ϕ ( x ))
D 2 F = ( A T H A − A T H − H A H ) , D 2 F ( ( v , w ) , ( v , w ) ) = H ( w − A v , w − A v ) . D^2F=\begin{pmatrix} A^{\mathsf T}HA & -A^{\mathsf T}H \\ -HA & H \end{pmatrix},\qquad D^2F((v,w),(v,w))=H(w-Av,w-Av). D 2 F = ( A T H A − H A − A T H H ) , D 2 F (( v , w ) , ( v , w )) = H ( w − A v , w − A v ) . Its kernel is exactly the tangent space to the graph, and its normal inertia is that of H H H x x x F x F_x F x g x g_x g x
Proposition 2.4 (Products). If F F F K K K M M M N N N
( F ⊕ K ) ( ( x , u ) , ( y , v ) ) = F ( x , y ) + K ( u , v ) (F\oplus K)((x,u),(y,v))=F(x,y)+K(u,v) ( F ⊕ K ) (( x , u ) , ( y , v )) = F ( x , y ) + K ( u , v ) is a critical kernel on M × N M\times N M × N
Proof. Since gradients and Hessians split, and criticality is the product of the two equivalence relations. The other axioms follow directly. □
Euler rigidity
Theorem 2.5 (Euler rigidity). If χ ( M ) ≠ 0 \chi(M) \ne0 χ ( M ) = 0
χ ( M ) = r > 0 , χ ( B F ) = 1. \chi(M)=r>0,\qquad\chi(B_F)=1. χ ( M ) = r > 0 , χ ( B F ) = 1. Every slice has only even-index critical points and is perfect over Z \mathbb{Z} Z M M M q q q G ≅ π 1 ( B F ) G \cong\pi_1(B_F) G ≅ π 1 ( B F )
∣ G ∣ = χ ( M ) = dim Q H ∗ ( M ; Q ) , H ∗ ( M ; Q ) G = Q . (1) |G|=\chi(M)=\dim_{\mathbb{Q}} H^*(M;\mathbb{Q}),\qquad H^*(M;\mathbb{Q})^G=\mathbb{Q}. \tag*{(1)} ∣ G ∣ = χ ( M ) = dim Q H ∗ ( M ; Q ) , H ∗ ( M ; Q ) G = Q . (1) The quotient B F B_F B F
p i ( T M ) = 0 ∈ H 4 i ( M ; Z ) ( i > 0 ) . (2) p_i(TM)=0\in H^{4i}(M;\mathbb{Z})\qquad(i>0). \tag*{(2)} p i ( T M ) = 0 ∈ H 4 i ( M ; Z ) ( i > 0 ) . (2) Proof. The covering formula gives χ ( M ) = r χ ( B F ) \chi(M)=r\chi(B_F) χ ( M ) = r χ ( B F ) χ ( M ) = ∑ j = 1 r ( − 1 ) λ j \chi(M)=\sum_{j=1}^{r}(-1)^{\lambda_j} χ ( M ) = ∑ j = 1 r ( − 1 ) λ j − r < χ ( M ) ≤ r -r<\chi(M)\le r − r < χ ( M ) ≤ r r r r r r r λ j \lambda_j λ j
The Morse CW decomposition has only even cells. Connectedness forces one zero-cell; there are no one-cells, so M M M ∣ G ∣ = r |G|=r ∣ G ∣ = r
Transfer injects H ∗ ( B F ; Q ) H^*(B_F;\mathbb{Q}) H ∗ ( B F ; Q ) H ∗ ( M ; Q ) H^*(M;\mathbb{Q}) H ∗ ( M ; Q ) H 0 = Q H^0=\mathbb{Q} H 0 = Q T M ≅ q ∗ T B F TM\cong q^*TB_F T M ≅ q ∗ T B F T M TM T M
Examples
The first two examples also cover χ ( M ) = 0 \chi(M)=0 χ ( M ) = 0 Remark 2.6 is not imposed in this subsection.
Example 2.7 (Spheres and their products). On the unit sphere S n ⊂ R n + 1 S^n\subset\mathbb{R}^{n+1} S n ⊂ R n + 1 n ≥ 1 n\ge1 n ≥ 1 F ( x , y ) = 1 − x ⋅ y F(x,y)=1-x\cdot y F ( x , y ) = 1 − x ⋅ y x x x − x -x − x n n n S n → R P n S^n\to\mathbb{R}P^n S n → R P n
F ( x , y ) = ∑ i = 1 k ( 1 − x i ⋅ y i ) on ∏ i = 1 k S n i . F(x,y)=\sum_{i=1}^{k}(1-x_i\cdot y_i)\qquad\text{on}\qquad\prod_{i=1}^{k}S^{n_i}. F ( x , y ) = i = 1 ∑ k ( 1 − x i ⋅ y i ) on i = 1 ∏ k S n i . The critical group is ( Z / 2 ) k (\mathbb{Z}/2)^k ( Z /2 ) k ∑ i ∈ I n i \sum_{i\in I} n_i ∑ i ∈ I n i I I I n i = d n_i=d n i = d d d d
Example 2.8 (Tori). On T m = ( R / 2 π Z ) m T^m=(\mathbb{R}/2\pi\mathbb{Z})^m T m = ( R /2 π Z ) m
F ( x , y ) = ∑ i = 1 m ( 1 − cos ( y i − x i ) ) . F(x,y)=\sum_{i=1}^{m}(1-\cos(y_i-x_i)). F ( x , y ) = i = 1 ∑ m ( 1 − cos ( y i − x i )) . Critical partners differ by 0 0 0 π \pi π x ↦ 2 x x\mapsto2x x ↦ 2 x 2 m 2^m 2 m j j j ( m j ) \binom{m}{j} ( j m ) χ ( T m ) = 0 \chi(T^m)=0 χ ( T m ) = 0
Example 2.9 (Complex full flags). Fix m ≥ 2 m\ge2 m ≥ 2 a 1 < ⋯ < a m a_1<\cdots<a_m a 1 < ⋯ < a m M M M M ≅ U ( m ) / T m M\cong U(m)/T^m M ≅ U ( m ) / T m
F ( X , Y ) = 1 2 tr ( ( X − Y ) 2 ) . F(X,Y)=\frac{1}{2}\operatorname{tr}((X-Y)^2). F ( X , Y ) = 2 1 tr (( X − Y ) 2 ) . Since tangent vectors at Y Y Y [ K , Y ] [K,Y] [ K , Y ] K K K [ X , Y ] = 0 [X,Y]=0 [ X , Y ] = 0 m ! m! m ! X X X Z Z Z Y Y Y
In that basis, the Hessian on the real two-dimensional ( i , j ) (i,j) ( i , j ) ( a i − a j ) ( a σ ( i ) − a σ ( j ) ) (a_i-a_j)(a_{\sigma(i)}-a_{\sigma(j)}) ( a i − a j ) ( a σ ( i ) − a σ ( j ) ) 2 inv ( σ ) 2\operatorname{inv}(\sigma) 2 inv ( σ ) S m S_m S m
For a i = i a_i=i a i = i c σ = 1 2 ∑ i ( i − σ ( i ) ) 2 c_\sigma=\frac{1}{2}\sum_i(i-\sigma(i))^2 c σ = 2 1 ∑ i ( i − σ ( i ) ) 2 1 1 1 S m S_m S m
H ∗ ( M ; Q ) ≅ Q [ x 1 , … , x m ] / ( σ 1 , … , σ m ) , ∣ x i ∣ = 2 , H^*(M;\mathbb{Q})\cong\mathbb{Q}[x_1,\ldots,x_m]/(\sigma_1,\ldots,\sigma_m),\qquad|x_i|=2, H ∗ ( M ; Q ) ≅ Q [ x 1 , … , x m ] / ( σ 1 , … , σ m ) , ∣ x i ∣ = 2 , where σ j \sigma_j σ j [1 ] . This family satisfies both hypotheses of Theorem 2.14 below.
First critical level
Assume again that χ ( M ) ≠ 0 \chi(M)\ne0 χ ( M ) = 0
c = min g ≠ 1 c g , S = { g ∈ G : c g = c } . c=\min_{g\ne1}c_g,\qquad S=\{g\in G:c_g=c\}. c = g = 1 min c g , S = { g ∈ G : c g = c } . For s ∈ S s\in S s ∈ S E s − → Γ s ≅ M E_s^-\to\Gamma_s\cong M E s − → Γ s ≅ M λ s \lambda_s λ s M M M e s = e ( E s − ) e_s=e(E_s^-) e s = e ( E s − )
Lemma 2.10 (First-level divisibility). For s ∈ S s\in S s ∈ S k k k
( 1 − s ∗ ) H k ( M ; Z ) ⊆ e s H k − λ s ( M ; Z ) . (4) (1-s^*)H^k(M;\mathbb{Z})\subseteq e_sH^{k-\lambda_s}(M;\mathbb{Z}). \tag*{(4)} ( 1 − s ∗ ) H k ( M ; Z ) ⊆ e s H k − λ s ( M ; Z ) . (4) Moreover e s ≠ 0 e_s\ne0 e s = 0 F F F M × M M\times M M × M
Proof. Let X − X_- X − X + X_+ X + c c c X − ≃ Δ X_-\simeq\Delta X − ≃ Δ [2 ] ,
H k ( X + , X − ; Z ) ≅ ⨁ h ∈ S H k − λ h ( M ; Z ) . H^k(X_+,X_-;\mathbb{Z})\cong\bigoplus_{h\in S}H^{k-\lambda_h}(M;\mathbb{Z}). H k ( X + , X − ; Z ) ≅ h ∈ S ⨁ H k − λ h ( M ; Z ) . For a ∈ H k ( M ; Z ) a\in H^k(M;\mathbb{Z}) a ∈ H k ( M ; Z ) π 1 ∗ a − π 2 ∗ a \pi_1^*a-\pi_2^*a π 1 ∗ a − π 2 ∗ a X − X_- X − X + X_+ X + Γ s \Gamma_s Γ s s s s e s e_s e s Γ s \Gamma_s Γ s s ∗ T M s^*TM s ∗ T M e ( E s − ) e ( E s + ) = ± e ( T M ) ≠ 0 e(E_s^- )e(E_s^+)=\pm e(TM)\ne0 e ( E s − ) e ( E s + ) = ± e ( T M ) = 0 e s ≠ 0 e_s\ne0 e s = 0
Theorem 2.11 (Quadratic extension at the first level). For every s ∈ S s\in S s ∈ S λ = λ s \lambda=\lambda_s λ = λ s B = A s ∗ B=A^{s^*} B = A s ∗
s 2 = 1 , s ∗ e s = − e s , A = B ⊕ e s B , (5) s^2=1,\qquad s^*e_s=-e_s,\qquad A=B\oplus e_sB, \tag*{(5)} s 2 = 1 , s ∗ e s = − e s , A = B ⊕ e s B , (5) where multiplication by e s e_s e s B j B^j B j A j + λ A^{j+\lambda} A j + λ
P M ( t ) = ( 1 + t λ ) P B ( t ) . (6) P_M(t)=(1+t^\lambda)P_B(t). \tag*{(6)} P M ( t ) = ( 1 + t λ ) P B ( t ) . (6) The transformation s s s B B B n − λ n-\lambda n − λ
Proof. By (4) , s ∗ s^* s ∗ λ \lambda λ s ∗ − 1 s^*-1 s ∗ − 1 λ \lambda λ s s s u ∈ H λ ( M × M ; Q ) u\in H^\lambda(M\times M;\mathbb{Q}) u ∈ H λ ( M × M ; Q ) 0 0 0 Δ \Delta Δ e s e_s e s Γ s \Gamma_s Γ s 0 0 0
Write the Künneth decomposition as
u = π 1 ∗ a + π 2 ∗ b + ∑ i π 1 ∗ a i π 2 ∗ b i , 0 < ∣ a i ∣ , ∣ b i ∣ < λ . u=\pi_1^*a+\pi_2^*b+\sum_i\pi_1^*a_i\pi_2^*b_i,\qquad0<|a_i|,|b_i|<\lambda. u = π 1 ∗ a + π 2 ∗ b + i ∑ π 1 ∗ a i π 2 ∗ b i , 0 < ∣ a i ∣ , ∣ b i ∣ < λ . Restriction to Δ \Delta Δ s ∗ s^* s ∗ Γ s \Gamma_s Γ s e s = ( s ∗ − 1 ) b ≠ 0 e_s=(s^*-1)b\ne0 e s = ( s ∗ − 1 ) b = 0 s ∗ − 1 s^*-1 s ∗ − 1 λ \lambda λ − 1 -1 − 1 s ∗ = ( s − 1 ) ∗ s^*=(s^{-1})^* s ∗ = ( s − 1 ) ∗ s ≠ s − 1 s\ne s^{-1} s = s − 1 u ∣ Γ s − 1 = e s u|_{\Gamma_{s^{-1}}}=e_s u ∣ Γ s − 1 = e s s 2 = 1 s^2=1 s 2 = 1 s ∗ e s = − e s s^*e_s=-e_s s ∗ e s = − e s
Let A + A^+ A + A − A^- A − A − = e s A + A^-=e_sA^+ A − = e s A + e s e_s e s (4) over Q \mathbb{Q} Q s s s dim A + = dim A − \dim A^+=\dim A^- dim A + = dim A − e s ⋅ − : A + → A − e_s\cdot-:A^+\to A^- e s ⋅ − : A + → A − (5) and (6) .
If m m m B B B n = m + λ n=m+\lambda n = m + λ A n = e s B m A^n=e_sB^m A n = e s B m s s s B m ≅ Q B^m\cong\mathbb{Q} B m ≅ Q a ∈ B i a\in B^i a ∈ B i A A A a a a e s b e_sb e s b b ∈ B m − i b\in B^{m-i} b ∈ B m − i a b ≠ 0 ab\ne0 ab = 0 B B B
Divided differences
Remark 2.12 (Divided differences). There is a unique rational operator ∂ s : A → B \partial_s:A\to B ∂ s : A → B − λ -\lambda − λ
e s ∂ s ( a ) = a − s ∗ a . e_s\partial_s(a)=a-s^*a. e s ∂ s ( a ) = a − s ∗ a . It satisfies ∂ s 2 = 0 \partial_s^2=0 ∂ s 2 = 0 ∂ s ( e s ) = 2 \partial_s(e_s)=2 ∂ s ( e s ) = 2
∂ s ( a b ) = ∂ s ( a ) b + s ∗ ( a ) ∂ s ( b ) . \partial_s(ab)=\partial_s(a)b+s^*(a)\partial_s(b). ∂ s ( ab ) = ∂ s ( a ) b + s ∗ ( a ) ∂ s ( b ) . Indeed, multiplication by e s e_s e s B B B a b − s ∗ ( a b ) ab-s^*(ab) ab − s ∗ ( ab ) e s e_s e s A ≅ B [ u ] / ( u 2 − e s 2 ) A\cong B[u]/(u^2-e_s^2) A ≅ B [ u ] / ( u 2 − e s 2 ) e s 2 = 0 e_s^2=0 e s 2 = 0 M / ⟨ s ⟩ M/\langle s\rangle M / ⟨ s ⟩ n n n n − λ n-\lambda n − λ
Corollary 2.13 (Topological and group obstructions). (i) If P M ( t ) = 1 + k t d + t 2 d P_M(t)=1+kt^d+t^{2d} P M ( t ) = 1 + k t d + t 2 d d > 0 d>0 d > 0 k > 0 k>0 k > 0 k = 2 k=2 k = 2 # k ( S 2 m × S 2 m ) \#_k(S^{2m}\times S^{2m}) # k ( S 2 m × S 2 m ) m ≥ 1 m\ge1 m ≥ 1 k ≥ 2 k\ge2 k ≥ 2
(ii) If G G G G ≅ Z / 2 G \cong\mathbb{Z}/2 G ≅ Z /2 M M M
Proof. For (i), the first positive index is d d d 2 d 2d 2 d 2 d 2d 2 d P M = 1 + t 2 d P_M=1+t^{2d} P M = 1 + t 2 d d d d 1 + t d 1+t^d 1 + t d 1 + k t d + t 2 d 1+kt^d+t^{2d} 1 + k t d + t 2 d k = 2 k=2 k = 2 2 k 2k 2 k
For (ii), the unique index-n n n λ = n \lambda=n λ = n (5) gives P M = 1 + t n P_M=1+t^n P M = 1 + t n ∣ G ∣ = 2 |G|=2 ∣ G ∣ = 2 M M M
Reflection groups and the 2, 4, 8 restriction
Theorem 2.14 (Conditional recognition). Assume, in addition to χ ( M ) ≠ 0 \chi(M)\ne0 χ ( M ) = 0
(a) A = H ∗ ( M ; Q ) A=H^*(M;\mathbb{Q}) A = H ∗ ( M ; Q ) V = H d ( M ; Q ) V=H^d(M;\mathbb{Q}) V = H d ( M ; Q ) d > 0 d>0 d > 0
(b) S S S G G G
Set ℓ = dim Q V \ell=\dim_{\mathbb{Q}}V ℓ = dim Q V d d d G G G V V V
A ≅ Sym Q ( V ) / ⟨ Sym Q ( V ) + G ⟩ , ∣ V ∣ = d . (7) A\cong\operatorname{Sym}_{\mathbb{Q}}(V)/\langle\operatorname{Sym}_{\mathbb{Q}}(V)^G_+\rangle,\qquad|V|=d. \tag*{(7)} A ≅ Sym Q ( V ) / ⟨ Sym Q ( V ) + G ⟩ , ∣ V ∣ = d . (7) The manifold is formal. If m 1 , … , m ℓ m_1,\ldots,m_\ell m 1 , … , m ℓ
P M ( t ) = ∏ i = 1 ℓ ( 1 + t d + ⋯ + t d ( m i − 1 ) ) , dim M = d ∑ i = 1 ℓ ( m i − 1 ) . (8) P_M(t)=\prod_{i=1}^{\ell}(1+t^d+\cdots+t^{d(m_i-1)}),\qquad\dim M=d\sum_{i=1}^{\ell}(m_i-1). \tag*{(8)} P M ( t ) = i = 1 ∏ ℓ ( 1 + t d + ⋯ + t d ( m i − 1 ) ) , dim M = d i = 1 ∑ ℓ ( m i − 1 ) . (8) Furthermore:
(i) If G G G d ∈ { 2 , 4 , 8 } d\in\{2,4,8\} d ∈ { 2 , 4 , 8 }
(ii) If d ∉ { 2 , 4 , 8 } d\notin\{2,4,8\} d ∈ / { 2 , 4 , 8 } G ≅ ( Z / 2 ) ℓ G\cong(\mathbb{Z}/2)^\ell G ≅ ( Z /2 ) ℓ M ≃ Q ( S d ) ℓ M\simeq_{\mathbb{Q}}(S^d)^\ell M ≃ Q ( S d ) ℓ
(iii) If d = 2 d=2 d = 2 M ≃ Q K / T M\simeq_{\mathbb{Q}}K/T M ≃ Q K / T K K K G G G T T T
Proof. Reflections. If a deck transformation fixes V V V A A A χ ( M ) > 0 \chi(M)>0 χ ( M ) > 0 V V V d d d s ∈ S s\in S s ∈ S λ s > d \lambda_s>d λ s > d (4) would make it fix V V V λ s = d \lambda_s=d λ s = d
In degree d d d (4) gives
a − s ∗ a = e s D s ( a ) , D s : H d ( M ; Z ) ⟶ Z . (9) a-s^*a=e_sD_s(a),\qquad D_s:H^d(M;\mathbb{Z})\longrightarrow\mathbb{Z}. \tag*{(9)} a − s ∗ a = e s D s ( a ) , D s : H d ( M ; Z ) ⟶ Z . (9) The integer coefficient is unique because e s ≠ 0 e_s\ne0 e s = 0 H d ( M ; Z ) H^d(M;\mathbb{Z}) H d ( M ; Z ) a a a 1 − s ∗ 1-s^* 1 − s ∗ V V V s ∗ s^* s ∗ G G G [3 ] . It is essential because V G = 0 V^G=0 V G = 0 (1) .
The spherical Euler obstruction. Choose a G G G V R V_{\mathbb{R}} V R e s e_s e s
D s ( e s ) = 2 , D s ( v ) = 2 ⟨ v , e s ⟩ ⟨ e s , e s ⟩ . D_s(e_s)=2,\qquad D_s(v)=\frac{2\langle v,e_s\rangle}{\langle e_s,e_s\rangle}. D s ( e s ) = 2 , D s ( v ) = ⟨ e s , e s ⟩ 2 ⟨ v , e s ⟩ . If G G G s , t ∈ S s,t \in S s , t ∈ S a = D s ( e t ) a = D_s(e_t) a = D s ( e t ) b = D t ( e s ) b = D_t(e_s) b = D t ( e s )
a b = 4 ⟨ e s , e t ⟩ 2 ⟨ e s , e s ⟩ ⟨ e t , e t ⟩ ∈ { 1 , 2 , 3 } . ab = \frac{4\langle e_s,e_t\rangle^2}{\langle e_s,e_s\rangle\langle e_t,e_t\rangle} \in\{1,2,3\}. ab = ⟨ e s , e s ⟩ ⟨ e t , e t ⟩ 4 ⟨ e s , e t ⟩ 2 ∈ { 1 , 2 , 3 } . Indeed, noncommuting reflection normals are neither parallel nor perpendicular, so 0 < a b < 4 0 < ab < 4 0 < ab < 4 D s ( e t ) D_s(e_t) D s ( e t )
Integral homology vanishes in degrees 0 < i < d 0 < i < d 0 < i < d M M M ( d − 1 ) (d-1) ( d − 1 ) D s D_s D s z s ∈ H d ( M ; Z ) z_s \in H_d(M;\mathbb{Z}) z s ∈ H d ( M ; Z ) f : S d → M f:S^d \to M f : S d → M d d d f ∗ E t f^*E_t f ∗ E t
⟨ e ( f ∗ E t ) , [ S d ] ⟩ = D s ( e t ) , \langle e(f^*E_t),[S^d]\rangle= D_s(e_t), ⟨ e ( f ∗ E t ) , [ S d ]⟩ = D s ( e t ) , which is odd. Its Thom space has cells in dimensions d d d 2 d 2d 2 d d ∈ { 2 , 4 , 8 } d \in\{2,4,8\} d ∈ { 2 , 4 , 8 } [4 , 5 ] . This proves (i).
Coinvariants. The surjection Sym ( V ) → A \operatorname{Sym}(V) \to A Sym ( V ) → A A G = Q A^G = \mathbb{Q} A G = Q ∣ G ∣ |G| ∣ G ∣ [6 ] , while dim A = ∣ G ∣ \dim A = |G| dim A = ∣ G ∣ (7) . The standard Hilbert series of the coinvariant complete intersection gives (8) .
Rational homotopy. Let f 1 , … , f ℓ f_1,\ldots,f_\ell f 1 , … , f ℓ m i ≥ 2 m_i \ge2 m i ≥ 2
( Λ ( x 1 , … , x ℓ , y 1 , … , y ℓ ) , D ) , { ∣ x i ∣ = d , D x i = 0 , ∣ y i ∣ = d m i − 1 , D y i = f i ( x ) , (10) (\Lambda(x_1,\ldots,x_\ell,y_1,\ldots,y_\ell),D), \qquad
\begin{cases}
|x_i|=d, & Dx_i=0,\\
|y_i|=dm_i-1, & Dy_i=f_i(x),
\end{cases} \tag*{(10)} ( Λ ( x 1 , … , x ℓ , y 1 , … , y ℓ ) , D ) , { ∣ x i ∣ = d , ∣ y i ∣ = d m i − 1 , D x i = 0 , D y i = f i ( x ) , (10) has cohomology A A A x i x_i x i A P L ( M ) A_{\mathrm{PL}}(M) A PL ( M ) f i ( x ) f_i(x) f i ( x ) A P L ( M ) A_{\mathrm{PL}}(M) A PL ( M ) ( A , 0 ) (A,0) ( A , 0 ) [7 ] . For d = 2 d=2 d = 2 G G G
If d ∉ { 2 , 4 , 8 } d \notin\{2,4,8\} d ∈ / { 2 , 4 , 8 } V V V G = ( Z / 2 ) ℓ G=(\mathbb{Z}/2)^\ell G = ( Z /2 ) ℓ (7) is Q [ x 1 , … , x ℓ ] / ( x 1 2 , … , x ℓ 2 ) \mathbb{Q}[x_1,\ldots,x_\ell]/(x_1^2,\ldots,x_\ell^2) Q [ x 1 , … , x ℓ ] / ( x 1 2 , … , x ℓ 2 ) ( S d ) ℓ (S^d)^\ell ( S d ) ℓ
Remark 2.15 (Limits of recognition). The generation assumptions are additional hypotheses, not consequences of the kernel axioms. The conclusion for d = 2 d=2 d = 2 d = 4 d=4 d = 4 8 8 8 d ∈ { 2 , 4 , 8 } d \in\{2,4,8\} d ∈ { 2 , 4 , 8 }
A twisted product
Example 2.16 (A nonabelian group in degree six). Let N = ( S 6 ) 4 N=(S^6)^4 N = ( S 6 ) 4 F 0 ( x , y ) = ∑ i = 1 4 ( 1 − x i ⋅ y i ) F_0(x,y)=\sum_{i=1}^4(1-x_i\cdot y_i) F 0 ( x , y ) = ∑ i = 1 4 ( 1 − x i ⋅ y i ) σ \sigma σ 1 , 2 1,2 1 , 2 3 , 4 3,4 3 , 4 M = N × S 6 M=N\times S^6 M = N × S 6 t = p ⋅ q t=p\cdot q t = p ⋅ q
Φ ( ( x , p ) , ( y , q ) ) = 10 ( 1 − t ) + 1 + t 2 F 0 ( x , y ) + 1 − t 2 F 0 ( x , σ y ) . (11) \Phi((x,p),(y,q))=10(1-t)+\frac{1+t}{2}F_0(x,y)+\frac{1-t}{2}F_0(x,\sigma y). \tag*{(11)} Φ (( x , p ) , ( y , q )) = 10 ( 1 − t ) + 2 1 + t F 0 ( x , y ) + 2 1 − t F 0 ( x , σ y ) . (11) This is a critical kernel whose critical group is the nonabelian group
G = ( Z / 2 ) 4 ⋊ Z / 2 , G=(\mathbb{Z}/2)^4\rtimes\mathbb{Z}/2, G = ( Z /2 ) 4 ⋊ Z /2 , where the last involution acts on the four sign changes by the double interchange. Nevertheless H ∗ ( M ; Q ) H^*(M;\mathbb{Q}) H ∗ ( M ; Q ) 6 6 6
Symmetry follows because σ \sigma σ F 0 F_0 F 0 q q q
[ − 10 + 1 2 ( F 0 ( x , y ) − F 0 ( x , σ y ) ) ] d q t . \left[-10+\frac{1}{2}\left(F_0(x,y)-F_0(x,\sigma y)\right)\right]\,\mathrm{d}q_t. [ − 10 + 2 1 ( F 0 ( x , y ) − F 0 ( x , σ y ) ) ] d q t . Since 0 ≤ F 0 ≤ 8 0\leq F_0\leq8 0 ≤ F 0 ≤ 8 q = ± p q=\pm p q = ± p
( k x , p ) or ( σ k x , − p ) , k ∈ ( Z / 2 ) 4 . (kx,p)\quad\text{or}\quad(\sigma kx,-p),\qquad k\in(\mathbb{Z}/2)^4. ( k x , p ) or ( σ k x , − p ) , k ∈ ( Z /2 ) 4 . They are the orbits of the stated free action: the additional involution is ( x , p ) ↦ ( σ x , − p ) (x,p)\mapsto(\sigma x,-p) ( x , p ) ↦ ( σ x , − p ) q q q
If k k k j j j
( value , index ) = ( 2 j , 6 j ) , ( 20 + 2 j , 6 j + 6 ) . (\text{value},\text{index})=(2j,6j),\qquad(20+2j,6j+6). ( value , index ) = ( 2 j , 6 j ) , ( 20 + 2 j , 6 j + 6 ) . The first positive level contains only the four individual sign changes on N N N ( Z / 2 ) 4 (\mathbb{Z}/2)^4 ( Z /2 ) 4 Theorem 2.14 cannot be omitted.
Related work
The geometric tool is Morse–Bott theory [2 ] ; the algebraic and homotopical inputs are Chevalley’s reflection invariant theory [6 ] , Adams’s Hopf invariant theorem [4 ] , and Sullivan’s rational homotopy theory [7 ] . The odd-Euler-number formulation used above is explained in [5 Section 3.2] .
The closest geometric antecedent is Hsiang–Palais–Terng [8 ] : Corollary 2.11 gives a 2, 4, 8 restriction for common even multiplicities under their irreducibility and rank hypotheses, and Theorem 6.12 describes cohomology by Weyl coinvariants generated by Euler classes of curvature distributions. Mare [1 Theorem 1.2 and Lemma 3.3] gives closely related formulas for real flag manifolds, including Euler-class reflection relations. These conclusions and divided-difference patterns are established mathematics. Under the hypotheses of Theorem 2.14 , the critical-kernel axioms and first-level Thom attachments yield an intrinsic recognition criterion. This does not give a complete classification.
We consider finite-dimensional linear spaces of functions modulo constants whose nonzero elements are Morse. The questions concern their possible ranks and their extension to larger such spaces.
Definitions and openness
Throughout this section, M = M n M=M^n M = M n n ≥ 1 n\geq1 n ≥ 1 M ( M ) ⊂ C ∞ ( M , R ) \mathcal{M}(M)\subset C^\infty(M,\mathbb{R}) M ( M ) ⊂ C ∞ ( M , R ) C ∞ C^\infty C ∞
Definition 3.1. A uniformly Morse linear system of rank r ≥ 1 r\geq1 r ≥ 1 r r r V ⊂ C ∞ ( M , R ) / R V\subset C^\infty(M,\mathbb{R})/\mathbb{R} V ⊂ C ∞ ( M , R ) / R V V V
Define μ min ( M ) \mu_{\min}(M) μ m i n ( M ) maximal by inclusion if it is contained in no strictly larger such system.
Adding constants does not affect Morseness. To discuss actual families of functions, choose representatives linearly, for instance by requiring zero mean with respect to a fixed positive smooth density. A basis [ f 1 ] , … , [ f r ] [f_1],\ldots,[f_r] [ f 1 ] , … , [ f r ] V V V
F = ( f 1 , … , f r ) : M ⟶ R r , f a = a ⋅ F ( a ∈ R r ) . F=(f_1,\ldots,f_r):M\longrightarrow\mathbb{R}^r,\qquad f_a=a\cdot F\quad(a\in\mathbb{R}^r). F = ( f 1 , … , f r ) : M ⟶ R r , f a = a ⋅ F ( a ∈ R r ) . The condition is precisely f a ∈ M ( M ) f_a \in\mathcal{M}(M) f a ∈ M ( M ) a ≠ 0 a \ne0 a = 0 a ∈ S r − 1 a \in S^{r-1} a ∈ S r − 1 F F F F F F A F + b AF+b A F + b A ∈ GL ( r , R ) A \in\operatorname{GL}(r,\mathbb{R}) A ∈ GL ( r , R ) V V V
Proposition 3.2 (Openness). For fixed r r r ( f 1 , … , f r ) (f_1,\ldots,f_r) ( f 1 , … , f r ) C 2 C^2 C 2
Proof. Choose a Riemannian metric and its covariant Hessian. On the compact set S r − 1 × M S^{r-1} \times M S r − 1 × M
∥ d f a ( x ) ∥ 2 + σ min ( ∇ 2 f a ( x ) ) 2 > 0 , \lVert df_a(x)\rVert^2+\sigma_{\min}\left(\nabla^2 f_a(x)\right)^2>0, ∥ d f a ( x ) ∥ 2 + σ m i n ( ∇ 2 f a ( x ) ) 2 > 0 , where σ min \sigma_{\min} σ m i n C 2 C^2 C 2
Critical incidence and the rank bound
Define
C ( V ) = { ( a , x ) ∈ S r − 1 × M : d f a ( x ) = 0 } , C i ( V ) = { ( a , x ) ∈ C ( V ) : ind x ( f a ) = i } . C(V)=\{(a,x)\in S^{r-1}\times M:df_a(x)=0\},\qquad C_i(V)=\{(a,x)\in C(V):\operatorname{ind}_x(f_a)=i\}. C ( V ) = {( a , x ) ∈ S r − 1 × M : d f a ( x ) = 0 } , C i ( V ) = {( a , x ) ∈ C ( V ) : ind x ( f a ) = i } . Let π ( a , x ) = a \pi(a,x)=a π ( a , x ) = a ev ( a , x ) = x \operatorname{ev}(a,x)=x ev ( a , x ) = x
Proposition 3.3 (Critical covering). The space C ( V ) C(V) C ( V ) ( r − 1 ) (r-1) ( r − 1 ) π \pi π C i ( V ) C_i(V) C i ( V ) r ≥ 2 r\ge2 r ≥ 2 m i m_i m i i i i a a a
m i = m n − i , χ ( M ) = ∑ i = 0 n ( − 1 ) i m i . m_i=m_{n-i},\qquad\chi(M)=\sum_{i=0}^{n}(-1)^i m_i. m i = m n − i , χ ( M ) = i = 0 ∑ n ( − 1 ) i m i . If r ≥ 3 r\ge3 r ≥ 3 S r − 1 S^{r-1} S r − 1
Proof. At a zero of d f a df_a d f a M M M π \pi π π \pi π r ≥ 2 r\ge2 r ≥ 2 f − a = − f a f_{-a}=-f_a f − a = − f a i i i n − i n-i n − i S r − 1 S^{r-1} S r − 1 r ≥ 3 r\ge3 r ≥ 3
In rank two, monodromy can permute critical points: a covering of S 1 S^1 S 1 c : S r − 1 → M c:S^{r-1}\to M c : S r − 1 → M
E c − ⊕ E c + ≅ c ∗ T M , rank E c − = ind ( c ) . E_c^- \oplus E_c^+ \cong c^*TM,\qquad\operatorname{rank}E_c^-=\operatorname{ind}(c). E c − ⊕ E c + ≅ c ∗ T M , rank E c − = ind ( c ) . More generally these bundles are defined over C i ( V ) C_i(V) C i ( V ) [9 §4.1] .
Lemma 3.4 (Minimum embedding). The map ev : C 0 ( V ) → M \operatorname{ev}:C_0(V)\to M ev : C 0 ( V ) → M x x x f a f_a f a
ker ( d F x ∗ ) = R a , rank d F x = r − 1. (12) \ker(dF_x^*)=\mathbb{R}a,\qquad\operatorname{rank}dF_x=r-1. \tag*{(12)} ker ( d F x ∗ ) = R a , rank d F x = r − 1. (12) Proof. Put K x = ker ( d F x ∗ ) K_x=\ker(dF_x^*) K x = ker ( d F x ∗ ) H b = Hess x f b H_b=\operatorname{Hess}_x f_b H b = Hess x f b b ∈ K x b\in K_x b ∈ K x H a H_a H a b ∈ K x b\in K_x b ∈ K x a a a H a − 1 / 2 H b H a − 1 / 2 H_a^{-1/2}H_bH_a^{-1/2} H a − 1/2 H b H a − 1/2
λ \lambda λ H b − λ H a H_b-\lambda H_a H b − λ H a [ f b − λ a ] [f_{b-\lambda a}] [ f b − λa ]
Two minimum directions at the same point must now be proportional, hence equal: opposite directions exchange minima and maxima. Therefore ev \operatorname{ev} ev ( b , v ) (b,v) ( b , v ) C 0 ( V ) C_0(V) C 0 ( V )
b ⊥ a , H a ( v , ⋅ ) + d F x ∗ b = 0. (13) b \perp a,\qquad H_a(v,\cdot)+dF_x^*b=0. \tag*{(13)} b ⊥ a , H a ( v , ⋅ ) + d F x ∗ b = 0. (13) If d ev ( b , v ) = v = 0 d\operatorname{ev}(b,v)=v=0 d ev ( b , v ) = v = 0 b ∈ R a ∩ a ⊥ = 0 b\in\mathbb{R}a\cap a^\perp=0 b ∈ R a ∩ a ⊥ = 0 ev \operatorname{ev} ev
The lemma applies to all local minima. The maximum locus has the same image in M M M ( a , x ) ↦ ( − a , x ) (a,x)\mapsto(-a,x) ( a , x ) ↦ ( − a , x )
Theorem 3.5 (Sharp bound and sphere rigidity). For every closed connected n n n
1 ≤ μ min ( M ) ≤ n + 1 , μ min ( M ) = n + 1 ⟺ M ≅ S n 1\leq\mu_{\min}(M)\leq n+1,\qquad\mu_{\min}(M)=n+1\quad\Longleftrightarrow\quad M\cong S^n 1 ≤ μ m i n ( M ) ≤ n + 1 , μ m i n ( M ) = n + 1 ⟺ M ≅ S n where ≅ \cong ≅ n ≥ 2 n\geq2 n ≥ 2 n + 1 n+1 n + 1 R n + 1 \mathbb{R}^{n+1} R n + 1
Proof. A Morse function gives rank one. At a minimum, (3.1) gives r − 1 ≤ n r-1\leq n r − 1 ≤ n r = n + 1 r=n+1 r = n + 1 C 0 ( V ) → M C_0(V)\to M C 0 ( V ) → M M M M M M M S n S^n S n S n S^n S n n ≥ 2 n\geq2 n ≥ 2 n = 1 n=1 n = 1 S n ⊂ R n + 1 S^n\subset\mathbb{R}^{n+1} S n ⊂ R n + 1 n + 1 n+1 n + 1
For n ≥ 2 n\geq2 n ≥ 2 M M M (12) then implies that any critical height at that point is a multiple of a definite one. There are no saddle points. Also d F dF d F n n n F ( x ) = F ( y ) F(x)=F(y) F ( x ) = F ( y ) x x x y = x y=x y = x F F F F ( M ) F(M) F ( M )
Remark 3.6. The final assertion requires n ≥ 2 n\geq2 n ≥ 2 S 1 S^1 S 1 cos ( k θ ) , sin ( k θ ) \cos(k\theta),\sin(k\theta) cos ( k θ ) , sin ( k θ ) k k k k k k
Obstructions to linear extension
A linear extension of V V V W ⊃ V W\supset V W ⊃ V r + 1 r+1 r + 1
α V : S r − 1 ⟶ M ( M ) , a ⟼ f a , \alpha_V:S^{r-1}\longrightarrow\mathcal{M}(M),\qquad a\longmapsto f_a, α V : S r − 1 ⟶ M ( M ) , a ⟼ f a , is the equatorial restriction of α W : S r → M ( M ) \alpha_W:S^r\to\mathcal{M}(M) α W : S r → M ( M ) α ( − a ) = − α ( a ) \alpha(-a)=-\alpha(a) α ( − a ) = − α ( a )
Proposition 3.7 (Family obstructions). If a rank-r r r α V \alpha_V α V D r D^r D r r > 2 r>2 r > 2 S r − 1 → M S^{r-1}\to M S r − 1 → M
S r − 1 ⟶ Conf N ( M ) , a ⟼ ( c 1 ( a ) , … , c N ( a ) ) , S^{r-1}\longrightarrow\operatorname{Conf}_N(M),\qquad a\longmapsto(c_1(a),\ldots,c_N(a)), S r − 1 ⟶ Conf N ( M ) , a ⟼ ( c 1 ( a ) , … , c N ( a )) , extends over D r D^r D r Conf N ( M ) \operatorname{Conf}_N(M) Conf N ( M ) N N N
Proof. Restrict the extended family to a closed hemisphere. For r ≥ 2 r \ge2 r ≥ 2 S r S^r S r
These necessary conditions do not suffice for linear extension, even when a smooth odd nonlinear extension exists, as Theorem 3.10 shows.
Proposition 3.8 (An unlink obstruction on S 3 S^3 S 3 For a rank-two system on S 3 S^3 S 3
L 0 ( V ) = ev ( C 0 ( V ) ) ⊂ S 3 L_0(V)=\operatorname{ev}(C_0(V))\subset S^3 L 0 ( V ) = ev ( C 0 ( V )) ⊂ S 3 is a smooth link. If V V V S 1 S^1 S 1 L 0 ( V ) L_0(V) L 0 ( V ) S 3 S^3 S 3
Proof. Lemma 3.4 embeds the finite union of covering circles. In an extension, minimum incidence is a disjoint union of copies of S 2 S^2 S 2 S 3 S^3 S 3
Examples and Hessian bundles
Spheres and tori
On S n S^n S n μ l i n ( S n ) = n + 1 \mu_{\mathrm{lin}}(S^n)=n+1 μ lin ( S n ) = n + 1 T n = ( R / 2 π Z ) n T^n=(\mathbb{R}/2\pi\mathbb{Z})^n T n = ( R /2 π Z ) n c j > 0 c_j>0 c j > 0
F ( θ 1 , … , θ n ) = ∑ j = 1 n c j e i θ j ∈ C . F(\theta_1,\ldots,\theta_n)=\sum_{j=1}^{n}c_j e^{i\theta_j}\in\mathbb{C}. F ( θ 1 , … , θ n ) = j = 1 ∑ n c j e i θ j ∈ C . The height Re ( e − i ϕ F ) = ∑ j c j cos ( θ j − ϕ ) \operatorname{Re}(e^{-i\phi}F)=\sum_j c_j\cos(\theta_j-\phi) Re ( e − i ϕ F ) = ∑ j c j cos ( θ j − ϕ ) 2 n 2^n 2 n m i = ( n i ) m_i=\binom{n}{i} m i = ( i n ) μ l i n ( T 2 ) = 2 \mu_{\mathrm{lin}}(T^2)=2 μ lin ( T 2 ) = 2 Theorem 3.5 . No value of μ l i n ( T n ) \mu_{\mathrm{lin}}(T^n) μ lin ( T n ) n > 2 n>2 n > 2
A maximal-rank system on C P 2 \mathbb{CP}^2 CP 2
For a = ( a 1 , a 2 ) ∈ C 2 ≅ R 4 a=(a_1,a_2)\in\mathbb{C}^2\cong\mathbb{R}^4 a = ( a 1 , a 2 ) ∈ C 2 ≅ R 4
A ( a ) = ( 0 0 a 1 0 0 a 2 a ‾ 1 a ‾ 2 0 ) , f a ( [ z ] ) = z ∗ A ( a ) z z ∗ z . (14) A(a)=\begin{pmatrix}0&0&a_1\\0&0&a_2\\\overline{a}_1&\overline{a}_2&0\end{pmatrix},\qquad f_a([z])=\frac{z^*A(a)z}{z^*z}. \tag*{(14)} A ( a ) = 0 0 a 1 0 0 a 2 a 1 a 2 0 , f a ([ z ]) = z ∗ z z ∗ A ( a ) z . (14) For a ≠ 0 a\ne0 a = 0 − ∥ a ∥ -\lVert a\rVert − ∥ a ∥ 0 0 0 ∥ a ∥ \lVert a\rVert ∥ a ∥ j j j
∑ ℓ ≠ j ( λ ℓ − λ j ) ∣ w ℓ ∣ 2 . \sum_{\ell\ne j}(\lambda_\ell-\lambda_j)|w_\ell|^2. ℓ = j ∑ ( λ ℓ − λ j ) ∣ w ℓ ∣ 2 . Hence (3.3) is a rank-four system with indices 0 , 2 , 4 0,2,4 0 , 2 , 4 C P 2 \mathbb{CP}^2 CP 2 S 4 S^4 S 4 Theorem 3.5 gives
μ l i n ( C P 2 ) = 4. (15) \mu_{\mathrm{lin}}(\mathbb{CP}^2)=4. \tag*{(15)} μ lin ( CP 2 ) = 4. (15) Its representing map is, up to a constant factor, the first four coordinates of the smooth collapse C P 2 → S 4 \mathbb{CP}^2\to S^4 CP 2 → S 4
[ z : w ] ⟼ ( 2 z w ‾ , ∣ z ∣ 2 − ∣ w ∣ 2 ) ∣ z ∣ 2 + ∣ w ∣ 2 , z ∈ C 2 , w ∈ C . [z:w]\longmapsto\frac{(2z\overline{w},\,|z|^2-|w|^2)}{|z|^2+|w|^2},\qquad z\in\mathbb{C}^2,\ w\in\mathbb{C}. [ z : w ] ⟼ ∣ z ∣ 2 + ∣ w ∣ 2 ( 2 z w , ∣ z ∣ 2 − ∣ w ∣ 2 ) , z ∈ C 2 , w ∈ C . This collapses the hyperplane w = 0 w=0 w = 0 [10 §16] for this construction.
Infinitely many Hessian-bundle types on S 2 × S 2 S^2 \times S^2 S 2 × S 2
Let h : S 2 → S 2 h:S^2 \to S^2 h : S 2 → S 2 d d d 0 < ε < 1 0 < \varepsilon< 1 0 < ε < 1
F h ( x , y ) = ( 1 + ε ⟨ h ( x ) , y ⟩ ) x ∈ R 3 . (16) F_h(x,y) = (1+\varepsilon\langle h(x),y\rangle)x \in\mathbb{R}^3. \tag*{(16)} F h ( x , y ) = ( 1 + ε ⟨ h ( x ) , y ⟩) x ∈ R 3 . (16) For a unit vector u u u u ⋅ F h u \cdot F_h u ⋅ F h
x = s u , y = τ h ( s u ) , s , τ ∈ { 1 , − 1 } . x=su,\qquad y=\tau h(su),\qquad s,\tau\in\{1,-1\}. x = s u , y = τ h ( s u ) , s , τ ∈ { 1 , − 1 } . Indeed u ⋅ x = 0 u\cdot x=0 u ⋅ x = 0 x x x d ( u ⋅ x ) d(u\cdot x) d ( u ⋅ x ) y y y y = τ h ( x ) y=\tau h(x) y = τ h ( x ) x x x x = s u x=su x = s u
Use local fiber coordinates centered on y = τ h ( x ) y=\tau h(x) y = τ h ( x ) T x S 2 ⊕ T h ( x ) S 2 T_xS^2\oplus T_{h(x)}S^2 T x S 2 ⊕ T h ( x ) S 2
− s ( 1 + ε τ ) g S 2 ⊕ ( − s ε τ ) g S 2 . (17) -s(1+\varepsilon\tau)g_{S^2}\oplus(-s\varepsilon\tau)g_{S^2}. \tag*{(17)} − s ( 1 + ε τ ) g S 2 ⊕ ( − s ε τ ) g S 2 . (17) Thus (16) gives a rank-three system with indices 0 , 2 , 2 , 4 0,2,2,4 0 , 2 , 2 , 4
{ 2 , 2 ∣ d ∣ } . (18) \{2,\;2|d|\}. \tag*{(18)} { 2 , 2∣ d ∣ } . (18) For ( s , τ ) = ( 1 , − 1 ) (s,\tau)=(1,-1) ( s , τ ) = ( 1 , − 1 ) T S 2 TS^2 T S 2 u u u ( − 1 , − 1 ) (-1,-1) ( − 1 , − 1 ) ( h ∘ ( − id ) ) ∗ T S 2 (h\circ(-\operatorname{id}))^*TS^2 ( h ∘ ( − id ) ) ∗ T S 2 2 ∣ d ∣ 2|d| 2∣ d ∣
In any smooth deformation through admissible triples, critical branches continue and their Hessian bundles extend over S 2 × [ 0 , 1 ] S^2\times[0,1] S 2 × [ 0 , 1 ] (18) is invariant. Taking d = 0 , 1 , 2 , … d=0,1,2,\ldots d = 0 , 1 , 2 , … [11 Theorem 5.7(3), Remark 5.8] , where the associated self-intersection data also appear.
Joint critical-point configurations
For the unit-parameter S 3 S^3 S 3 (14) , the maximum and minimum branches are
c + ( a ) = [ a 1 : a 2 : 1 ] , c − ( a ) = [ − a 1 : − a 2 : 1 ] . c_+(a)=[a_1:a_2:1],\qquad c_-(a)=[-a_1:-a_2:1]. c + ( a ) = [ a 1 : a 2 : 1 ] , c − ( a ) = [ − a 1 : − a 2 : 1 ] . The middle branch is [ − a ‾ 2 : a ‾ 1 : 0 ] [-\overline{a}_2:\overline{a}_1:0] [ − a 2 : a 1 : 0 ] S 1 → S 5 → C P 2 S^1\to S^5\to\mathbb{CP}^2 S 1 → S 5 → CP 2 π 3 ( C P 2 ) = 0 \pi_3(\mathbb{CP}^2)=0 π 3 ( CP 2 ) = 0 S 3 S^3 S 3 H 2 ( S 3 ; Z ) = 0 H^2(S^3;\mathbb{Z})=0 H 2 ( S 3 ; Z ) = 0 T C P 2 T\mathbb{CP}^2 T CP 2
Proposition 3.9. Nevertheless the joint extrema map ( c + , c − ) : S 3 → Conf 2 ( C P 2 ) (c_+,c_-):S^3\to\operatorname{Conf}_2(\mathbb{CP}^2) ( c + , c − ) : S 3 → Conf 2 ( CP 2 ) π 3 Conf 2 ( C P 2 ) ≅ Z \pi_3\operatorname{Conf}_2(\mathbb{CP}^2)\cong\mathbb{Z} π 3 Conf 2 ( CP 2 ) ≅ Z (14) has no disk filling through Morse functions.
Proof. Use the affine chart [ z 1 : z 2 : 1 ] ↔ ( z 1 , z 2 ) [z_1:z_2:1]\leftrightarrow(z_1,z_2) [ z 1 : z 2 : 1 ] ↔ ( z 1 , z 2 ) ( a , − a ) (a,-a) ( a , − a ) ( 0 , − a ) (0,-a) ( 0 , − a )
( a , t ) ⟼ ( ( 1 − t ) a , − a ) , 0 ≤ t ≤ 1. (a,t)\longmapsto((1-t)a,-a),\qquad0\le t\le1. ( a , t ) ⟼ (( 1 − t ) a , − a ) , 0 ≤ t ≤ 1. The Fadell–Neuwirth fibration [12 ] is
C P 2 ∖ { p } ⟶ Conf 2 ( C P 2 ) → pr 1 C P 2 , p = [ 0 : 0 : 1 ] . \mathbb{CP}^2\setminus\{p\}\longrightarrow\operatorname{Conf}_2(\mathbb{CP}^2)\xrightarrow{\operatorname{pr}_1}\mathbb{CP}^2,\qquad p=[0:0:1]. CP 2 ∖ { p } ⟶ Conf 2 ( CP 2 ) pr 1 CP 2 , p = [ 0 : 0 : 1 ] . Its fiber deformation retracts to C P 1 \mathbb{CP}^1 CP 1 [ z 1 : z 2 : w ] ↦ [ z 1 : z 2 : 0 ] [z_1:z_2:w]\mapsto[z_1:z_2:0] [ z 1 : z 2 : w ] ↦ [ z 1 : z 2 : 0 ] π 4 ( C P 2 ) = π 3 ( C P 2 ) = 0 \pi_4(\mathbb{CP}^2)=\pi_3(\mathbb{CP}^2)=0 π 4 ( CP 2 ) = π 3 ( CP 2 ) = 0 π 3 \pi_3 π 3 a ↦ ( 0 , − a ) a\mapsto(0,-a) a ↦ ( 0 , − a ) a ↦ [ a 1 : a 2 ] a\mapsto[a_1:a_2] a ↦ [ a 1 : a 2 ] π 3 ( S 2 ) \pi_3(S^2) π 3 ( S 2 ) □ \square □
The preceding obstruction concerns fillings by general Morse families. The next result separates nonlinear extension from linear extension.
Linear and nonlinear extensions
Theorem 3.10 (Linked minima). There is a rank-two uniformly Morse system V V V S 3 S^3 S 3 S 2 S^2 S 2 S 3 S^3 S 3 V V V ( 2 , 3 , 3 , 2 ) (2,3,3,2) ( 2 , 3 , 3 , 2 )
Proof. Write S 3 = { ( z 1 , z 2 ) ∈ C 2 : ∣ z 1 ∣ 2 + ∣ z 2 ∣ 2 = 1 } S^3=\{(z_1,z_2)\in\mathbb{C}^2:|z_1|^2+|z_2|^2=1\} S 3 = {( z 1 , z 2 ) ∈ C 2 : ∣ z 1 ∣ 2 + ∣ z 2 ∣ 2 = 1 }
F ( z 1 , z 2 ) = z 1 [ 1 + ∣ z 2 ∣ 2 + 1 10 Re ( ( z 1 z ‾ 2 ) 2 ) ] , f = Re F , g = Im F . (19) F(z_1,z_2)=z_1\left[1+|z_2|^2+\frac{1}{10}\operatorname{Re}\left((z_1\overline{z}_2)^2\right)\right],\qquad f=\operatorname{Re}F,\quad g=\operatorname{Im}F. \tag*{(19)} F ( z 1 , z 2 ) = z 1 [ 1 + ∣ z 2 ∣ 2 + 10 1 Re ( ( z 1 z 2 ) 2 ) ] , f = Re F , g = Im F . (19) On z 2 = 0 z_2=0 z 2 = 0 F = z 1 F=z_1 F = z 1 f , g f,g f , g
F ( e i θ z 1 , e i θ z 2 ) = e i θ F ( z 1 , z 2 ) , f θ : = cos θ f + sin θ g = f ∘ R − θ , (20) F(e^{i\theta}z_1,e^{i\theta}z_2)=e^{i\theta}F(z_1,z_2),\qquad f_\theta:=\cos\theta f+\sin\theta g=f\circ R_{-\theta}, \tag*{(20)} F ( e i θ z 1 , e i θ z 2 ) = e i θ F ( z 1 , z 2 ) , f θ := cos θ f + sin θ g = f ∘ R − θ , (20) where R θ ( z 1 , z 2 ) = ( e i θ z 1 , e i θ z 2 ) R_\theta(z_1,z_2)=(e^{i\theta}z_1,e^{i\theta}z_2) R θ ( z 1 , z 2 ) = ( e i θ z 1 , e i θ z 2 )
Step 1: the Morse calculation. Away from the coordinate circles use t = ∣ z 1 ∣ 2 t=|z_1|^2 t = ∣ z 1 ∣ 2 ψ = arg ( z 1 z ‾ 2 ) \psi=\arg(z_1\overline{z}_2) ψ = arg ( z 1 z 2 ) ϕ = arg z 1 \phi=\arg z_1 ϕ = arg z 1 ε = 1 / 10 \varepsilon=1/10 ε = 1/10
f = ρ ( t , ψ ) cos ϕ , ρ = t [ 2 − t + ε t ( 1 − t ) cos 2 ψ ] > 0. f=\rho(t,\psi)\cos\phi,\qquad\rho=\sqrt{t}\left[2-t+\varepsilon t(1-t)\cos2\psi\right]>0. f = ρ ( t , ψ ) cos ϕ , ρ = t [ 2 − t + εt ( 1 − t ) cos 2 ψ ] > 0. Interior critical points have ϕ = 0 , π \phi=0,\pi ϕ = 0 , π d ρ = 0 d\rho=0 d ρ = 0 s = cos 2 ψ ∈ { 1 , − 1 } s=\cos2\psi\in\{1,-1\} s = cos 2 ψ ∈ { 1 , − 1 }
P s ( t ) = 2 − 3 t + ε s ( 3 t − 5 t 2 ) = 0. P_s(t)=2-3t+\varepsilon s(3t-5t^2)=0. P s ( t ) = 2 − 3 t + ε s ( 3 t − 5 t 2 ) = 0. For both signs, P s P_s P s [ 0 , 1 ] [0,1] [ 0 , 1 ] t s t_s t s
ρ t t = P s ′ ( t s ) 2 t s < 0 , ρ ψ ψ = − 4 ε s t s 3 / 2 ( 1 − t s ) , ρ t ψ = 0. \rho_{tt}=\frac{P_s'(t_s)}{2\sqrt{t_s}}<0,\qquad\rho_{\psi\psi}=-4\varepsilon s t_s^{3/2}(1-t_s),\qquad\rho_{t\psi}=0. ρ tt = 2 t s P s ′ ( t s ) < 0 , ρ ψ ψ = − 4 ε s t s 3/2 ( 1 − t s ) , ρ t ψ = 0. Thus the base amplitude has two nondegenerate maxima ( ψ = 0 , π ) (\psi=0,\pi) ( ψ = 0 , π ) ( ψ = π / 2 , 3 π / 2 ) (\psi=\pi/2,3\pi/2) ( ψ = π /2 , 3 π /2 ) z 2 = 0 z_2=0 z 2 = 0 ζ = z 2 / z 1 = x + i y \zeta=z_2/z_1=x+iy ζ = z 2 / z 1 = x + i y
ρ = 1 + ( 1 / 2 + ε ) x 2 + ( 1 / 2 − ε ) y 2 + O ( ∣ ζ ∣ 4 ) , \rho=1+(1/2+\varepsilon)x^2+(1/2-\varepsilon)y^2+O(|\zeta|^4), ρ = 1 + ( 1/2 + ε ) x 2 + ( 1/2 − ε ) y 2 + O ( ∣ ζ ∣ 4 ) , giving one base minimum. At z 1 = 0 z_1=0 z 1 = 0 d F = 2 d z 1 dF=2\,dz_1 d F = 2 d z 1 ϕ = 0 , π \phi=0,\pi ϕ = 0 , π
Hess ρ ⊕ ( − ρ ) , − Hess ρ ⊕ ρ . \operatorname{Hess}\rho\oplus(-\rho),\qquad-\operatorname{Hess}\rho\oplus\rho. Hess ρ ⊕ ( − ρ ) , − Hess ρ ⊕ ρ . All are nonsingular. The base maxima give indices 3, 0, the saddles 2, 1, and the minimum 1, 2, yielding ( 2 , 3 , 3 , 2 ) (2,3,3,2) ( 2 , 3 , 3 , 2 ) (20) proves uniform Morseness.
Step 2: an odd nonlinear extension. Identify S 3 S^3 S 3 q = z 1 + z 2 j q=z_1+z_2j q = z 1 + z 2 j f f f a = ( a 1 , a 2 , a 3 ) ∈ S 2 a=(a_1,a_2,a_3)\in S^2 a = ( a 1 , a 2 , a 3 ) ∈ S 2
u ( a ) = a 1 + i a 2 + j a 3 , Φ a ( q ) = f ( u ( a ) − 1 q ) . (21) u(a)=a_1+ia_2+ja_3,\qquad\Phi_a(q)=f\left(u(a)^{-1}q\right). \tag*{(21)} u ( a ) = a 1 + i a 2 + j a 3 , Φ a ( q ) = f ( u ( a ) − 1 q ) . (21) Every Φ a \Phi_a Φ a f f f Φ − a = − Φ a \Phi_{-a}=-\Phi_a Φ − a = − Φ a a = ( cos θ , sin θ , 0 ) a=(\cos\theta,\sin\theta,0) a = ( cos θ , sin θ , 0 ) f θ f_\theta f θ u ∈ Sp ( 1 ) = S 3 u\in\operatorname{Sp}(1)=S^3 u ∈ Sp ( 1 ) = S 3
Step 3: the linear obstruction. The two local minima of f θ f_\theta f θ
L ± = { e i θ ( t + , ± 1 − t + ) : θ ∈ R / 2 π Z } . L_\pm=\left\{e^{i\theta}\left(\sqrt{t_+},\pm\sqrt{1-t_+}\right):\theta\in\mathbb{R}/2\pi\mathbb{Z}\right\}. L ± = { e i θ ( t + , ± 1 − t + ) : θ ∈ R /2 π Z } . These are two distinct Hopf fibers, so ∣ lk ( L + , L − ) ∣ = 1 |\operatorname{lk}(L_+,L_-)|=1 ∣ lk ( L + , L − ) ∣ = 1 Proposition 3.8 would force a zero linking number if a rank-three linear extension existed. This contradiction proves the theorem.
Corollary 3.11 (Maximality and affine unavoidability). The system in Theorem 3.10 is maximal by inclusion although μ l i n ( S 3 ) = 4 \mu_{\mathrm{lin}}(S^3)=4 μ lin ( S 3 ) = 4 h ∈ C ∞ ( S 3 ) h\in C^\infty(S^3) h ∈ C ∞ ( S 3 ) a , b ∈ R a,b\in\mathbb{R} a , b ∈ R h + a f + b g h+af+bg h + a f + b g
Proof. Any larger containing system would contain a rank-three extension. If [ h ] ∈ V [h]\in V [ h ] ∈ V h + a f + b g h+af+bg h + a f + b g span { [ f ] , [ g ] , [ h ] } \operatorname{span}\{[f],[g],[h]\} span {[ f ] , [ g ] , [ h ]} h h h
Corollary 3.12 (Deformation obstruction and an infinite family). The system of Theorem 3.10 cannot be deformed through admissible ordered pairs to a linearly extendible system. Moreover, for every k ≥ 2 k\geq2 k ≥ 2 ( k , k + 1 , k + 1 , k ) (k,k+1,k+1,k) ( k , k + 1 , k + 1 , k )
Proof. In a smooth deformation of admissible pairs, the minimum covering extends over S 1 × [ 0 , 1 ] S^1\times[0,1] S 1 × [ 0 , 1 ]
For the infinite family replace the square in (19) by the k k k ε k \varepsilon_k ε k ρ 0 ( t ) = t ( 2 − t ) \rho_0(t)=\sqrt{t}(2-t) ρ 0 ( t ) = t ( 2 − t ) t = 2 / 3 t=2/3 t = 2/3 ψ \psi ψ
ε k t ( k + 1 ) / 2 ( 1 − t ) k / 2 cos ( k ψ ) . \varepsilon_k t^{(k+1)/2}(1-t)^{k/2}\cos(k\psi). ε k t ( k + 1 ) /2 ( 1 − t ) k /2 cos ( k ψ ) . Its angular critical equation is sin ( k ψ ) = 0 \sin(k\psi)=0 sin ( k ψ ) = 0 2 / 3 2/3 2/3 ε k \varepsilon_k ε k k k k k k k k k k
A general Morse-family filling gives critical-point maps of disks into M M M entire minimum incidence to embed in M M M M ( S 3 ) \mathcal{M}(S^3) M ( S 3 )
Related work
The basic language belongs to parametrized Morse theory: critical coverings and Hessian bundles are explicit in Igusa’s framing principle [9 ] . Product maps of two functions and their discriminants are studied by Takao [13 ] . The sphere-bundle examples and the circular singular-value geometry are closely related to round fold maps [14 , 11 ] ; the degree data in Subsection 3.4 already have precedents in that classification. The projective collapse used above is described in [10 §16] , and Proposition 3.9 uses the classical configuration-space fibration [12 ] . Linking also has established roles in Morse theory, including Usher’s results for a single Morse function [15 ] .
These results do not provide a general classification or existence theory for uniformly Morse linear systems.
Presymplectic dependency algebras
We study positive families of closed rank-two forms through their finite wedge algebras and geometric deletion and contraction.
Definitions
Throughout this section, manifolds are nonempty, smooth, connected and without boundary. A closed manifold is compact without boundary. Kernels of two-forms are taken as kernels of the maps v ↦ ι v ω v \mapsto\iota_v\omega v ↦ ι v ω
Definition 4.1 (Positive rank-two system). Let E E E M M M ω = ( ω e ) e ∈ E \boldsymbol{\omega} = (\omega_e)_{e\in E} ω = ( ω e ) e ∈ E
(i) each ω e \omega_e ω e
(ii) K S : = ⋂ e ∈ S ker ω e K_S := \bigcap_{e\in S} \ker\omega_e K S := ⋂ e ∈ S ker ω e S ⊆ E S \subseteq E S ⊆ E K ∅ = T M K_{\varnothing} = TM K ∅ = T M
(iii) positive sums introduce no additional kernel directions:
ker ( ∑ e ∈ S t e ω e ) = K S ( t e > 0 ) . (22) \ker\left(\sum_{e\in S} t_e\omega_e\right) = K_S \qquad(t_e > 0). \tag*{(22)} ker ( e ∈ S ∑ t e ω e ) = K S ( t e > 0 ) . (22) The system is effective if K E = 0 K_E = 0 K E = 0 dependency algebra is the unital graded real algebra
A ω : = R [ ω e : e ∈ E ] ⊆ Ω e v e n ( M ) , deg ω e = 2. (23) A_{\omega} := \mathbb{R}[\omega_e : e \in E] \subseteq\Omega^{\mathrm{even}}(M), \qquad\deg\omega_e = 2. \tag*{(23)} A ω := R [ ω e : e ∈ E ] ⊆ Ω even ( M ) , deg ω e = 2. (23) The coefficients in A ω A_{\omega} A ω
Each rank-two form is pointwise decomposable, so ω e 2 = 0 \omega_e^2 = 0 ω e 2 = 0 A ω A_{\omega} A ω
ω S : = ⋀ e ∈ S ω e , ω ∅ = 1 , dim R A ω ≤ 2 ∣ E ∣ . \omega_S := \bigwedge_{e\in S}\omega_e, \qquad\omega_{\varnothing} = 1, \qquad\dim_{\mathbb{R}} A_{\omega} \leq2^{|E|}. ω S := e ∈ S ⋀ ω e , ω ∅ = 1 , dim R A ω ≤ 2 ∣ E ∣ . If the system is effective, ∑ e ω e \sum_e \omega_e ∑ e ω e dim M \dim M dim M
Matroid structure and foliations
Proposition 4.2 (Matroid rank). The function
r ( S ) = 1 2 codim K S r(S) = \frac{1}{2}\operatorname{codim} K_S r ( S ) = 2 1 codim K S is the rank function of an ordinary matroid on E E E
r ( S ) = ∣ S ∣ ⟺ ω S ( p ) ≠ 0 for every p ∈ M . (24) r(S) = |S| \quad\Longleftrightarrow\quad\omega_S(p) \ne0 \text{ for every } p \in M. \tag*{(24)} r ( S ) = ∣ S ∣ ⟺ ω S ( p ) = 0 for every p ∈ M . (24) Proof. At p p p U e = ( ker ω e ( p ) ) ∘ ⊆ T p ∗ M U_e = (\ker\omega_e(p))^\circ\subseteq T_p^*M U e = ( ker ω e ( p ) ) ∘ ⊆ T p ∗ M U S = ∑ e ∈ S U e U_S = \sum_{e\in S} U_e U S = ∑ e ∈ S U e U S = K S ( p ) ∘ U_S = K_S(p)^\circ U S = K S ( p ) ∘ (22) , dim U S \dim U_S dim U S r ( S ) r(S) r ( S ) p p p U e U_e U e
dim U A + dim U B = dim U A ∪ B + dim ( U A ∩ U B ) ≥ dim U A ∪ B + dim U A ∩ B . \dim U_A + \dim U_B = \dim U_{A\cup B} + \dim(U_A \cap U_B) \geq\dim U_{A\cup B} + \dim U_{A\cap B}. dim U A + dim U B = dim U A ∪ B + dim ( U A ∩ U B ) ≥ dim U A ∪ B + dim U A ∩ B . After division by two these are the matroid rank axioms. Finally, a squarefree wedge is nonzero exactly when its two-dimensional support planes sum directly, equivalently when r ( S ) = ∣ S ∣ r(S) = |S| r ( S ) = ∣ S ∣
A subset satisfying (24) is independent ; a basis of S S S S S S r ( S ) r(S) r ( S ) U e U_e U e [16 ] .
Lemma 4.3 (Coherent orientation of bases). For any U ⊆ E U \subseteq E U ⊆ E T p M / K U ( p ) T_pM/K_U(p) T p M / K U ( p ) symplectic form Ω U = ∑ e ∈ U ω e \Omega_U=\sum_{e\in U}\omega_e Ω U = ∑ e ∈ U ω e For every basis B B B of U U U ω B \omega_B ω B is positive in this orientation.
Proof. The path Ω B + s ∑ e ∈ U ∖ B ω e \Omega_B+s\sum_{e\in U\setminus B}\omega_e Ω B + s ∑ e ∈ U ∖ B ω e 0 ≤ s ≤ 1 0\le s\le1 0 ≤ s ≤ 1 T p M / K U ( p ) T_pM/K_U(p) T p M / K U ( p ) (22) , since K B = K U K_B=K_U K B = K U Ω B r ( U ) = r ( U ) ! ω B \Omega_B^{r(U)}=r(U)!\omega_B Ω B r ( U ) = r ( U )! ω B
Proposition 4.4 (Foliations and basis coordinates). Every K S K_S K S is integrable. If the system is effective on a 2 n 2n 2 n manifold and B = { b 1 , … , b n } B=\{b_1,\ldots,b_n\} B = { b 1 , … , b n } is a basis, then locally there are coordinates ( p 1 , q 1 , … , p n , q n ) (p_1,q_1,\ldots,p_n,q_n) ( p 1 , q 1 , … , p n , q n ) such that
ω b i = d p i ∧ d q i ( 1 ≤ i ≤ n ) . \omega_{b_i}=dp_i\wedge dq_i \qquad(1\le i\le n). ω b i = d p i ∧ d q i ( 1 ≤ i ≤ n ) . Proof. The closed form Ω S \Omega_S Ω S K S K_S K S X , Y ∈ Γ ( K S ) X,Y\in\Gamma(K_S) X , Y ∈ Γ ( K S ) L X Ω S = 0 \mathcal{L}_X\Omega_S=0 L X Ω S = 0 ι [ X , Y ] Ω S = 0 \iota_{[X,Y]}\Omega_S=0 ι [ X , Y ] Ω S = 0
Each nonzero ω b i \omega_{b_i} ω b i p i , q i p_i,q_i p i , q i 2 n 2n 2 n
Deletion and contraction
Deleting labels preserves Definition 4.1; algebraically it takes the subalgebra generated by the retained forms. Contraction uses characteristic leaves, rather than an arbitrary submanifold.
Theorem 4.5 (Hereditary compatibility and geometric contraction). For S , T ⊆ E S,T\subseteq E S , T ⊆ E and t e > 0 t_e>0 t e > 0
ker ( ∑ e ∈ T t e ω e ∣ K S ) = K S ∪ T . (25) \operatorname{ker}\left(\left.\sum_{e\in T}t_e\omega_e\right|_{K_S}\right)=K_{S\cup T}. \tag*{(25)} ker e ∈ T ∑ t e ω e K S = K S ∪ T . (25) Consequently, on every connected leaf L L L of K S K_S K S the forms ω e ∣ L \omega_e|_L ω e ∣ L e ∈ E ∖ S e\in E\setminus S e ∈ E ∖ S form a positive rank-two system with rank function
r / S ( T ) = r ( S ∪ T ) − r ( S ) , T ⊆ E ∖ S . (26) r/S(T)=r(S\cup T)-r(S),\qquad T\subseteq E\setminus S. \tag*{(26)} r / S ( T ) = r ( S ∪ T ) − r ( S ) , T ⊆ E ∖ S . (26) This is matroid contraction.
Proof. Put U = S ∪ T U=S\cup T U = S ∪ T s = r ( S ) s=r(S) s = r ( S ) q = r ( U ) − s q=r(U)-s q = r ( U ) − s B B B S S S β = ω B \beta=\omega_B β = ω B η = ∑ e ∈ T t e ω e \eta=\sum_{e\in T}t_e\omega_e η = ∑ e ∈ T t e ω e T M / K U TM/K_U T M / K U q > 0 q>0 q > 0 β ∧ η q \beta\wedge\eta^q β ∧ η q U U U B B B U U U T T T β ∧ η q ≠ 0 \beta\wedge\eta^q\ne0 β ∧ η q = 0
Since β \beta β K S K_S K S ( η ∣ K S ) q ≠ 0 (\eta|_{K_S})^q\ne0 ( η ∣ K S ) q = 0 K U K_U K U η ∣ K S \eta|_{K_S} η ∣ K S 2 q 2q 2 q q = 0 q=0 q = 0 K U = K S K_U=K_S K U = K S
Taking T T T K S ∪ { e } K_{S\cup\{e\}} K S ∪ { e }
Theorem 4.6 (Algebraic contraction). Choose a basis B S B_S B S of S S S and set β S = ω B S \beta_S=\omega_{B_S} β S = ω B S The image of A ω A_\omega A ω under restriction to the distribution K S K_S K S is canonically isomorphic to
A ω / S : = A ω / Ann A ω ( β S ) , Ann A ω ( β S ) = { a ∈ A ω : β S ∧ a = 0 } . (27) A_{\omega/S}:=A_\omega/\operatorname{Ann}_{A_\omega}(\beta_S),\qquad\operatorname{Ann}_{A_\omega}(\beta_S)=\{a\in A_\omega:\beta_S\wedge a=0\}. \tag*{(27)} A ω / S := A ω / Ann A ω ( β S ) , Ann A ω ( β S ) = { a ∈ A ω : β S ∧ a = 0 } . (27) The ideal in (27) is independent of the chosen basis B S B_S B S
Proof. Pointwise, for any exterior form α \alpha α
β S ∧ α = 0 ⟺ α ∣ K S = 0. \beta_S \wedge\alpha= 0 \Longleftrightarrow\alpha|_{K_S} = 0. β S ∧ α = 0 ⟺ α ∣ K S = 0. Indeed, choose a cotangent splitting with K S ∘ K_S^\circ K S ∘ K S ∘ K_S^\circ K S ∘ K S K_S K S
A ω ⟶ Γ ( Λ e v e n K S ∗ ) A_\omega\longrightarrow\Gamma(\Lambda^{\mathrm{even}} K_S^*) A ω ⟶ Γ ( Λ even K S ∗ ) is Ann A ω ( β S ) \operatorname{Ann}_{A_\omega}(\beta_S) Ann A ω ( β S )
Remark 4.7. The quotient describes the image along the whole foliation , not the full algebra of leafwise forms. Restriction to a single leaf can have additional kernel. The rank formula (26) , however, holds on every leaf. Multiplication by β S \beta_S β S β S A ω \beta_S A_\omega β S A ω
Products supply a second elementary construction. On M × N M \times N M × N (22) holds, ranks add, and the matroid is the direct sum. Their algebras satisfy
A M × N ≅ A M ⊗ R A N . A_{M \times N} \cong A_M \otimes_{\mathbb{R}} A_N. A M × N ≅ A M ⊗ R A N . The external-product map is injective: evaluations of forms on tangent multivectors separate any finite-dimensional space of forms, so applying such evaluations on one factor detects every coefficient on the other.
Cohomology and volume
Throughout this subsection assume that M M M dim M = 2 n \dim M = 2n dim M = 2 n M M M ∑ e ω e \sum_e \omega_e ∑ e ω e
Theorem 4.8 (Nonvanishing of independent monomials). For every S ⊆ E S \subseteq E S ⊆ E
∏ e ∈ S [ ω e ] ≠ 0 ⟺ S is independent. (28) \prod_{e \in S} [\omega_e] \ne0 \Longleftrightarrow S\text{ is independent.} \tag*{(28)} e ∈ S ∏ [ ω e ] = 0 ⟺ S is independent. (28) The classes are in real de Rham cohomology.
Proof. For dependent S S S ω S = 0 \omega_S = 0 ω S = 0 S S S B B B E E E ∫ M ω B > 0 \int_M \omega_B > 0 ∫ M ω B > 0 ω S = d α \omega_S = d\alpha ω S = d α ω B = d ( α ∧ ω B ∖ S ) \omega_B = d(\alpha\wedge\omega_{B \setminus S}) ω B = d ( α ∧ ω B ∖ S )
Corollary 4.9 (Obstructions). The following assertions hold.
(i) b 2 k ( M ) ≥ ( n k ) b_{2k}(M) \ge\binom{n}{k} b 2 k ( M ) ≥ ( k n ) 0 ≤ k ≤ n 0 \le k \le n 0 ≤ k ≤ n b 2 ( M ) ≥ n b_2(M) \ge n b 2 ( M ) ≥ n H 2 ( M ; R ) H^2(M;\mathbb{R}) H 2 ( M ; R )
(ii) If n > 1 n > 1 n > 1 C P n \mathbb{C}P^n C P n
(iii) A system of three pairwise independent generators on a closed four-manifold requires b 2 ( M ) ≥ 3 b_2(M) \ge3 b 2 ( M ) ≥ 3 S 2 × S 2 S^2 \times S^2 S 2 × S 2
Proof. For (i), fix a basis B B B [ ω S ] [\omega_S] [ ω S ] S ⊆ B S \subseteq B S ⊆ B ∣ S ∣ = k |S| = k ∣ S ∣ = k ω B ∖ S 0 \omega_{B \setminus S_0} ω B ∖ S 0 S 0 S_0 S 0 H 2 ( C P n ; R ) H^2(\mathbb{C}P^n;\mathbb{R}) H 2 ( C P n ; R )
( 0 a b a 0 c b c 0 ) , a , b , c > 0 , \begin{pmatrix}
0 & a & b \\
a & 0 & c \\
b & c & 0
\end{pmatrix},
\qquad a,b,c > 0, 0 a b a 0 c b c 0 , a , b , c > 0 , @ Marunadan
whose determinant is 2 a b c > 0 2abc > 0 2 ab c > 0
The associated volume polynomial is
P ω ( t ) = 1 n ! ∫ M ( ∑ e t e ω e ) n = ∑ B a basis c B ∏ e ∈ B t e , c B = ∫ M ω B > 0. (29) P_{\omega}(t)=\frac{1}{n!}\int_M\left(\sum_e t_e\omega_e\right)^n=\sum_{B\text{ a basis}}c_B\prod_{e\in B}t_e,\qquad c_B=\int_M\omega_B>0. \tag*{(29)} P ω ( t ) = n ! 1 ∫ M ( e ∑ t e ω e ) n = B a basis ∑ c B e ∈ B ∏ t e , c B = ∫ M ω B > 0. (29) Thus it is multiaffine, with support exactly the matroid bases. Theorem 4.8 is a statement about monomials; the algebra map A ω → H e v e n ( M ; R ) A_\omega\to H^{\mathrm{even}}(M;\mathbb{R}) A ω → H even ( M ; R )
Examples and separation
Product and linear models
Example 4.10 (Products of surfaces). Let M = Σ 1 × ⋯ × Σ n M=\Sigma_1\times\cdots\times\Sigma_n M = Σ 1 × ⋯ × Σ n σ i \sigma_i σ i ω i = π i ∗ σ i \omega_i=\pi_i^*\sigma_i ω i = π i ∗ σ i
A ω ≅ R [ x 1 , … , x n ] / ( x 1 2 , … , x n 2 ) , deg x i = 2. A_\omega\cong\mathbb{R}[x_1,\ldots,x_n]/(x_1^2,\ldots,x_n^2),\qquad\deg x_i=2. A ω ≅ R [ x 1 , … , x n ] / ( x 1 2 , … , x n 2 ) , deg x i = 2. The squarefree products are linearly independent by their distinct degrees in the individual factors.
Example 4.11 (Compact realizations of complex-representable matroids). Let ( c e ) e ∈ E (c_e)_{e\in E} ( c e ) e ∈ E n n n C n \mathbb{C}^n C n T 2 n \mathbb{T}^{2n} T 2 n
α e = ∑ j = 1 n c e j ( d x j + i d y j ) , ω e = i 2 α e ∧ α e ‾ . \alpha_e=\sum_{j=1}^{n}c_{ej}(dx_j+i\,dy_j),\qquad\omega_e=\frac{i}{2}\alpha_e\wedge\overline{\alpha_e}. α e = j = 1 ∑ n c e j ( d x j + i d y j ) , ω e = 2 i α e ∧ α e . The system is effective and its matroid is that of the vectors c e c_e c e J J J
( ∑ e t e ω e ) ( v , J v ) = ∑ e t e ∣ α e ( v ) ∣ 2 , \left(\sum_e t_e\omega_e\right)(v,Jv)=\sum_e t_e\lvert\alpha_e(v)\rvert^2, ( e ∑ t e ω e ) ( v , J v ) = e ∑ t e ∣ α e ( v ) ∣ 2 , so (22) holds; common complex kernels have the required real codimensions. Thus every complex-representable matroid has a compact model. No rationality assumption on the coefficients is required.
A four-generator torus family
On T 4 = ( R / Z ) 4 \mathbb{T}^4=(\mathbb{R}/\mathbb{Z})^4 T 4 = ( R / Z ) 4 x , y , u , v x,y,u,v x , y , u , v
V = d x ∧ d y ∧ d u ∧ d v , a ε ( u ) = 2 + ε sin ( 2 π u ) , 0 ≤ ε < 1 , V=dx\wedge dy\wedge du\wedge dv,\qquad a_\varepsilon(u)=2+\varepsilon\sin(2\pi u),\qquad0\leq\varepsilon<1, V = d x ∧ d y ∧ d u ∧ d v , a ε ( u ) = 2 + ε sin ( 2 π u ) , 0 ≤ ε < 1 , and define
ω 1 = d x ∧ d y , ω 2 = d u ∧ d v , \omega_1=dx\wedge dy,\qquad\omega_2=du\wedge dv, ω 1 = d x ∧ d y , ω 2 = d u ∧ d v , ω 3 = ( d x + d u ) ∧ ( d y + d v ) , ω 4 = ( d x + a ε d u ) ∧ ( d y + 4 d v ) . (30) \omega_3=(dx+du)\wedge(dy+dv),\qquad\omega_4=(dx+a_\varepsilon du)\wedge(dy+4\,dv). \tag*{(30)} ω 3 = ( d x + d u ) ∧ ( d y + d v ) , ω 4 = ( d x + a ε d u ) ∧ ( d y + 4 d v ) . (30) All four forms are closed and of rank two. Writing a = a ε a=a_\varepsilon a = a ε
Table 2.
Every coefficient is positive. Hence every positive sum involving at least two labels is symplectic, while a singleton has rank two. The system is effective and has matroid U 2 , 4 U_{2,4} U 2 , 4
Theorem 4.12 (Same cohomology data, different finite algebras). For the family (4.9), all labelled de Rham classes, the matroid, and the volume polynomial are independent of ε \varepsilon ε
Table 4.12.
Consequently the constant system is not isomorphic to any nonconstant member, even after relabelling.
Proof. Only ω 4 \omega_4 ω 4 F ( u ) = − ε cos ( 2 π u ) / ( 2 π ) F(u)=-\varepsilon\cos(2\pi u)/(2\pi) F ( u ) = − ε cos ( 2 π u ) / ( 2 π )
ω 4 ε − ω 4 0 = ( a ε − 2 ) d u ∧ ( d y + 4 d v ) = d ( F ( u ) ( d y + 4 d v ) ) . \omega_4^\varepsilon-\omega_4^0=(a_\varepsilon-2)du\wedge(dy+4\,dv)=d\bigl(F(u)(dy+4\,dv)\bigr). ω 4 ε − ω 4 0 = ( a ε − 2 ) d u ∧ ( d y + 4 d v ) = d ( F ( u ) ( d y + 4 d v ) ) . Thus the labelled classes agree, and (4.8) gives the same volume polynomial.
The four degree-two generators are linearly independent: the coefficients of d x ∧ d v dx\wedge dv d x ∧ d v d u ∧ d y du\wedge dy d u ∧ d y c 3 + 4 c 4 = 0 c_3+4c_4=0 c 3 + 4 c 4 = 0 c 3 + a c 4 = 0 c_3+ac_4=0 c 3 + a c 4 = 0 a ≠ 4 a\ne4 a = 4 c 3 = c 4 = 0 c_3=c_4=0 c 3 = c 4 = 0 c 1 = c 2 = 0 c_1=c_2=0 c 1 = c 2 = 0
A 4 = span R { V , a V } , A^4=\operatorname{span}_{\mathbb{R}}\{V,aV\}, A 4 = span R { V , aV } , which has dimension one exactly when a a a
Proposition 4.13 (Local moduli and absence of a common complex structure). For (4.9):
(i) where d a ≠ 0 da\ne0 d a = 0
(ii) no almost complex structure preserves all four kernels. In particular the forms cannot all be of type (1, 1) for a common almost complex structure.
Proof. For (i), the intrinsic ratio of top forms is
ω 1 ∧ ω 4 ω 1 ∧ ω 2 = 4 a ( u ) . \frac{\omega_1\wedge\omega_4}{\omega_1\wedge\omega_2}=4a(u). ω 1 ∧ ω 2 ω 1 ∧ ω 4 = 4 a ( u ) . It would be constant in any simultaneous constant-coefficient model.
For (ii), write P = ker ω 2 = span R { ∂ x , ∂ y } P=\ker\omega_2=\operatorname{span}_{\mathbb{R}}\{\partial_x,\partial_y\} P = ker ω 2 = span R { ∂ x , ∂ y } Q = ker ω 1 = span R { ∂ u , ∂ v } Q=\ker\omega_1=\operatorname{span}_{\mathbb{R}}\{\partial_u,\partial_v\} Q = ker ω 1 = span R { ∂ u , ∂ v } Q → P Q\to P Q → P − I -I − I − D -D − D D = diag ( a , 4 ) D=\operatorname{diag}(a,4) D = diag ( a , 4 ) P , Q P,Q P , Q − I -I − I − D -D − D D D D a ≠ 4 a\ne4 a = 4 − I -I − I
Remark 4.14 (Scope of the separation). Theorem 4.12 distinguishes chosen structures on the same manifold; it does not yet distinguish underlying manifolds with the same homotopy type. The abstract algebra also does not classify all local moduli: for nonconstant a a a V , a V V,aV V , aV a a a
Related work
The pointwise support planes form 2-arrangements, hence multilinear matroid representations [16 ] . Complementary two-form examples in dimension four are symplectic pairs [17 ] . An effective tuple, viewed as an R E \mathbb{R}^{E} R E [18 ] ; its kernel foliations also connect it to symplectic webs [19 ] .
The finite wedge algebra is retained before passage to cohomology; positive compatibility supplies hereditary restriction and annihilator contraction along characteristic foliations. The raw algebra is an invariant of the structured manifold and may jump under small deformations, as Theorem 4.12 shows.
Diffusion transport defects
We study the rank defects of connection heat kernels and coverings by the open sets on which these kernels trivialize a bundle.
Heat kernels and defects
Throughout this section, ( M n , g ) (M^n,g) ( M n , g ) E → M E \to M E → M r ≥ 1 r \ge1 r ≥ 1 F = R \mathbb{F} = \mathbb{R} F = R C \mathbb{C} C ∇ \nabla ∇
L = ∇ ∗ ∇ ≥ 0. L = \nabla^{*}\nabla\ge0. L = ∇ ∗ ∇ ≥ 0. Its heat kernel, normalized against Riemannian volume, is
H t ( x , p ) ∈ Hom ( E p , E x ) , ( e − t L s ) ( x ) = ∫ M H t ( x , p ) s ( p ) d v g ( p ) , t > 0. H_t(x,p) \in\operatorname{Hom}(E_p,E_x), \qquad(e^{-tL}s)(x) = \int_M H_t(x,p)s(p)\,\mathrm{d}v_g(p), \qquad t > 0. H t ( x , p ) ∈ Hom ( E p , E x ) , ( e − t L s ) ( x ) = ∫ M H t ( x , p ) s ( p ) d v g ( p ) , t > 0. Thus a source vector v ∈ E p v \in E_p v ∈ E p x ↦ H t ( x , p ) v x \mapsto H_t(x,p)v x ↦ H t ( x , p ) v
Definition 5.1 (Defect correspondence). For 1 ≤ k ≤ r 1 \le k \le r 1 ≤ k ≤ r
D ( k ) = { ( t , x , p ) ∈ ( 0 , ∞ ) × M × M : dim F ker H t ( x , p ) ≥ k } . \mathcal{D}^{(k)} = \{(t,x,p) \in(0,\infty) \times M \times M : \dim_{\mathbb{F}} \ker H_t(x,p) \ge k\}. D ( k ) = {( t , x , p ) ∈ ( 0 , ∞ ) × M × M : dim F ker H t ( x , p ) ≥ k } . Write D t ( k ) = { ( x , p ) : ( t , x , p ) ∈ D ( k ) } \mathcal{D}^{(k)}_t = \{(x,p) : (t,x,p) \in\mathcal{D}^{(k)}\} D t ( k ) = {( x , p ) : ( t , x , p ) ∈ D ( k ) } D t ( 1 ) ( p ) = { x : ( x , p ) ∈ D t ( 1 ) } \mathcal{D}^{(1)}_t(p) = \{x : (x,p) \in\mathcal{D}^{(1)}_t\} D t ( 1 ) ( p ) = { x : ( x , p ) ∈ D t ( 1 ) }
I = { ( t , x , p , [ v ] ) : [ v ] ∈ P ( E p ) , H t ( x , p ) v = 0 } . \mathcal{I} = \{(t,x,p,[v]) : [v] \in\mathbb{P}(E_p),\ H_t(x,p)v = 0\}. I = {( t , x , p , [ v ]) : [ v ] ∈ P ( E p ) , H t ( x , p ) v = 0 } . Here P \mathbb{P} P
Definition 5.2 (Heat covering number). Set
U t ( p ) = M ∖ D t ( 1 ) ( p ) = { x : H t ( x , p ) is invertible } , U_t(p) = M \setminus\mathcal{D}^{(1)}_t(p) = \{x : H_t(x,p)\text{ is invertible}\}, U t ( p ) = M ∖ D t ( 1 ) ( p ) = { x : H t ( x , p ) is invertible } , and define
h t ( M , g , E , ∇ ) = min { q ≥ 1 : ∃ p 1 , … , p q ∈ M , M = ⋃ j = 1 q U t ( p j ) } . (31) h_t(M,g,E,\nabla) = \min\left\{q \ge1 : \exists p_1,\ldots,p_q \in M,\quad M = \bigcup_{j=1}^{q} U_t(p_j)\right\}. \tag*{(31)} h t ( M , g , E , ∇ ) = min { q ≥ 1 : ∃ p 1 , … , p q ∈ M , M = j = 1 ⋃ q U t ( p j ) } . (31) When the data are fixed, write simply h t h_t h t
Finiteness is proved below. The sets U t ( p ) U_t(p) U t ( p ) D ( k ) \mathcal{D}^{(k)} D ( k ) I \mathcal{I} I
Basic properties Proposition 5.3 (Symmetry, positivity, and invariance). For every t > 0 t > 0 t > 0
H t ( x , p ) ∗ = H t ( p , x ) , H t ( p , p ) > 0. H_t(x,p)^* = H_t(p,x), \qquad H_t(p,p) > 0. H t ( x , p ) ∗ = H t ( p , x ) , H t ( p , p ) > 0. Consequently, D t ( k ) \mathcal{D}^{(k)}_t D t ( k ) is invariant under ( x , p ) ↦ ( p , x ) (x,p) \mapsto(p,x) ( x , p ) ↦ ( p , x ) and is disjoint from the diagonal. These loci and h t h_t h t are invariant under metric-preserving bundle isomorphisms intertwining the connections and covering isometries of the base.
Proof. Let ( ϕ j ) (\phi_j) ( ϕ j ) L L L λ j ≥ 0 \lambda_j \ge0 λ j ≥ 0
H t ( x , p ) = ∑ j e − t λ j ϕ j ( x ) ϕ j ( p ) ∗ . H_t(x,p)=\sum_j e^{-t\lambda_j}\phi_j(x)\phi_j(p)^*. H t ( x , p ) = j ∑ e − t λ j ϕ j ( x ) ϕ j ( p ) ∗ . It gives symmetry and
⟨ H t ( p , p ) v , v ⟩ = ∑ j e − t λ j ∣ ⟨ ϕ j ( p ) , v ⟩ ∣ 2 . \langle H_t(p,p)v,v\rangle=\sum_j e^{-t\lambda_j}\lvert\langle\phi_j(p),v\rangle\rvert^2. ⟨ H t ( p , p ) v , v ⟩ = j ∑ e − t λ j ∣⟨ ϕ j ( p ) , v ⟩ ∣ 2 . If this vanishes, v v v ϕ j ( p ) \phi_j(p) ϕ j ( p ) p p p C ∞ C^\infty C ∞ v = 0 v=0 v = 0 L L L
Proposition 5.4 (Trivialization and topological obstruction). For all t > 0 t>0 t > 0 1 ≤ h t < ∞ 1\le h_t<\infty 1 ≤ h t < ∞ U t ( p ) U_t(p) U t ( p ) E E E E E E D t ( p ) ≠ ∅ D_t(p)\ne\varnothing D t ( p ) = ∅ p p p t t t
More generally, let γ 1 ( E ) , … , γ q ( E ) \gamma_1(E),\ldots,\gamma_q(E) γ 1 ( E ) , … , γ q ( E )
γ 1 ( E ) ⌣ ⋯ ⌣ γ q ( E ) ≠ 0 , then h t ≥ q + 1. \gamma_1(E)\smile\cdots\smile\gamma_q(E)\ne0,\qquad\text{then}\qquad h_t\ge q+1. γ 1 ( E ) ⌣ ⋯ ⌣ γ q ( E ) = 0 , then h t ≥ q + 1. Proof. The map ( x , v ) ↦ H t ( x , p ) v (x,v)\mapsto H_t(x,p)v ( x , v ) ↦ H t ( x , p ) v U t ( p ) × E p U_t(p)\times E_p U t ( p ) × E p E ∣ U t ( p ) E|_{U_t(p)} E ∣ U t ( p ) p ∈ U t ( p ) p\in U_t(p) p ∈ U t ( p ) M M M E E E
If q q q M M M γ i ( E ) \gamma_i(E) γ i ( E ) i i i H ∗ ( M , M ) = 0 H^*(M,M)=0 H ∗ ( M , M ) = 0 [5 ] .
Proposition 5.5 (Stability and rescaling). The integer-valued function t ↦ h t t\mapsto h_t t ↦ h t ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) s > 0 s>0 s > 0
H t ∇ , s g ( x , p ) = s − n / 2 H t / s ∇ , g ( x , p ) , h t ( M , s g , E , ∇ ) = h t / s ( M , g , E , ∇ ) . (32) H_t^{\nabla,sg}(x,p)=s^{-n/2}H_{t/s}^{\nabla,g}(x,p),\qquad h_t(M,sg,E,\nabla)=h_{t/s}(M,g,E,\nabla). \tag*{(32)} H t ∇ , s g ( x , p ) = s − n /2 H t / s ∇ , g ( x , p ) , h t ( M , s g , E , ∇ ) = h t / s ( M , g , E , ∇ ) . (32) Proof. Choose a covering tuple p 1 , … , p q p_1,\ldots,p_q p 1 , … , p q t 0 t_0 t 0
min x ∈ M max 1 ≤ i ≤ q σ min ( H t 0 ( x , p i ) ) > 0. \min_{x\in M}\max_{1\le i\le q}\sigma_{\min}(H_{t_0}(x,p_i))>0. x ∈ M min 1 ≤ i ≤ q max σ m i n ( H t 0 ( x , p i )) > 0. Continuity of the heat kernel preserves this inequality nearby, so h t ≤ h t 0 h_t\le h_{t_0} h t ≤ h t 0 L s g = s − 1 L g L_{sg}=s^{-1}L_g L s g = s − 1 L g d vol s g = s n / 2 d vol g d\operatorname{vol}_{sg}=s^{n/2}d\operatorname{vol}_g d vol s g = s n /2 d vol g
Theorem 5.6 (Small-time localization). Fix p ∈ M p\in M p ∈ M V V V Cut ( p ) \operatorname{Cut}(p) Cut ( p ) ε > 0 \varepsilon>0 ε > 0
D t ( p ) ⊂ V ( 0 < t < ε ) . D_t(p)\subset V\qquad(0<t<\varepsilon). D t ( p ) ⊂ V ( 0 < t < ε ) . Equivalently, if t j ↓ 0 t_j\downarrow0 t j ↓ 0 x j ∈ D t j ( p ) x_j\in D_{t_j}(p) x j ∈ D t j ( p ) x j → x x_j\to x x j → x x ∈ Cut ( p ) x\in\operatorname{Cut}(p) x ∈ Cut ( p )
Proof. On a compact set K ⊂ M ∖ Cut ( p ) K\subset M\setminus\operatorname{Cut}(p) K ⊂ M ∖ Cut ( p ) [20 Theorem 1.1] gives
H t ( x , p ) = ( 4 π t ) − n / 2 e − d ( x , p ) 2 / ( 4 t ) ( a ( x , p ) P x ← p + O K ( t ) ) , (33) H_t(x,p)=(4\pi t)^{-n/2}e^{-d(x,p)^2/(4t)}\left(a(x,p)P_{x\leftarrow p}+O_K(t)\right), \tag*{(33)} H t ( x , p ) = ( 4 π t ) − n /2 e − d ( x , p ) 2 / ( 4 t ) ( a ( x , p ) P x ← p + O K ( t ) ) , (33) where a > 0 a>0 a > 0 P x ← p P_{x\leftarrow p} P x ← p K K K H t ( x , p ) H_t(x,p) H t ( x , p ) t t t K = M ∖ V K=M\setminus V K = M ∖ V
This is a one-sided localization statement: it does not assert that every cut point is a limit of defects.
Examples
Example 5.7 (A global parallel frame). If E E E
H t ( x , p ) = h t ( x , p ) I r H_t(x,p)=h_t(x,p)I_r H t ( x , p ) = h t ( x , p ) I r in that frame, where h t > 0 h_t>0 h t > 0 D ( 1 ) = ∅ \mathcal{D}^{(1)}=\varnothing D ( 1 ) = ∅ h t = 1 \mathfrak{h}_t=1 h t = 1
Proposition 5.8 (Rectangular tori with sign holonomy). On T m = ( R / 2 π Z ) m \mathbb{T}^m=(\mathbb{R}/2\pi\mathbb{Z})^m T m = ( R /2 π Z ) m
∇ = d + i 2 ∑ j = 1 k d θ j , 0 ≤ k ≤ m . \nabla=d+\frac{i}{2}\sum_{j=1}^{k}d\theta_j,\qquad0\leq k\leq m. ∇ = d + 2 i j = 1 ∑ k d θ j , 0 ≤ k ≤ m . Then, for every t > 0 t>0 t > 0
U t ( p ) = ∏ j = 1 k ( S 1 ∖ { p j + π } ) × ( S 1 ) m − k , h t = k + 1. (34) U_t(p)=\prod_{j=1}^{k}\left(S^1\setminus\{p_j+\pi\}\right)\times(S^1)^{m-k},\qquad\mathfrak{h}_t=k+1. \tag*{(34)} U t ( p ) = j = 1 ∏ k ( S 1 ∖ { p j + π } ) × ( S 1 ) m − k , h t = k + 1. (34) Proof. For one twisted circle, a kernel representative on the universal cover, up to a nowhere-zero phase, is
k t − ( u ) = 1 4 π t ∑ ℓ ∈ Z ( − 1 ) ℓ e − ( u + 2 π ℓ ) 2 / ( 4 t ) . k_t^-(u)=\frac{1}{\sqrt{4\pi t}}\sum_{\ell\in\mathbb{Z}}(-1)^\ell e^{-(u+2\pi\ell)^2/(4t)}. k t − ( u ) = 4 π t 1 ℓ ∈ Z ∑ ( − 1 ) ℓ e − ( u + 2 π ℓ ) 2 / ( 4 t ) . On ( − π , π ) (-\pi,\pi) ( − π , π ) 0 0 0 2 π 2\pi 2 π (34) .
Given q ≤ k q\leq k q ≤ k p ( 1 ) , … , p ( q ) p^{(1)},\ldots,p^{(q)} p ( 1 ) , … , p ( q ) θ j = p j ( j ) + π \theta_j=p_j^{(j)}+\pi θ j = p j ( j ) + π 1 ≤ j ≤ q 1\leq j\leq q 1 ≤ j ≤ q k + 1 k+1 k + 1 k k k ( a ℓ , … , a ℓ ) (a_\ell,\ldots,a_\ell) ( a ℓ , … , a ℓ ) a ℓ a_\ell a ℓ k k k k + 1 k+1 k + 1 k = 0 k=0 k = 0
Thus even a topologically trivial line bundle with zero curvature can have h t > 1 \mathfrak{h}_t>1 h t > 1
Example 5.9 (Antipodal defects on the round sphere). For the Levi-Civita connection on T S 2 TS^2 T S 2
H t ( − p , p ) = 0 ( t > 0 ) . H_t(-p,p)=0\qquad(t>0). H t ( − p , p ) = 0 ( t > 0 ) . Indeed, regard the oriented tangent planes as complex lines, with J p v = p × v J_pv=p\times v J p v = p × v J J J p p p − p -p − p e i θ e^{i\theta} e i θ e − i θ e^{-i\theta} e − i θ H t ( − p , p ) H_t(-p,p) H t ( − p , p )
Line bundles on surfaces
Let Σ \Sigma Σ L → Σ L\to\Sigma L → Σ d = ⟨ c 1 ( L ) , [ Σ ] ⟩ d=\langle c_1(L),[\Sigma]\rangle d = ⟨ c 1 ( L ) , [ Σ ]⟩
Proposition 5.10 (Fixed-source index formula). Fix t > 0 t>0 t > 0 p ∈ Σ p\in\Sigma p ∈ Σ 0 ≠ v ∈ L p 0\ne v\in L_p 0 = v ∈ L p s ( x ) = H t ( x , p ) v s(x)=H_t(x,p)v s ( x ) = H t ( x , p ) v
∑ x : s ( x ) = 0 ind x ( s ) = d . \sum_{x:s(x)=0}\operatorname{ind}_x(s)=d. x : s ( x ) = 0 ∑ ind x ( s ) = d . If all zeros are transverse, there are at least ∣ d ∣ |d| ∣ d ∣ L = T Σ L=T\Sigma L = T Σ χ ( Σ ) \chi(\Sigma) χ ( Σ )
Proof. The local index is the winding number of the local section around a zero. The sum of these indices is the Euler number of the underlying oriented real plane bundle, which is its first Chern number [5 ] . A transverse zero has index + 1 +1 + 1 − 1 -1 − 1 c 1 ( T Σ ) [ Σ ] = χ ( Σ ) c_1(T\Sigma)[\Sigma] = \chi(\Sigma) c 1 ( T Σ ) [ Σ ] = χ ( Σ )
Theorem 5.11 (The two-point defect cycle). Suppose, at a fixed t > 0 t > 0 t > 0 H t H_t H t
F = π 1 ∗ L ⊗ π 2 ∗ L ∗ ⟶ Σ × Σ . F = \pi_1^*L \otimes\pi_2^*L^* \longrightarrow\Sigma\times\Sigma. F = π 1 ∗ L ⊗ π 2 ∗ L ∗ ⟶ Σ × Σ. Then D t ( 1 ) D_t^{(1)} D t ( 1 )
PD [ D t ( 1 ) ] = π 1 ∗ c 1 ( L ) − π 2 ∗ c 1 ( L ) , (35) \operatorname{PD}[D_t^{(1)}] = \pi_1^*c_1(L) - \pi_2^*c_1(L), \tag*{(35)} PD [ D t ( 1 ) ] = π 1 ∗ c 1 ( L ) − π 2 ∗ c 1 ( L ) , (35) [ D t ( 1 ) ] ⋅ [ D t ( 1 ) ] = − 2 d 2 . (36) [D_t^{(1)}]\cdot[D_t^{(1)}] = -2d^2. \tag*{(36)} [ D t ( 1 ) ] ⋅ [ D t ( 1 ) ] = − 2 d 2 . (36) Interchanging the two factors restricts to a free orientation-reversing involution on the defect surface.
Proof. A transverse zero set of a complex line-bundle section represents its first Chern class [5 ] . Since c 1 ( F ) = π 1 ∗ c 1 ( L ) − π 2 ∗ c 1 ( L ) c_1(F) = \pi_1^*c_1(L) - \pi_2^*c_1(L) c 1 ( F ) = π 1 ∗ c 1 ( L ) − π 2 ∗ c 1 ( L ) (35) . Squaring and integrating over the product gives (36) ; the squares of classes pulled back from either surface vanish, and the mixed product integrates to d 2 d^2 d 2
The heat-kernel adjoint identity preserves the zero set. Factor interchange preserves the orientation of the four-dimensional ambient product but induces complex conjugation on the normal complex line, reversing its orientation. It therefore reverses the induced orientation of the zero surface. It has no fixed points there by diagonal positivity.
Transversality is a hypothesis, not a genericity result proved here. A general smooth heat-kernel section is not holomorphic; its local indices need not be positive.
A transition on a flat torus
The following example changes with t t t
Theorem 5.12 (A two-dimensional transition). Let
a = ( 1 , 0 ) , b = ( 1 / 2 , 3 / 2 ) , Λ = Z a + Z b , M 0 = R 2 / Λ , a=(1,0),\qquad b=(1/2,\sqrt{3}/2),\qquad\Lambda=\mathbb{Z}a+\mathbb{Z}b,\qquad M_0=\mathbb{R}^2/\Lambda, a = ( 1 , 0 ) , b = ( 1/2 , 3 /2 ) , Λ = Z a + Z b , M 0 = R 2 /Λ , with the Euclidean quotient metric g 0 g_0 g 0 x = q 1 a + q 2 b x=q_1a+q_2b x = q 1 a + q 2 b q j ∈ R / Z q_j\in\mathbb{R}/\mathbb{Z} q j ∈ R / Z
∇ 0 = d + 2 π i ( 5 12 d q 1 − 1 3 d q 2 ) . (37) \nabla_0=d+2\pi i\left(\frac{5}{12}\,dq_1-\frac{1}{3}\,dq_2\right). \tag*{(37)} ∇ 0 = d + 2 π i ( 12 5 d q 1 − 3 1 d q 2 ) . (37) There exists ϵ ∈ ( 0 , 1 ) \epsilon\in(0,1) ϵ ∈ ( 0 , 1 )
h t = 2 ( 0 < t < ϵ ) , h t = 1 ( t ≥ 1 ) . h_t=2\quad(0<t<\epsilon),\qquad h_t=1\quad(t\ge1). h t = 2 ( 0 < t < ϵ ) , h t = 1 ( t ≥ 1 ) . For each source and every sufficiently small t t t
Proof. 1. Two useful representations. Put α = ( 5 / 12 , − 1 / 3 ) \alpha=(5/12,-1/3) α = ( 5/12 , − 1/3 )
ρ ( m a + n b ) = e 2 π i ( m α 1 + n α 2 ) . \rho(ma+nb)=e^{2\pi i(m\alpha_1+n\alpha_2)}. ρ ( ma + nb ) = e 2 π i ( m α 1 + n α 2 ) . The image sum
K t ( x ) = 1 4 π t ∑ λ ∈ Λ ρ ( λ ) e − ∣ x − λ ∣ 2 / ( 4 t ) (38) K_t(x)=\frac{1}{4\pi t}\sum_{\lambda\in\Lambda}\rho(\lambda)e^{-\lvert x-\lambda\rvert^2/(4t)} \tag*{(38)} K t ( x ) = 4 π t 1 λ ∈ Λ ∑ ρ ( λ ) e − ∣ x − λ ∣ 2 / ( 4 t ) (38) obeys K t ( x + λ ) = ρ ( λ ) K t ( x ) K_t(x+\lambda)=\rho(\lambda)K_t(x) K t ( x + λ ) = ρ ( λ ) K t ( x )
H t ( x , 0 ) = e − 2 π i α ⋅ q K t ( x ) . H_t(x,0)=e^{-2\pi i\alpha\cdot q}K_t(x). H t ( x , 0 ) = e − 2 π i α ⋅ q K t ( x ) . This follows either by gauging the Euclidean heat equation or by Fourier expansion. In particular, K t K_t K t H t ( ⋅ , 0 ) H_t(\cdot,0) H t ( ⋅ , 0 ) H t ( x , p ) = H t ( x − p , 0 ) H_t(x,p)=H_t(x-p,0) H t ( x , p ) = H t ( x − p , 0 )
K t ( − x ) = K t ( x ) ‾ . (39) K_t(-x)=\overline{K_t(x)}. \tag*{(39)} K t ( − x ) = K t ( x ) . (39) A nondegenerate small-time cancellation
At the Voronoi vertex v = ( a + b ) / 3 v=(a+b)/3 v = ( a + b ) /3 0 , a , b 0,a,b 0 , a , b
A = ρ ( a ) = − 3 2 + i 2 , B = ρ ( b ) = − 1 2 − 3 2 i , A=\rho(a)=-\frac{\sqrt{3}}{2}+\frac{i}{2},\qquad B=\rho(b)=-\frac{1}{2}-\frac{\sqrt{3}}{2}i, A = ρ ( a ) = − 2 3 + 2 i , B = ρ ( b ) = − 2 1 − 2 3 i , and satisfy
1 + 3 2 A + 1 2 B = 0. (40) 1+\frac{\sqrt{3}}{2}A+\frac{1}{2}B=0. \tag*{(40)} 1 + 2 3 A + 2 1 B = 0. (40) For x = v + t z x=v+tz x = v + t z (38) by ( 4 π t ) − 1 e − ∣ x ∣ 2 / ( 4 t ) (4\pi t)^{-1}e^{-|x|^2/(4t)} ( 4 π t ) − 1 e − ∣ x ∣ 2 / ( 4 t ) z z z
F ( z ) + R t ( z ) , F ( z ) = 1 + A e a ⋅ z / 2 + B e b ⋅ z / 2 , ∥ R t ∥ C 1 = O ( e − c / t ) (41) F(z)+R_t(z),\qquad F(z)=1+A e^{a\cdot z/2}+B e^{b\cdot z/2},\qquad\lVert R_t\rVert_{C^1}=O(e^{-c/t}) \tag*{(41)} F ( z ) + R t ( z ) , F ( z ) = 1 + A e a ⋅ z /2 + B e b ⋅ z /2 , ∥ R t ∥ C 1 = O ( e − c / t ) (41) for some c > 0 c>0 c > 0
Since a , b a,b a , b (40) gives the unique zero of F F F
e a ⋅ z 0 / 2 = 3 2 , e b ⋅ z 0 / 2 = 1 2 , z 0 = ( log 3 4 , − log 12 3 ) . e^{a\cdot z_0/2}=\frac{\sqrt{3}}{2},\qquad e^{b\cdot z_0/2}=\frac{1}{2},\qquad z_0=\left(\log\frac{3}{4},-\frac{\log12}{\sqrt{3}}\right). e a ⋅ z 0 /2 = 2 3 , e b ⋅ z 0 /2 = 2 1 , z 0 = ( log 4 3 , − 3 log 12 ) . With the usual orientations of R 2 \mathbb{R}^2 R 2 C \mathbb{C} C
det R D F ( z 0 ) = 3 32 > 0. \det_{\mathbb{R}} DF(z_0)=\frac{3}{32}>0. R det D F ( z 0 ) = 32 3 > 0. The implicit function theorem applied to (41) gives a simple zero
x + ( t ) = v + t z 0 + O ( t e − c / t ) . (42) x_+(t)=v+tz_0+O(te^{-c/t}). \tag*{(42)} x + ( t ) = v + t z 0 + O ( t e − c / t ) . (42) The exponentially small remainder extends smoothly to t = 0 t=0 t = 0 z z z (39) gives a second zero x − ( t ) = − x + ( t ) x_-(t)=-x_+(t) x − ( t ) = − x + ( t ) Λ \Lambda Λ
Exclusion of all other small-time zeros
There are precisely two Voronoi-vertex classes modulo Λ \Lambda Λ v v v − v -v − v
{ ρ ( a ) , ρ ( b ) , ρ ( a − b ) , ρ ( − a ) , ρ ( − b ) , ρ ( b − a ) } . \{\rho(a),\rho(b),\rho(a-b),\rho(-a),\rho(-b),\rho(b-a)\}. { ρ ( a ) , ρ ( b ) , ρ ( a − b ) , ρ ( − a ) , ρ ( − b ) , ρ ( b − a )} . None is − 1 -1 − 1 u , w u,w u , w c 0 > 0 c_0>0 c 0 > 0
∣ r u + s w ∣ ≥ c 0 ( r + s ) ( r , s ≥ 0 ) . |ru+sw|\ge c_0(r+s)\qquad(r,s\ge0). ∣ r u + s w ∣ ≥ c 0 ( r + s ) ( r , s ≥ 0 ) . The single leading term or the sum of the two leading terms consequently dominates the exponentially smaller tail. There are no zeros in these neighborhoods for sufficiently small t t t
For completeness, uniqueness near v v v v v v
w 0 = e − ∣ x ∣ 2 / ( 4 t ) , w a = e − ∣ x − a ∣ 2 / ( 4 t ) , w b = e − ∣ x − b ∣ 2 / ( 4 t ) , W = w 0 + w a + w b . w_0=e^{-|x|^2/(4t)},\quad w_a=e^{-|x-a|^2/(4t)},\quad w_b=e^{-|x-b|^2/(4t)},\quad W=w_0+w_a+w_b. w 0 = e − ∣ x ∣ 2 / ( 4 t ) , w a = e − ∣ x − a ∣ 2 / ( 4 t ) , w b = e − ∣ x − b ∣ 2 / ( 4 t ) , W = w 0 + w a + w b . The sum of all omitted absolute values is at most C e − c / t W Ce^{-c/t}W C e − c / t W
∣ q 0 + A q a + B q b ∣ ≤ C e − c / t , ( q 0 , q a , q b ) = W − 1 ( w 0 , w a , w b ) . |q_0+Aq_a+Bq_b|\le Ce^{-c/t},\qquad(q_0,q_a,q_b)=W^{-1}(w_0,w_a,w_b). ∣ q 0 + A q a + B q b ∣ ≤ C e − c / t , ( q 0 , q a , q b ) = W − 1 ( w 0 , w a , w b ) . The affine map from q 0 + q a + q b = 1 q_0+q_a+q_b=1 q 0 + q a + q b = 1 q 0 + A q a + B q b ∈ C q_0+Aq_a+Bq_b\in\mathbb{C} q 0 + A q a + B q b ∈ C ( 1 , 3 / 2 , 1 / 2 ) (1,\sqrt{3}/2,1/2) ( 1 , 3 /2 , 1/2 ) ( q 0 , q a , q b ) (q_0,q_a,q_b) ( q 0 , q a , q b )
x − v t = z 0 + O ( e − c / t ) . \frac{x-v}{t}=z_0+O(e^{-c/t}). t x − v = z 0 + O ( e − c / t ) . Every zero near v v v − v -v − v
4. Large-time nonvanishing. In the global periodic frame, the eigenfunctions are Fourier characters and the eigenvalues are
λ k = 4 π 2 Q ( k + α ) , Q ( u , v ) = 4 3 ( u 2 − u v + v 2 ) , k ∈ Z 2 . \lambda_k=4\pi^2Q(k+\alpha),\qquad Q(u,v)=\frac{4}{3}(u^2-uv+v^2),\qquad k\in\mathbb{Z}^2. λ k = 4 π 2 Q ( k + α ) , Q ( u , v ) = 3 4 ( u 2 − uv + v 2 ) , k ∈ Z 2 . Let k 0 = ( − 1 , 0 ) k_0=(-1,0) k 0 = ( − 1 , 0 ) ℓ = ( u , v ) ∈ Z 2 \ell=(u,v)\in\mathbb{Z}^2 ℓ = ( u , v ) ∈ Z 2 B 0 = u 2 − u v + v 2 B_0=u^2-uv+v^2 B 0 = u 2 − uv + v 2
λ k 0 + ℓ − λ k 0 = 4 π 2 9 ( 12 B 0 − 10 u − v ) ≥ 2 π 2 9 ( u 2 + v 2 ) . (43) \lambda_{k_0+\ell}-\lambda_{k_0}=\frac{4\pi^2}{9}(12B_0-10u-v)\ge\frac{2\pi^2}{9}(u^2+v^2). \tag*{(43)} λ k 0 + ℓ − λ k 0 = 9 4 π 2 ( 12 B 0 − 10 u − v ) ≥ 9 2 π 2 ( u 2 + v 2 ) . (43) Indeed, for integer u , v u,v u , v B 0 ≥ max ( ∣ u ∣ , ∣ v ∣ ) B_0\ge\max(|u|,|v|) B 0 ≥ max ( ∣ u ∣ , ∣ v ∣ ) B 0 ≥ ( u 2 + v 2 ) / 2 B_0\ge(u^2+v^2)/2 B 0 ≥ ( u 2 + v 2 ) /2 B 0 ≥ 3 u 2 / 4 B_0\ge3u^2/4 B 0 ≥ 3 u 2 /4 ∣ u ∣ ≤ 1 |u|\le1 ∣ u ∣ ≤ 1 v . v. v .
Thus k 0 k_0 k 0 1 + S t ( q ) 1+S_t(q) 1 + S t ( q )
sup q ∣ S t ( q ) ∣ ≤ ∑ ( u , v ) ≠ ( 0 , 0 ) e − ( 2 π 2 / 9 ) t ( u 2 + v 2 ) . \sup_q|S_t(q)|\le\sum_{(u,v)\ne(0,0)}e^{-(2\pi^2/9)t(u^2+v^2)}. q sup ∣ S t ( q ) ∣ ≤ ( u , v ) = ( 0 , 0 ) ∑ e − ( 2 π 2 /9 ) t ( u 2 + v 2 ) . For t ≥ 1 t\ge1 t ≥ 1 r = e − 2 π 2 / 9 < 1 / 8 r=e^{-2\pi^2/9}<1/8 r = e − 2 π 2 /9 < 1/8 u 2 ≥ ∣ u ∣ u^2\ge|u| u 2 ≥ ∣ u ∣ u u u
sup q ∣ S t ( q ) ∣ ≤ ( ∑ u ∈ Z e − ( 2 π 2 / 9 ) u 2 ) 2 − 1 < ( 9 7 ) 2 − 1 = 32 49 < 1. \sup_q|S_t(q)|\le\left(\sum_{u\in\mathbb{Z}}e^{-(2\pi^2/9)u^2}\right)^2-1<\left(\frac{9}{7}\right)^2-1=\frac{32}{49}<1. q sup ∣ S t ( q ) ∣ ≤ ( u ∈ Z ∑ e − ( 2 π 2 /9 ) u 2 ) 2 − 1 < ( 7 9 ) 2 − 1 = 49 32 < 1. Consequently, the heat kernel is nowhere zero for every t ≥ 1 t\ge1 t ≥ 1
5. Covering numbers. At small times, let Z t = { x + ( t ) , x − ( t ) } Z_t=\{x_+(t),x_-(t)\} Z t = { x + ( t ) , x − ( t )} p + Z t p+Z_t p + Z t p 2 − p 1 ∉ Z t − Z t p_2-p_1\notin Z_t-Z_t p 2 − p 1 ∈ / Z t − Z t M 0 M_0 M 0 h t = 2 h_t=2 h t = 2
Products and covering levels
Lemma 5.13 (Finite defects in product factors). Consider a product of flat tori, trivial Hermitian line bundles with constant unitary connections, and the product connection and metric. Fix a time. Suppose that exactly k k k h t = k + 1 h_t=k+1 h t = k + 1
Proof. The product heat kernel is the product of the factor kernels. For each of q ≤ k q \le k q ≤ k q q q
For the upper bound, choose k + 1 k+1 k + 1 k k k k + 1 k+1 k + 1 k = 0 k=0 k = 0
Theorem 5.14 (Arbitrarily many distinct levels). For each N ≥ 1 N \ge1 N ≥ 1 2 N 2N 2 N
I 0 < I 1 < ⋯ < I N I_0 < I_1 < \cdots< I_N I 0 < I 1 < ⋯ < I N such that
h t = N + 1 − j ( t ∈ I j , 0 ≤ j ≤ N ) . \mathfrak{h}_t = N+1-j \qquad(t \in I_j,\ 0 \le j \le N). h t = N + 1 − j ( t ∈ I j , 0 ≤ j ≤ N ) . The ordering means that every time in I j I_j I j I j + 1 I_{j+1} I j + 1
Proof. Take N N N s j g 0 s_jg_0 s j g 0 s 1 > 0 s_1 > 0 s 1 > 0
s j + 1 > s j / ε ( 1 ≤ j < N ) . s_{j+1} > s_j/\varepsilon\qquad(1 \le j < N). s j + 1 > s j / ε ( 1 ≤ j < N ) . The effective time in factor j j j t / s j t/s_j t / s j (32) . Define
I 0 = ( 0 , ε s 1 ) , I j = ( s j , ε s j + 1 ) ( 1 ≤ j < N ) , I N = ( s N , ∞ ) . I_0 = (0,\varepsilon s_1), \qquad I_j = (s_j,\varepsilon s_{j+1}) \quad(1 \le j < N), \qquad I_N = (s_N,\infty). I 0 = ( 0 , ε s 1 ) , I j = ( s j , ε s j + 1 ) ( 1 ≤ j < N ) , I N = ( s N , ∞ ) . On I j I_j I j j j j N − j N-j N − j
This theorem determines values on the displayed intervals. It makes no claim about the intervening times or monotonicity of the complete profile. The dimension grows with N N N
Corollary 5.15 (Persistence of a two-time distinction). For the two-dimensional example, choose t s ∈ ( 0 , ε ) t_s \in(0,\varepsilon) t s ∈ ( 0 , ε ) t ℓ ≥ 1 t_\ell\ge1 t ℓ ≥ 1
h t s = 2 , h t ℓ = 1. \mathfrak{h}_{t_s} = 2, \qquad\mathfrak{h}_{t_\ell} = 1. h t s = 2 , h t ℓ = 1. Proof. At t s t_s t s h t s ≥ 2 \mathfrak{h}_{t_s} \ge2 h t s ≥ 2 t ℓ t_\ell t ℓ M 0 × M 0 M_0 \times M_0 M 0 × M 0
Related work
Connection heat kernels already underlie vector diffusion maps [21 ] and the vector heat method [22 ] . The off-cut-locus estimate used here is the standard expansion in the precise uniform form proved by Ludewig [20 ] . The topological arguments apply classical bundle and characteristic-class theory [5 ] .
Orthogonality loci of coherent states and their polar divisors provide a close structural precedent for kernel-zero geometry [23 ] . Knöppel studies zero currents of random line-bundle sections distributed using heat flow [24 ] ; here the sections are deterministic and have a fixed source. The lattice sums in Subsection 5.5 belong to the established theta-function description of torus heat kernels; see, for scalar rectangular and hexagonal tori, Faulhuber [25 ] .
The results concern the rank-defect correspondence and covering by heat trivializations; they do not include a general transversality theorem or a classification of possible time profiles.
Tangent shapes and knots
We study embedded shapes in tangent spaces and knots in spheres of tangent directions, with particular attention to oriented four-manifolds.
Tangent shapes
Throughout this section, manifolds are smooth, Hausdorff, and second countable. The base M n M^n M n F m F^m F m m ≤ n m \le n m ≤ n
Definition 6.1 (Tangent shape). A tangent F F F is an embedded submanifold, with boundary when appropriate,
S ⊂ T M ∖ 0 M , p = π ∣ S : S ⟶ M , S \subset TM \setminus0_M,\qquad p = \left.\pi\right|_S:S \longrightarrow M, S ⊂ T M ∖ 0 M , p = π ∣ S : S ⟶ M , such that p p p F F F
S x = p − 1 ( x ) S_x=p^{-1}(x) S x = p − 1 ( x ) is an embedded copy of F F F T x M ∖ { 0 } T_xM \setminus\{0\} T x M ∖ { 0 } π : T M → M \pi:TM \to M π : T M → M 0 M 0_M 0 M
A diffeomorphism f : M → N f:M \to N f : M → N isomorphism of shape fields when
d f ( S ) = S ′ . df(S)=S'. df ( S ) = S ′ . A deformation is a smooth family of such fields over a fixed base. More precisely, it is an F F F
pr M ∗ T M ∖ 0 ⟶ M × [ 0 , 1 ] . \operatorname{pr}_M^*TM \setminus0 \longrightarrow M \times[0,1]. pr M ∗ T M ∖ 0 ⟶ M × [ 0 , 1 ] . A parametrized field additionally specifies a smooth map
ι : M × F ⟶ T M , π ( ι ( x , u ) ) = x , \iota:M \times F \longrightarrow TM,\qquad\pi(\iota(x,u))=x, ι : M × F ⟶ T M , π ( ι ( x , u )) = x , such that each ι x \iota_x ι x
The tautological section
The pullback p ∗ T M p^*TM p ∗ T M
s ( x , v ) = v ∈ T x M , ( x , v ) ∈ S . (44) s(x,v)=v\in T_xM,\qquad(x,v)\in S. \tag*{(44)} s ( x , v ) = v ∈ T x M , ( x , v ) ∈ S . (44) With a metric, this gives a splitting
p ∗ T M ≅ R ⊕ s ⊥ . p^*TM \cong\mathbb{R} \oplus s^\perp. p ∗ T M ≅ R ⊕ s ⊥ . Consequently,
p ∗ e ( T M ) = 0. p^*e(TM)=0. p ∗ e ( T M ) = 0. whenever T M TM T M [26 ] . This does not , in general, give a nowhere-zero vector field on M M M
The space of structures
Set
Q F n = Emb ( F , R n ∖ { 0 } ) / Diff ( F ) . Q_F^n = \operatorname{Emb}(F,\mathbb{R}^n \setminus\{0\})/\operatorname{Diff}(F). Q F n = Emb ( F , R n ∖ { 0 }) / Diff ( F ) . Here embeddings are identified by reparametrization, and GL ( n , R ) \operatorname{GL}(n,\mathbb{R}) GL ( n , R )
Shape fields are precisely smooth sections of the associated bundle
Fr ( T M ) × GL ( n , R ) Q F n ⟶ M . (45) \operatorname{Fr}(TM) \times_{\operatorname{GL}(n,\mathbb{R})} Q_F^n \longrightarrow M. \tag*{(45)} Fr ( T M ) × GL ( n , R ) Q F n ⟶ M . (45) Indeed, a frame identifies a fibre shape with an element of Q F n Q_F^n Q F n [27 ] .
Basic examples
A one-point shape is a nowhere-zero vector field.
Every Riemannian metric gives the unit-sphere field
S x = { v ∈ T x M : ∣ v ∣ g = 1 } S_x = \{v \in T_xM : |v|_g = 1\} S x = { v ∈ T x M : ∣ v ∣ g = 1 } and the shell field
S x = { v ∈ T x M : 1 ≤ ∣ v ∣ g ≤ 2 } , S_x = \{v \in T_xM : 1 \le|v|_g \le2\}, S x = { v ∈ T x M : 1 ≤ ∣ v ∣ g ≤ 2 } , with fibres S n − 1 S^{n-1} S n − 1 S n − 1 × [ 1 , 2 ] S^{n-1} \times[1,2] S n − 1 × [ 1 , 2 ]
If T M TM T M
F ⊂ R n ∖ { 0 } F \subset\mathbb{R}^n \setminus\{0\} F ⊂ R n ∖ { 0 } gives a constant field in a chosen framing.
No convexity, radiality, or distinguished point of F F F
Euler obstruction and surfaces
Theorem 6.2 (Euler obstruction). Let M n M^n M n M M M F F F
χ ( F ) e ( T M ) = 0 in H n ( M ; Z ) \chi(F)e(TM) = 0 \quad\text{in } H^n(M;\mathbb{Z}) χ ( F ) e ( T M ) = 0 in H n ( M ; Z ) and therefore
χ ( F ) χ ( M ) = 0. (46) \chi(F)\chi(M) = 0. \tag*{(46)} χ ( F ) χ ( M ) = 0. (46) The numerical conclusion also holds for closed nonorientable M M M
Proof. Equation (44) implies
p ∗ e ( T M ) = 0. p^*e(TM) = 0. p ∗ e ( T M ) = 0. The Becker–Gottlieb transfer for the compact-fibre bundle p p p
tr p : H ∗ ( S ; Z ) ⟶ H ∗ ( M ; Z ) \operatorname{tr}_p : H^*(S;\mathbb{Z}) \longrightarrow H^*(M;\mathbb{Z}) tr p : H ∗ ( S ; Z ) ⟶ H ∗ ( M ; Z ) satisfying
tr p ∘ p ∗ = χ ( F ) id \operatorname{tr}_p \circ p^* = \chi(F)\operatorname{id} tr p ∘ p ∗ = χ ( F ) id [28 ] . Compact smooth manifolds with boundary have finite CW type, so this applies to the allowed fibres.
Applying this identity to e ( T M ) e(TM) e ( T M )
0 = tr p ( p ∗ e ( T M ) ) = χ ( F ) e ( T M ) . 0 = \operatorname{tr}_p(p^*e(TM)) = \chi(F)e(TM). 0 = tr p ( p ∗ e ( T M )) = χ ( F ) e ( T M ) . Pairing with the fundamental class and using
⟨ e ( T M ) , [ M ] ⟩ = χ ( M ) \langle e(TM), [M]\rangle= \chi(M) ⟨ e ( T M ) , [ M ]⟩ = χ ( M ) proves the numerical assertion [26 ] . For nonorientable M M M M ~ \widetilde{M} M
χ ( M ~ ) = 2 χ ( M ) . \chi(\widetilde{M}) = 2\chi(M). χ ( M ) = 2 χ ( M ) . Thus a base with nonzero Euler characteristic cannot support a shape with nonzero Euler characteristic. This obstruction uses the topology of the fibre, not its knotting.
Theorem 6.3 (Planar domains in tangent planes). Let M M M F F F R 2 \mathbb{R}^2 R 2
M admits a tangent F -shape field ⟺ χ ( M ) χ ( F ) = 0. (47) M\text{ admits a tangent }F\text{-shape field}\quad\Longleftrightarrow\quad\chi(M)\chi(F)=0. \tag*{(47)} M admits a tangent F -shape field ⟺ χ ( M ) χ ( F ) = 0. (47) Proof. Necessity is Theorem 6.2. Since F F F b ≥ 1 b \ge1 b ≥ 1
χ ( F ) = 2 − b . \chi(F)=2-b. χ ( F ) = 2 − b . Suppose first that χ ( F ) = 0 \chi(F)=0 χ ( F ) = 0 b = 2 b=2 b = 2 F F F
1 ≤ ∣ v ∣ g ≤ 2 1\leq\lvert v\rvert_g\leq2 1 ≤ ∣ v ∣ g ≤ 2 constructs the required field on every surface.
Suppose instead that χ ( M ) = 0 \chi(M)=0 χ ( M ) = 0 V V V
T M = R V ⊕ L , L = V ⊥ . TM=\mathbb{R}V\oplus L,\qquad L=V^\perp. T M = R V ⊕ L , L = V ⊥ . Choose a planar model A A A F F F a > 0 a>0 a > 0
( a , b ) ⟼ ( a , − b ) . (a,b)\longmapsto(a,-b). ( a , b ) ⟼ ( a , − b ) . Such a model is a disk centred on the positive horizontal axis with b − 1 b-1 b − 1
In a local unit frame ( V , W ) (V,W) ( V , W ) R V ⊕ L \mathbb{R}V\oplus L R V ⊕ L
S x = { a V x + b W x : ( a , b ) ∈ A } . S_x=\{aV_x+bW_x:(a,b)\in A\}. S x = { a V x + b W x : ( a , b ) ∈ A } . Replacing W W W − W -W − W a > 0 a>0 a > 0
Table 1.
Equation (6.4) classifies existence . It does not classify deformation classes or the embeddings of the individual fibres.
Three-dimensional knot fields
Throughout this subsection, dim M = 3 \dim M = 3 dim M = 3 F = S 1 F = S^1 F = S 1
Knot type means ordinary unparametrized ambient-isotopy type in R 3 \mathbb{R}^3 R 3
A knot type K K K chiral if it is not isotopic to its mirror image, even after forgetting the orientation of the knot.
A field of unoriented ambient type K K K K K K
Theorem 6.4 (Chirality detects orientability). Fix a chiral knot type K K K K K K
Proof. Fix a handedness of K K K x x x T x M T_xM T x M K x K_x K x
Knot type is locally constant in a smooth family of compact embeddings: nearby embeddings are isotopic, and isotopy extension supplies an ambient isotopy. Hence the selected orientations vary continuously and orient M M M
Conversely, a closed orientable three-manifold is parallelizable [29 ] . Choose a trivialization
T M ≅ M × R 3 TM \cong M \times\mathbb{R}^3 T M ≅ M × R 3 and place a representative
K ⊂ R 3 ∖ { 0 } K \subset\mathbb{R}^3 \setminus\{0\} K ⊂ R 3 ∖ { 0 } in every fibre.
Tangent trefoils
Every closed orientable three-manifold admits tangent trefoils. In a framing of T 3 T^3 T 3 S 3 S^3 S 3 ( 2 , 3 ) (2,3) ( 2 , 3 )
κ ( t ) = ( ( 2 + cos 3 t ) cos 2 t , ( 2 + cos 3 t ) sin 2 t , sin 3 t ) , t ∈ R / 2 π Z . (48) \kappa(t) = \left((2+\cos3t)\cos2t,\,(2+\cos3t)\sin2t,\,\sin3t\right), \qquad t \in\mathbb{R}/2\pi\mathbb{Z}. \tag*{(48)} κ ( t ) = ( ( 2 + cos 3 t ) cos 2 t , ( 2 + cos 3 t ) sin 2 t , sin 3 t ) , t ∈ R /2 π Z . (48) Its distance from the vertical axis is at least 1 1 1
Holonomy rigidity
Definition 6.5 (Parallel field). On a Riemannian manifold ( M , g ) (M,g) ( M , g ) parallel if Levi–Civita parallel transport satisfies
P γ ( S γ ( 0 ) ) = S γ ( 1 ) P_\gamma(S_{\gamma(0)}) = S_{\gamma(1)} P γ ( S γ ( 0 ) ) = S γ ( 1 ) for every piecewise smooth path γ \gamma γ
Lemma 6.6. For every nontrivial smooth knot K ⊂ R 3 K \subset\mathbb{R}^3 K ⊂ R 3
Stab O ( 3 ) ( K ) = { A ∈ O ( 3 ) : A ( K ) = K } \operatorname{Stab}_{O(3)}(K) = \{A \in O(3) : A(K) = K\} Stab O ( 3 ) ( K ) = { A ∈ O ( 3 ) : A ( K ) = K } is finite.
Proof. The stabilizer is a closed subgroup of the compact Lie group O ( 3 ) O(3) O ( 3 )
If its identity component had positive dimension, it would contain a circle of rotations about an axis. The knot cannot lie entirely in that axis, so it contains the circular orbit of an off-axis point.
This orbit is an embedded circle in the connected one-manifold K K K K K K K K K
Thus K K K
Proof. At every x ∈ M x \in M x ∈ M
Hol x ( g ) ⊂ Stab O ( T x M , g x ) ( K x ) . \operatorname{Hol}_{x}(g) \subset\operatorname{Stab}_{\mathrm{O}(T_xM,g_x)}(K_x). Hol x ( g ) ⊂ Stab O ( T x M , g x ) ( K x ) . The right-hand side is finite by the lemma. The restricted holonomy group is connected, hence trivial.
By the Ambrose–Singer theorem, curvature endomorphisms belong to the restricted holonomy Lie algebra. That Lie algebra is zero, so the curvature vanishes [30 ] .
The same argument works in any dimension when a parallel shape has finite orthogonal stabilizer.
The standard flat three-torus has parallel knot fields. Although S 3 S^3 S 3
Deformation and normal variation
Parametrized deformation classes
Theorem 6.8 (Infinitely many parametrized deformation classes). For every smooth knot type K K K K K K S 3 S^3 S 3
Proof. Fix a framing
τ x : R 3 ⟶ T x S 3 , \tau_x : \mathbb{R}^3 \longrightarrow T_xS^3, τ x : R 3 ⟶ T x S 3 , an orientation of S 3 S^3 S 3
κ : S 1 ⟶ R 3 ∖ { 0 } \kappa: S^1 \longrightarrow\mathbb{R}^3 \setminus\{0\} κ : S 1 ⟶ R 3 ∖ { 0 } of type K K K
Choose smooth maps
R k : S 3 ⟶ S O ( 3 ) , k ∈ Z , R_k : S^3 \longrightarrow\mathrm{SO}(3), \qquad k \in\mathbb{Z}, R k : S 3 ⟶ SO ( 3 ) , k ∈ Z , representing the elements of
π 3 ( S O ( 3 ) ) ≅ Z . \pi_3(\mathrm{SO}(3)) \cong\mathbb{Z}. π 3 ( SO ( 3 )) ≅ Z . Define
γ k ( x , t ) = τ x ( R k ( x ) κ ( t ) ) . (49) \gamma_k(x,t) = \tau_x\bigl(R_k(x)\kappa(t)\bigr). \tag*{(49)} γ k ( x , t ) = τ x ( R k ( x ) κ ( t ) ) . (49) For any parametrized field γ \gamma γ t 0 ∈ S 1 t_0 \in S^1 t 0 ∈ S 1
u γ ( x ) = τ x − 1 ( ∂ t γ ( x , t 0 ) ) ∣ τ x − 1 ( ∂ t γ ( x , t 0 ) ) ∣ ∈ S 2 . (50) u_\gamma(x) =
\frac{\tau_x^{-1}(\partial_t\gamma(x,t_0))}
{\left\lvert\tau_x^{-1}(\partial_t\gamma(x,t_0)) \right\rvert}
\in S^2. \tag*{(50)} u γ ( x ) = ∣ τ x − 1 ( ∂ t γ ( x , t 0 )) ∣ τ x − 1 ( ∂ t γ ( x , t 0 )) ∈ S 2 . (50) The denominator is nonzero because each fibre map is an embedding. A deformation of parametrized fields induces a homotopy of these maps.
For (6.6),
u γ k ( x ) = R k ( x ) a , a = κ ′ ( t 0 ) ∣ κ ′ ( t 0 ) ∣ u_{\gamma_k}(x) = R_k(x)a, \qquad a = \frac{\kappa'(t_0)}{\left\lvert\kappa'(t_0) \right\rvert} u γ k ( x ) = R k ( x ) a , a = ∣ κ ′ ( t 0 ) ∣ κ ′ ( t 0 ) The fibration
S O ( 2 ) ⟶ S O ( 3 ) → R ↦ R a S 2 \mathrm{SO}(2) \longrightarrow\mathrm{SO}(3) \xrightarrow{R\mapsto Ra} S^2 SO ( 2 ) ⟶ SO ( 3 ) R ↦ R a S 2 induces an isomorphism on π 3 \pi_3 π 3
π 3 ( S O ( 2 ) ) = π 2 ( S O ( 2 ) ) = 0. \pi_3(\mathrm{SO}(2))=\pi_2(\mathrm{SO}(2))=0. π 3 ( SO ( 2 )) = π 2 ( SO ( 2 )) = 0. Therefore the maps u γ k u_{\gamma_k} u γ k
π 3 ( S 2 ) ≅ Z . \pi_3(S^2)\cong\mathbb{Z}. π 3 ( S 2 ) ≅ Z . They are detected by the Hopf invariant [31 ] . With compatible choices of generators,
H ( u γ k ) = k . H(u_{\gamma_k})=k. H ( u γ k ) = k . Thus the fields γ k \gamma_k γ k
The integer labels use the chosen framing. This proves neither a classification of all parametrized fields nor a classification after forgetting parametrizations or quotienting by diffeomorphisms of the base.
Normal variation
Fix a Riemannian metric g g g
∂ F = ∅ . \partial F=\varnothing. ∂ F = ∅ . Let
ι : U × F ⟶ T M \iota:U\times F\longrightarrow TM ι : U × F ⟶ T M be a local parametrization of a shape field. For v = ι x ( u ) v=\iota_x(u) v = ι x ( u ) X ∈ T x M X\in T_xM X ∈ T x M
B v ( X ) = ( ∇ X ι ( ⋅ , u ) ) ⊥ T v S x . (51) B_v(X)=\left(\nabla_X\iota(\mathord{\cdot},u)\right)^{\perp_{T_vS_x}} . \tag*{(51)} B v ( X ) = ( ∇ X ι ( ⋅ , u ) ) ⊥ T v S x . (51) Here ∇ \nabla ∇ T v S x T_vS_x T v S x T x M T_xM T x M
T v ( T x M ) ≅ T x M , T_v(T_xM)\cong T_xM, T v ( T x M ) ≅ T x M , and the orthogonal projection uses g x g_x g x
Thus B B B
p ∗ T ∗ M ⊗ ν v e r t S , p^*T^*M\otimes\nu_{\mathrm{vert}}S, p ∗ T ∗ M ⊗ ν vert S , where ν v e r t S \nu_{\mathrm{vert}}S ν vert S
Proposition 6.9. For compact boundaryless fibres , (6.8) is independent of local parametrization, and
B = 0 ⟺ the shape field is parallel. B=0\quad\Longleftrightarrow\quad\text{the shape field is parallel.} B = 0 ⟺ the shape field is parallel. Proof. An x x x ∇ X ι \nabla_X\iota ∇ X ι S x S_x S x B = 0 B=0 B = 0
Conversely, consider a smooth path in the base and trivialize T M TM T M ι t \iota_t ι t F F F
If B = 0 B=0 B = 0
∂ t ι t = d ι t ( Z t ) \partial_t\iota_t=d\iota_t(Z_t) ∂ t ι t = d ι t ( Z t ) for a smooth time-dependent vector field Z t Z_t Z t F F F ϕ t \phi_t ϕ t
∂ t ϕ t = − Z t ∘ ϕ t , ϕ 0 = id . (52) \partial_t\phi_t=-Z_t\circ\phi_t,\qquad\phi_0=\operatorname{id}. \tag*{(52)} ∂ t ϕ t = − Z t ∘ ϕ t , ϕ 0 = id . (52) ::: Compactness and absence of boundary ensure that this flow exists throughout each compact time interval. The chain rule gives
d d t ( ι t ∘ ϕ t ) = 0. \frac{d}{dt}(\iota_t \circ\phi_t)=0. d t d ( ι t ∘ ϕ t ) = 0. Hence the images of the ι t \iota_t ι t
The same conclusion holds along piecewise smooth paths by concatenation. Thus parallel transport preserves the shape field.
For fibres with boundary, (6.8) alone is insufficient: it does not detect boundary motion tangent to the fibre. For example, an expanding interval in a one-dimensional vector space has identically zero normal variation. No corresponding equivalence is asserted in that case.
Four-dimensional direction knots
In Subsections 6.5–6.11, unless otherwise specified, M M M a ˉ \bar{a} a ˉ a ∈ H 2 ( M ; Z ) a \in H^2(M;\mathbb{Z}) a ∈ H 2 ( M ; Z ) 2 2 2
The oriented direction bundle is
S ( T M ) = ( T M ∖ 0 ) / R > 0 , π : S ( T M ) ⟶ M . S(TM)=(TM\setminus0)/\mathbb{R}_{>0},\qquad\pi:S(TM)\longrightarrow M. S ( T M ) = ( T M ∖ 0 ) / R > 0 , π : S ( T M ) ⟶ M . Its fiber is an oriented 3-sphere. A Riemannian metric identifies it with the unit tangent sphere bundle by radial normalization, but no metric is part of the definition.
Definition 6.10 (Tangent-knot field). An oriented tangent-knot field is a smooth embedded submanifold K ⊂ S ( T M ) K\subset S(TM) K ⊂ S ( T M ) π ∣ K \pi|_K π ∣ K type K K K K x ⊂ S ( T x M ) K_x\subset S(T_xM) K x ⊂ S ( T x M ) K ⊂ S 3 K\subset S^3 K ⊂ S 3
The fibers are unparametrized : the data do not include a map M × S 1 → S ( T M ) M\times S^1\to S(TM) M × S 1 → S ( T M )
Proposition 6.11 (Basic properties). The total space K K K K → M K\to M K → M
Proof. Ehresmann’s theorem makes the proper submersion a smooth circle bundle. Local trivializations and isotopy extension identify nearby embedded fibers by ambient isotopies. An oriented frame on an open set identifies its direction bundle with U × S 3 U\times S^3 U × S 3 U × K U\times K U × K
Define the charge of a field by
a ( K ) = e ( K → M ) ∈ H 2 ( M ; Z ) . a(K)=e(K\to M)\in H^2(M;\mathbb{Z}). a ( K ) = e ( K → M ) ∈ H 2 ( M ; Z ) . Here the Euler class of an oriented circle bundle is defined using Diff + ( S 1 ) ≃ SO ( 2 ) \operatorname{Diff}^{+}(S^1)\simeq\operatorname{SO}(2) Diff + ( S 1 ) ≃ SO ( 2 ) e ( S ( L ) ) = c 1 ( L ) e(S(L))=c_1(L) e ( S ( L )) = c 1 ( L ) L L L
For a chosen metric, let P → M P\to M P → M C K \mathcal{C}_K C K K K K
Emb ( S 1 , S 3 ) / Diff + ( S 1 ) , (53) \operatorname{Emb}(S^1,S^3)/\operatorname{Diff}^{+}(S^1), \tag*{(53)} Emb ( S 1 , S 3 ) / Diff + ( S 1 ) , (53) with its smooth topology. A field is equivalently a smooth family of points in the associated bundle
P × S O ( 4 ) C K ⟶ M . P \mathbin{\times_{\mathrm{SO}(4)}} \mathcal{C}_K \longrightarrow M. P × SO ( 4 ) C K ⟶ M . This description is the bridge to obstruction theory.
Relation to tangent shapes
The direction construction is a restricted variant of the tangent-shape construction. A metric identifies each positive ray with its unit vector, so a direction-knot field lifts to a tangent S 1 S^1 S 1 T x M ∖ { 0 } T_xM \setminus\{0\} T x M ∖ { 0 }
The orientation of the circle fibers does not specify a global parametrization. The latter would trivialize the circle bundle. Also, the Euler obstruction in Theorem 6.2 is vacuous for F = S 1 F=S^1 F = S 1 S 3 S^3 S 3
Torus knots and existence
Fix relatively prime integers p , q ≥ 2 p,q \ge2 p , q ≥ 2
K p , q = { 2 − 1 / 2 ( z p , z q ) : ∣ z ∣ = 1 } ⊂ S 3 ⊂ C 2 , K_{p,q}=\left\{2^{-1/2}(z^p,z^q): |z|=1\right\}\subset S^3\subset\mathbb{C}^2, K p , q = { 2 − 1/2 ( z p , z q ) : ∣ z ∣ = 1 } ⊂ S 3 ⊂ C 2 , H p , q = { diag ( u p , u q ) : ∣ u ∣ = 1 } ⊂ S O ( 4 ) . H_{p,q}=\{\operatorname{diag}(u^p,u^q): |u|=1\}\subset\mathrm{SO}(4). H p , q = { diag ( u p , u q ) : ∣ u ∣ = 1 } ⊂ SO ( 4 ) . The knot has the orientation induced by z = e i t z=e^{it} z = e i t H p , q ≅ S 1 H_{p,q}\cong S^1 H p , q ≅ S 1
Lemma 6.12 (External knot-space input: Hatcher). The natural rotation-orbit inclusion
S O ( 4 ) / H p , q ⟶ C K p , q (54) \mathrm{SO}(4)/H_{p,q}\longrightarrow\mathcal{C}_{K_{p,q}} \tag*{(54)} SO ( 4 ) / H p , q ⟶ C K p , q (54) is an S O ( 4 ) \mathrm{SO}(4) SO ( 4 )
This is the oriented version of Hatcher’s theorem [32 ] : the unoriented stabilizer is a weighted copy of O ( 2 ) O(2) O ( 2 ) (54) . The map, not just the abstract homotopy type of its target, is important.
Theorem 6.13 (Weighted circle reduction). A field of type K p , q K_{p,q} K p , q a a a M M M L L L c 1 ( L ) = a c_1(L)=a c 1 ( L ) = a
T M ≅ ( L ⊗ p ⊕ L ⊗ q ) R . (55) TM \cong\left(L^{\otimes p}\oplus L^{\otimes q}\right)_{\mathbb{R}}. \tag*{(55)} T M ≅ ( L ⊗ p ⊕ L ⊗ q ) R . (55) Moreover, every such field is homotopic through sections of the knot-space bundle to one arising from an H p , q H_{p,q} H p , q
Proof. The equivariant map (6.9) induces a fiberwise homotopy equivalence
P / H p , q ⟶ P × S O ( 4 ) C K p , q . P/H_{p,q}\longrightarrow P\mathbin{\times_{\mathrm{SO}(4)}}\mathcal{C}_{K_{p,q}}. P / H p , q ⟶ P × SO ( 4 ) C K p , q . Thus a section on the right determines, up to homotopy, a section on the left; this is the standard homotopy lifting argument over a CW base. Such a section is an H p , q H_{p,q} H p , q P P P S ( L ) S(L) S ( L ) u ∈ H p , q u\in H_{p,q} u ∈ H p , q z ↦ u z z\mapsto uz z ↦ u z a a a
Conversely, choose a Hermitian metric on L L L
S ( L ) ⟶ S ( ( L ⊗ p ⊕ L ⊗ q ) R ) , v ⟼ [ v ⊗ p ⊕ v ⊗ q ] S(L) \longrightarrow S((L^{\otimes p}\oplus L^{\otimes q})_{\mathbb{R}}),\qquad v\longmapsto[v^{\otimes p}\oplus v^{\otimes q}] S ( L ) ⟶ S (( L ⊗ p ⊕ L ⊗ q ) R ) , v ⟼ [ v ⊗ p ⊕ v ⊗ q ] is fiberwise an embedded K p , q K_{p,q} K p , q gcd ( p , q ) = 1 \gcd(p,q)=1 g cd( p , q ) = 1 a a a
Remark 6.14. The theorem is a statement about existence and homotopy, not a linear normal form for every given fiber and not a classification up to diffeomorphism. The class a a a a = 0 a=0 a = 0 T M TM T M
For any such field, the characteristic-class identities are
e ( T M ) = p q a 2 , e(TM)=pq\,a^2, e ( T M ) = pq a 2 , p 1 ( T M ) = ( p 2 + q 2 ) a 2 , (56) p_1(TM)=(p^2+q^2)a^2, \tag*{(56)} p 1 ( T M ) = ( p 2 + q 2 ) a 2 , (56) w 2 ( T M ) = ( p + q ) a ˉ . w_2(TM)=(p+q)\bar{a}. w 2 ( T M ) = ( p + q ) a ˉ . They follow directly from the Chern classes of the two line summands. In particular, the reduction induces an almost complex structure with c 1 = ( p + q ) a c_1=(p+q)a c 1 = ( p + q ) a
Theorem 6.15 (Existence on a closed four-manifold). Let M M M a ∈ H 2 ( M ; Z ) a\in H^2(M;\mathbb{Z}) a ∈ H 2 ( M ; Z ) K p , q K_{p,q} K p , q a a a N = ⟨ a 2 , [ M ] ⟩ N=\langle a^2,[M]\rangle N = ⟨ a 2 , [ M ]⟩
( p + q ) a ˉ = w 2 ( T M ) , (p+q)\bar{a}=w_2(TM), ( p + q ) a ˉ = w 2 ( T M ) , p q N = χ ( M ) , pqN=\chi(M), pq N = χ ( M ) , ( p 2 + q 2 ) N = 3 σ ( M ) . (57) (p^2+q^2)N=3\sigma(M). \tag*{(57)} ( p 2 + q 2 ) N = 3 σ ( M ) . (57) Proof. Necessity follows from (56) , the Euler formula and the signature theorem. For sufficiency, choose L L L c 1 ( L ) = a c_1(L)=a c 1 ( L ) = a
E = ( L ⊗ p ⊕ L ⊗ q ) R . E=(L^{\otimes p}\oplus L^{\otimes q})_{\mathbb{R}}. E = ( L ⊗ p ⊕ L ⊗ q ) R . Equations (57) give equal w 2 , e , p 1 w_2,e,p_1 w 2 , e , p 1 E E E T M TM T M H 4 ( M ; Z ) ≅ Z H^4(M;\mathbb{Z})\cong\mathbb{Z} H 4 ( M ; Z ) ≅ Z [33 ] . Hence E ≅ T M E\cong TM E ≅ T M
The same argument can be expressed through the Hirzebruch–Hopf criterion [34 ] : c = ( p + q ) a c=(p+q)a c = ( p + q ) a c ˉ = w 2 \bar{c}=w_2 c ˉ = w 2
c 2 [ M ] = 2 χ + 3 σ . c^2[M]=2\chi+3\sigma. c 2 [ M ] = 2 χ + 3 σ . Topology and arithmetic
Corollary 6.16 (Slope identity). A closed manifold carrying a K p , q K_{p,q} K p , q
3 p q σ = ( p 2 + q 2 ) χ , 3pq\,\sigma=(p^2+q^2)\chi, 3 pq σ = ( p 2 + q 2 ) χ , 2 χ − 3 σ = − ( p − q ) 2 N . (58) 2\chi-3\sigma=-(p-q)^2N. \tag*{(58)} 2 χ − 3 σ = − ( p − q ) 2 N . (58) If M M M N > 0 N>0 N > 0 σ > 0 \sigma>0 σ > 0
b 2 = p q N − 2 , σ = p 2 + q 2 3 N . b_2=pqN-2,\qquad\sigma=\frac{p^2+q^2}{3}N. b 2 = pq N − 2 , σ = 3 p 2 + q 2 N . If p p p q q q M M M
Proof. Use (57) , χ = 2 + b 2 \chi=2+b_2 χ = 2 + b 2 p + q ≡ 0 ( m o d 2 ) p+q\equiv0\pmod{2} p + q ≡ 0 ( mod 2 )
Corollary 6.17 (Trefoil divisibility). For a closed manifold with a K 2 , 3 K_{2,3} K 2 , 3
N ∈ 12 Z , ( χ , σ ) = ( 72 k , 52 k ) for some k ∈ Z . N \in12\mathbb{Z}, \qquad(\chi,\sigma)=(72k,52k) \quad\text{for some } k\in\mathbb{Z}. N ∈ 12 Z , ( χ , σ ) = ( 72 k , 52 k ) for some k ∈ Z . If it is simply connected, χ ≥ 72 \chi\ge72 χ ≥ 72 N ∈ 48 Z N\in48\mathbb{Z} N ∈ 48 Z χ ≥ 288 \chi\ge288 χ ≥ 288
Proof. The identities are χ = 6 N \chi=6N χ = 6 N 3 σ = 13 N 3\sigma=13N 3 σ = 13 N a ˉ = w 2 \bar{a}=w_2 a ˉ = w 2 N = 3 m N=3m N = 3 m σ = 13 m \sigma=13m σ = 13 m a a a H 2 ( M ; Z ) / torsion H^2(M;\mathbb{Z})/\text{torsion} H 2 ( M ; Z ) / torsion [35 ] gives N ≡ σ ( m o d 8 ) N\equiv\sigma\pmod{8} N ≡ σ ( mod 8 )
3 m ≡ 13 m ( m o d 8 ) , 3m\equiv13m\pmod{8}, 3 m ≡ 13 m ( mod 8 ) , so 4 ∣ m 4\mid m 4 ∣ m 16 ∣ σ 16\mid\sigma 16 ∣ σ [36 ] , introduction), whence 16 ∣ m 16\mid m 16 ∣ m N > 0 N>0 N > 0
The slope identity already excludes S 4 S^4 S 4 C P 2 \mathbb{CP}^2 CP 2 S 2 × S 2 S^2\times S^2 S 2 × S 2 ( χ , σ ) (\chi,\sigma) ( χ , σ ) ( 2 , 0 ) (2,0) ( 2 , 0 ) ( 3 , 1 ) (3,1) ( 3 , 1 ) ( 4 , 0 ) (4,0) ( 4 , 0 )
Proposition 6.18 (Asymmetric hyperbolic type). Let K ⊂ S 3 K\subset S^3 K ⊂ S 3 χ = σ = 0 \chi=\sigma=0 χ = σ = 0 T M TM T M
Proof. For this asymmetric case Hatcher’s orbit theorem [32 ] gives a homotopy equivalence
SO ( 4 ) ⟶ C K . \operatorname{SO}(4)\longrightarrow\mathcal{C}_K. SO ( 4 ) ⟶ C K . The associated bundle map identifies the existence of a section with that of a section of
P × SO ( 4 ) SO ( 4 ) ≅ P . P\times_{\operatorname{SO}(4)}\operatorname{SO}(4)\cong P. P × SO ( 4 ) SO ( 4 ) ≅ P . This is a framing. The converse follows by placing a fixed copy of K K K
Einstein metrics
Let g g g
E = Ric g − 1 4 s g g E=\operatorname{Ric}_g-\frac{1}{4}s_g g E = Ric g − 4 1 s g g for traceless Ricci curvature, and W ± W^\pm W ±
∥ T ∥ 2 2 = ∫ M ∣ T ∣ 2 d μ g . \lVert T\rVert_2^2=\int_M |T|^2\,d\mu_g. ∥ T ∥ 2 2 = ∫ M ∣ T ∣ 2 d μ g . We use the Hilbert–Schmidt Weyl norm convention
∣ W ∣ 2 = 1 4 W i j k l W i j k l , ∣ E ∣ 2 = E i j E i j . |W|^2=\frac{1}{4}W_{ijkl}W^{ijkl}, \qquad|E|^2=E_{ij}E^{ij}. ∣ W ∣ 2 = 4 1 W ij k l W ij k l , ∣ E ∣ 2 = E ij E ij . Theorem 6.19 (Curvature identity and Einstein rigidity). If M M M K p , q K_{p,q} K p , q N N N g g g
∥ E ∥ 2 2 = 8 π 2 ( p − q ) 2 N + 4 ∥ W − ∥ 2 2 + 1 12 ∥ s g ∥ 2 2 . (59) \lVert E\rVert_2^2=8\pi^2(p-q)^2N+4\lVert W^-\rVert_2^2+\frac{1}{12}\lVert s_g\rVert_2^2. \tag*{(59)} ∥ E ∥ 2 2 = 8 π 2 ( p − q ) 2 N + 4 ∥ W − ∥ 2 2 + 12 1 ∥ s g ∥ 2 2 . (59) For N > 0 N>0 N > 0
∥ E ∥ 2 2 ≥ 8 π 2 ( p − q ) 2 N . \lVert E\rVert_2^2\ge8\pi^2(p-q)^2N. ∥ E ∥ 2 2 ≥ 8 π 2 ( p − q ) 2 N . Every Einstein metric on M M M
Proof. The Chern–Gauss–Bonnet and signature formulas, with these conventions, give
8 π 2 χ = ∥ W + ∥ 2 2 + ∥ W − ∥ 2 2 + 1 24 ∥ s g ∥ 2 2 − 1 2 ∥ E ∥ 2 2 , 8\pi^2\chi= \lVert W^+\rVert_2^2 + \lVert W^-\rVert_2^2 + \frac{1}{24}\lVert s_g\rVert_2^2 - \frac{1}{2}\lVert E\rVert_2^2, 8 π 2 χ = ∥ W + ∥ 2 2 + ∥ W − ∥ 2 2 + 24 1 ∥ s g ∥ 2 2 − 2 1 ∥ E ∥ 2 2 , 12 π 2 σ = ∥ W + ∥ 2 2 − ∥ W − ∥ 2 2 12\pi^2\sigma= \lVert W^+\rVert_2^2 - \lVert W^-\rVert_2^2 12 π 2 σ = ∥ W + ∥ 2 2 − ∥ W − ∥ 2 2 (see [37 ] , Section 2, in the smooth case). Subtracting yields
4 π 2 ( 2 χ − 3 σ ) = 2 ∥ W − ∥ 2 2 + 1 24 ∥ s g ∥ 2 2 − 1 2 ∥ E ∥ 2 2 . 4\pi^2(2\chi- 3\sigma) = 2\lVert W^-\rVert_2^2 + \frac{1}{24}\lVert s_g\rVert_2^2 - \frac{1}{2}\lVert E\rVert_2^2. 4 π 2 ( 2 χ − 3 σ ) = 2 ∥ W − ∥ 2 2 + 24 1 ∥ s g ∥ 2 2 − 2 1 ∥ E ∥ 2 2 . Substitute (58) . Since p ≠ q p \ne q p = q N > 0 N > 0 N > 0 E = 0 E = 0 E = 0 N < 0 N < 0 N < 0 χ = p q N < 0 \chi= pqN < 0 χ = pq N < 0 N = 0 N = 0 N = 0 E = 0 E = 0 E = 0 W + = W − = 0 W^+ = W^- = 0 W + = W − = 0 s g = 0 s_g = 0 s g = 0
This is a consequence of classical four-dimensional curvature identities, not a new curvature tensor attached to the knot field. The metric need not be compatible with the field.
Four-dimensional examples
In a hyperbolic summand of an intersection form, use generators u , v u,v u , v
u 2 = v 2 = 0 , u ⋅ v = 1. u^2 = v^2 = 0,\qquad u\cdot v = 1. u 2 = v 2 = 0 , u ⋅ v = 1. Then
( r u + s v ) 2 [ M ] = 2 r s . (ru+sv)^2[M] = 2rs. ( r u + s v ) 2 [ M ] = 2 r s . We write K 3 ‾ \overline{\mathrm{K3}} K3
b 2 = 22 , χ = 24 , σ = 16 , b_2 = 22,\qquad\chi= 24,\qquad\sigma= 16, b 2 = 22 , χ = 24 , σ = 16 , with hyperbolic summands in its intersection form (cf. [36 , 38 ] ).
Example 6.20 (A field of every knot type). A global frame on T 4 T^4 T 4
S ( T T 4 ) ≅ T 4 × S 3 . S(TT^4) \cong T^4 \times S^3. S ( T T 4 ) ≅ T 4 × S 3 . For every knot K K K T 4 × K T^4 \times K T 4 × K K 2 , 3 K_{2,3} K 2 , 3
t ⟼ 2 − 1 / 2 ( e 2 i t , e 3 i t ) , t \longmapsto2^{-1/2}(e^{2it},e^{3it}), t ⟼ 2 − 1/2 ( e 2 i t , e 3 i t ) , with $a = 0.
Example 6.21 (Nonparallelizable with zero characteristic numbers). Let
M 0 = ( C P 2 # C P ‾ 2 ) # 2 ( S 1 × S 3 ) . M_0 = (\mathbb{CP}^2 \mathbin{\#} \overline{\mathbb{CP}}{}^2) \mathbin{\#} 2(S^1 \times S^3). M 0 = ( CP 2 # CP 2 ) # 2 ( S 1 × S 3 ) . Its form is ⟨ 1 ⟩ ⊕ ⟨ − 1 ⟩ \langle1\rangle\oplus\langle-1\rangle ⟨ 1 ⟩ ⊕ ⟨ − 1 ⟩ χ = σ = 0 \chi= \sigma= 0 χ = σ = 0 h , e h,e h , e a = h + e a = h+e a = h + e
a 2 [ M 0 ] = 0 , a ‾ = w 2 ( T M 0 ) ≠ 0. a^2[M_0] = 0,\qquad\overline{a} = w_2(TM_0) \ne0. a 2 [ M 0 ] = 0 , a = w 2 ( T M 0 ) = 0. The criterion gives a trefoil field, although M 0 M_0 M 0
Example 6.22 (Sharp simply connected trefoil bound). The manifold
M 1 = # 61 C P 2 # # 9 C P ‾ 2 M_1 = \mathbin{\#}^{61}\mathbb{CP}^2 \mathbin{\#} \mathbin{\#}^{9}\overline{\mathbb{CP}}{}^2 M 1 = # 61 CP 2 # # 9 CP 2 has ( χ , σ ) = ( 72 , 52 ) (\chi,\sigma) = (72,52) ( χ , σ ) = ( 72 , 52 ) a a a
a 2 [ M 1 ] = 61 − 5 ⋅ 9 − 4 = 12 , a ‾ = w 2 ( T M 1 ) . a^2[M_1] = 61 - 5\cdot9 - 4 = 12,\qquad\overline{a} = w_2(TM_1). a 2 [ M 1 ] = 61 − 5 ⋅ 9 − 4 = 12 , a = w 2 ( T M 1 ) . Theorem 6.15 gives a trefoil field, attaining χ = 72 \chi= 72 χ = 72
Example 6.23 (Sharp simply connected spin trefoil bound). Let
M 2 = # 13 K 3 ‾ . M_2=\#^{13}\overline{K3}. M 2 = # 13 K 3 . Then ( χ , σ ) = ( 288 , 208 ) (\chi,\sigma)=(288,208) ( χ , σ ) = ( 288 , 208 ) a = 2 u + 12 v a=2u+12v a = 2 u + 12 v
a 2 [ M 2 ] = 48 , a ˉ = 0. a^2[M_2]=48,\qquad\bar a=0. a 2 [ M 2 ] = 48 , a ˉ = 0. The trefoil criterion holds, attaining the spin bound χ = 288 \chi=288 χ = 288
Example 6.24 (An odd–odd torus-knot field). Let
M 3 = # 74 K 3 ‾ # # 25 ( S 2 × S 2 ) . M_3=\#^{74}\overline{K3}\mathbin{\#}\#^{25}(S^2\times S^2). M 3 = # 74 K 3 # # 25 ( S 2 × S 2 ) . It is spin, with ( χ , σ ) = ( 1680 , 1184 ) (\chi,\sigma)=(1680,1184) ( χ , σ ) = ( 1680 , 1184 ) a = u + 24 v a=u+24v a = u + 24 v a 2 [ M 3 ] = 48 a^2[M_3]=48 a 2 [ M 3 ] = 48 ( p , q ) = ( 5 , 7 ) (p,q)=(5,7) ( p , q ) = ( 5 , 7 )
p q ⋅ 48 = 1680 pq\cdot48=1680 pq ⋅ 48 = 1680 and
( p 2 + q 2 ) 48 = 3552 = 3 ⋅ 1184. (p^2+q^2)48=3552=3\cdot1184. ( p 2 + q 2 ) 48 = 3552 = 3 ⋅ 1184. Thus M 3 M_3 M 3 K 5 , 7 K_{5,7} K 5 , 7
All connected-sum numbers follow from
χ ( X # Y ) = χ ( X ) + χ ( Y ) − 2 \chi(X\mathbin{\#}Y)=\chi(X)+\chi(Y)-2 χ ( X # Y ) = χ ( X ) + χ ( Y ) − 2 and
σ ( X # Y ) = σ ( X ) + σ ( Y ) . \sigma(X\mathbin{\#}Y)=\sigma(X)+\sigma(Y). σ ( X # Y ) = σ ( X ) + σ ( Y ) . The last three examples admit no Einstein metric by Theorem 6.19 .
Spin geography and the 11/8 conjecture
Here p , q ≥ 3 p,q\ge3 p , q ≥ 3
A = p 2 + q 2 , B = p q , ρ p , q = A 3 B . A=p^2+q^2,\qquad B=pq,\qquad\rho_{p,q}=\frac{A}{3B}. A = p 2 + q 2 , B = pq , ρ p , q = 3 B A . Let C 11 / 8 C_{11/8} C 11/8 X X X
b 2 ( X ) ≥ 11 8 ∣ σ ( X ) ∣ . b_2(X)\ge\frac{11}{8}|\sigma(X)|. b 2 ( X ) ≥ 8 11 ∣ σ ( X ) ∣. This is the simply connected form of the 11/8 conjecture discussed in [36 , 38 ] ; the reformulation below does not prove or disprove it.
Proposition 6.25 (Existence and nonexistence regions). If ρ p , q < 8 / 11 \rho_{p,q}<8/11 ρ p , q < 8/11 K p , q K_{p,q} K p , q ρ p , q ≥ 4 / 5 \rho_{p,q}\ge4/5 ρ p , q ≥ 4/5
( 12 B − 5 A ) N ≥ 48. (60) (12B-5A)N\ge48. \tag*{(60)} ( 12 B − 5 A ) N ≥ 48. (60) Proof. For existence, put
Δ = 24 B − 11 A > 0. \Delta=24B-11A>0. Δ = 24 B − 11 A > 0. Choose an integer t ≥ 1 t\ge1 t ≥ 1 m = Δ t − 1 ≥ 1 m=\Delta t-1\ge1 m = Δ t − 1 ≥ 1 n = A t n=At n = A t
M = # n K 3 ‾ # # m ( S 2 × S 2 ) M=\#^n\overline{K3}\mathbin{\#}\#^m(S^2\times S^2) M = # n K 3 # # m ( S 2 × S 2 ) one has
χ = 48 B t , σ = 16 A t , \chi= 48Bt,\qquad\sigma= 16At, χ = 48 B t , σ = 16 A t , and a hyperbolic-summand class a = u + 24 t v a=u+24tv a = u + 24 t v
a 2 [ M ] = 48 t . a^2[M]=48t. a 2 [ M ] = 48 t . Since p + q p+q p + q
For nonexistence, any candidate is spin and has nonzero b 2 b_2 b 2 [36 ] therefore applies:
B N − 2 = b 2 ≥ 5 4 σ + 2 = 5 A 12 N + 2. BN-2=b_2\ge\frac{5}{4}\sigma+2=\frac{5A}{12}N+2. B N − 2 = b 2 ≥ 4 5 σ + 2 = 12 5 A N + 2. This is (60) . In particular, 12 B − 5 A > 0 12B-5A>0 12 B − 5 A > 0 ρ p , q < 4 / 5 \rho_{p,q}<4/5 ρ p , q < 4/5
Theorem 6.26 (11/8 detection by torus-knot fields). The assertion C 11 / 8 C_{11/8} C 11/8 p , q ≥ 3 p,q\ge3 p , q ≥ 3 K p , q K_{p,q} K p , q
11 ( p 2 + q 2 ) > 24 p q , or equivalently ρ p , q > 8 11 . (61) 11(p^2+q^2)>24pq,\qquad\text{or equivalently}\qquad\rho_{p,q}>\frac{8}{11}. \tag*{(61)} 11 ( p 2 + q 2 ) > 24 pq , or equivalently ρ p , q > 11 8 . (61) Proof. Given such a field, the manifold is spin, N > 0 N>0 N > 0
b 2 σ = 3 B A − 6 A N < 3 B A < 11 8 . \frac{b_2}{\sigma}=\frac{3B}{A}-\frac{6}{AN}<\frac{3B}{A}<\frac{11}{8}. σ b 2 = A 3 B − A N 6 < A 3 B < 8 11 . Thus it violates C 11 / 8 C_{11/8} C 11/8
Conversely, let X X X s = σ ( X ) > 0 s=\sigma(X)>0 s = σ ( X ) > 0 b = b 2 ( X ) b=b_2(X) b = b 2 ( X ) p , q ≥ 3 p,q\ge3 p , q ≥ 3
8 11 < A 3 B < s b . (62) \frac{8}{11}<\frac{A}{3B}<\frac{s}{b}. \tag*{(62)} 11 8 < 3 B A < b s . (62) This is possible: ratios of coprime odd integers q / p > 1 q/p>1 q / p > 1 ( 1 , ∞ ) (1,\infty) ( 1 , ∞ )
r ⟼ r + r − 1 3 r\longmapsto\frac{r+r^{-1}}{3} r ⟼ 3 r + r − 1 maps that interval continuously and strictly increasingly onto ( 2 / 3 , ∞ ) (2/3,\infty) ( 2/3 , ∞ )
D = 3 B s − A b > 0. D=3Bs-Ab>0. D = 3 B s − A b > 0. For an integer t t t ℓ = D t − 1 ≥ 1 \ell=Dt-1\ge1 ℓ = D t − 1 ≥ 1 k = 2 A t k=2At k = 2 A t
M = # k X # # ℓ ( S 2 × S 2 ) . M=\mathbin{\#}^{k}X\mathbin{\#}\mathbin{\#}^{\ell}(S^2\times S^2). M = # k X # # ℓ ( S 2 × S 2 ) . This manifold is smooth, simply connected and spin. Direct calculation gives
χ ( M ) = k b + 2 ℓ + 2 = 6 B s t , σ ( M ) = k s = 2 A s t . \chi(M)=kb+2\ell+2=6Bst,\qquad\sigma(M)=ks=2Ast. χ ( M ) = k b + 2 ℓ + 2 = 6 B s t , σ ( M ) = k s = 2 A s t . In a hyperbolic summand choose a = u + 3 s t v a=u+3stv a = u + 3 s t v
a 2 [ M ] = 6 s t . a^2[M]=6st. a 2 [ M ] = 6 s t . These numbers satisfy (57) , including w 2 = ( p + q ) a ˉ = 0 w_2=(p+q)\bar a=0 w 2 = ( p + q ) a ˉ = 0 M M M
Incidence data
Let Conf 4 ( K / M ) \operatorname{Conf}_4(\mathcal K/M) Conf 4 ( K / M ) ξ i = [ v i ] ∈ K x \xi_i=[v_i]\in\mathcal K_x ξ i = [ v i ] ∈ K x ω x \omega_x ω x v i v_i v i
I K ( x ; ξ 1 , ξ 2 , ξ 3 , ξ 4 ) = sgn ω x ( v 1 , v 2 , v 3 , v 4 ) ∈ { − 1 , 0 , 1 } . \mathcal I_{\mathcal K}(x;\xi_1,\xi_2,\xi_3,\xi_4)=\operatorname{sgn}_{\omega_x}(v_1,v_2,v_3,v_4)\in\{-1,0,1\}. I K ( x ; ξ 1 , ξ 2 , ξ 3 , ξ 4 ) = sgn ω x ( v 1 , v 2 , v 3 , v 4 ) ∈ { − 1 , 0 , 1 } . Positive rescaling of any v i v_i v i ω x \omega_x ω x
D K = { I K = 0 } ⊂ Conf 4 ( K / M ) \mathcal D_{\mathcal K}=\{\mathcal I_{\mathcal K}=0\}\subset\operatorname{Conf}_4(\mathcal K/M) D K = { I K = 0 } ⊂ Conf 4 ( K / M ) is the set of linearly dependent quadruples. After choosing a metric and volume form it is the zero locus of a smooth determinant function, but it need not be a smooth hypersurface.
This data is not a knot-isotopy invariant: an ambient isotopy of a fiber can change linear incidence. No additional characteristic-class obstruction follows merely from writing down D K \mathcal D_{\mathcal K} D K
For torus knots, existence is already exhausted by Theorem 6.15. To obtain information beyond that criterion, one must study deformation or diffeomorphism classes, impose extra differential conditions, or pass to knot/link types with richer embedding spaces. One cannot obtain a stronger obstruction to unrestricted torus-knot-field existence while keeping the same definition and all of (57) satisfied.
Related work
The quotient Q F n \mathcal Q_{\mathcal F}^{n} Q F n (45) is a standard type of nonlinear Grassmannian. Gay-Balmaz and Vizman study its principal-bundle and smooth-manifold foundations, including compact source manifolds with boundary [27 ] . Fibrewise shapes therefore belong naturally to the established geometry of embedding spaces.
Suppose all shapes are locally smooth linear images of a fixed model
A ⊂ R n ∖ { 0 } . A\subset\mathbb R^n\setminus\{0\}. A ⊂ R n ∖ { 0 } . Then the adapted frames form a reduction of Fr ( T M ) \operatorname{Fr}(TM) Fr ( T M )
G = Stab GL ( n , R ) ( A ) . G=\operatorname{Stab}_{\operatorname{GL}(n,\mathbb R)}(A). G = Stab GL ( n , R ) ( A ) . This is an ordinary G G G
Theorem 6.2 is an application of the Becker–Gottlieb transfer [28 ] and Euler-class theory [26 ] . Theorem 6.4 uses classical three-manifold parallelizability [29 ] . Theorem 6.7 is a finite-stabilizer application of Ambrose–Singer holonomy theory [30 ] .
Hatcher and Budney study homotopy types of spaces of knots [32 , 39 ] . Their work is a natural starting point for analyzing the section space in (6.2).
Results for long knots or knots in S 3 S^3 S 3 T M TM T M Theorem 6.8 uses only the classical Hopf invariant [31 ] .
A further problem is to compute components and obstruction classes of the section space in (6.2) for a prescribed knot or higher-dimensional embedding type.
The question is whether coupling the embedding space to T M TM T M G G G
Havens–Koytcheff [40 ] treat related link and solid-torus embedding spaces.
The tangent-bundle reduction belongs to classical G G G [33 ] and Hirzebruch–Hopf [34 ] provide the relevant four-dimensional classification results.
The arithmetic uses van der Blij [35 ] and Rokhlin; the spin inequalities come from Furuta [36 ] , with stronger related results in Hopkins–Lin–Shi–Xu [38 ] .
The Einstein obstruction uses the classical Hitchin–Thorpe mechanism, with curvature conventions recorded in Atiyah–LeBrun [37 ] .
Finite-scale bundle comparisons
A bundle comparison identifies fibers over pairs of points. We study its extension radius from the diagonal and the restriction maps between spaces of comparisons.
Definitions and basic properties
Throughout this section, M M M g g g E → M E \to M E → M p i : M × M → M p_i: M \times M \to M p i : M × M → M
Δ = { ( x , x ) : x ∈ M } , U r ( g ) = { ( x , y ) ∈ M 2 : d g ( x , y ) < r } ( r > 0 ) . \Delta= \{(x,x) : x \in M\}, \qquad U_r(g) = \{(x,y) \in M^2 : d_g(x,y) < r\} \quad(r > 0). Δ = {( x , x ) : x ∈ M } , U r ( g ) = {( x , y ) ∈ M 2 : d g ( x , y ) < r } ( r > 0 ) . The strict inequality in this definition matters at critical radii.
Definition 7.1 (Comparisons and their extension radius). A comparison at scale r r r
P : p 1 ∗ E ∣ U r ( g ) ⟶ p 2 ∗ E ∣ U r ( g ) , P x x = id E x . P : p_1^*E|_{U_r(g)} \longrightarrow p_2^*E|_{U_r(g)}, \qquad P_{xx} = \operatorname{id}_{E_x}. P : p 1 ∗ E ∣ U r ( g ) ⟶ p 2 ∗ E ∣ U r ( g ) , P xx = id E x . Thus P x y : E x → E y P_{xy}: E_x \to E_y P x y : E x → E y ( x , y ) (x,y) ( x , y ) C r ( E , g ) \mathcal{C}_r(E,g) C r ( E , g ) C ∞ C^\infty C ∞
R E ( g ) = sup { r > 0 : C r ( E , g ) ≠ ∅ } ∈ ( 0 , ∞ ] . R_E(g) = \sup\{r > 0 : \mathcal{C}_r(E,g) \ne\varnothing\} \in(0,\infty]. R E ( g ) = sup { r > 0 : C r ( E , g ) = ∅ } ∈ ( 0 , ∞ ] . For 0 < s < r 0 < s < r 0 < s < r res r , s : C r ( E , g ) → C s ( E , g ) \operatorname{res}_{r,s} : \mathcal{C}_r(E,g) \to\mathcal{C}_s(E,g) res r , s : C r ( E , g ) → C s ( E , g ) comparison system C ( M , g , E ) \mathcal{C}(M,g,E) C ( M , g , E )
We impose neither P y x = P x y − 1 P_{yx}=P_{xy}^{-1} P y x = P x y − 1 P y z P x y = P x z P_{yz}P_{xy}=P_{xz} P y z P x y = P x z
Proposition 7.2 (Local and global existence). The following statements hold.
(i) C r ( E , g ) ≠ ∅ \mathcal{C}_r(E,g) \ne\varnothing C r ( E , g ) = ∅ p 1 ∗ E ∣ U r ( g ) ≅ p 2 ∗ E ∣ U r ( g ) p_1^*E|_{U_r(g)} \cong p_2^*E|_{U_r(g)} p 1 ∗ E ∣ U r ( g ) ≅ p 2 ∗ E ∣ U r ( g )
(ii) If 0 < r < inj ( M , g ) 0<r<\operatorname{inj}(M,g) 0 < r < inj ( M , g ) C r ( E , g ) \mathcal{C}_r(E,g) C r ( E , g )
(iii) R E ( g ) = ∞ R_E(g)=\infty R E ( g ) = ∞ E E E E E E
inj ( M , g ) ≤ R E ( g ) ≤ rad ( M , g ) ≤ diam ( M , g ) , rad ( M , g ) : = min x max y d g ( x , y ) . \operatorname{inj}(M,g) \le R_E(g) \le\operatorname{rad}(M,g) \le\operatorname{diam}(M,g), \qquad\operatorname{rad}(M,g):=\min_{x}\max_{y}d_g(x,y). inj ( M , g ) ≤ R E ( g ) ≤ rad ( M , g ) ≤ diam ( M , g ) , rad ( M , g ) := x min y max d g ( x , y ) . Proof. (i) Given an unnormalized isomorphism Q Q Q P x y = Q x y ( Q x x ) − 1 P_{xy}=Q_{xy}(Q_{xx})^{-1} P x y = Q x y ( Q xx ) − 1
(ii) Choose a connection on E E E P x y 0 P^0_{xy} P x y 0 P x y = P x y 0 A x y P_{xy}=P^0_{xy}A_{xy} P x y = P x y 0 A x y A x y ∈ Aut ( E x ) A_{xy}\in\operatorname{Aut}(E_x) A x y ∈ Aut ( E x ) A x x = id A_{xx}=\operatorname{id} A xx = id y t = exp x ( t exp x − 1 y ) y_t=\exp_x(t\exp_x^{-1}y) y t = exp x ( t exp x − 1 y )
H t ( P ) x y = P x y 0 A x , y t , 0 ≤ t ≤ 1 , H_t(P)_{xy}=P^0_{xy}A_{x,y_t}, \qquad0\le t\le1, H t ( P ) x y = P x y 0 A x , y t , 0 ≤ t ≤ 1 , is a continuous contraction: H 1 ( P ) = P H_1(P)=P H 1 ( P ) = P H 0 ( P ) = P 0 H_0(P)=P^0 H 0 ( P ) = P 0
(iii) A trivialization defines a comparison on all of M 2 M^2 M 2 r > rad ( M , g ) r>\operatorname{rad}(M,g) r > rad ( M , g ) x 0 x_0 x 0 { x 0 } × M ⊂ U r ( g ) \{x_0\}\times M\subset U_r(g) { x 0 } × M ⊂ U r ( g ) P x 0 , y P_{x_0,y} P x 0 , y E E E
For E = T M E=TM E = T M r = R E ( g ) r=R_E(g) r = R E ( g )
Stability and functoriality
Proposition 7.3 (Metric stability). For two metrics g , h g,h g , h M M M E E E
ϵ = sup x , y ∈ M ∣ d g ( x , y ) − d h ( x , y ) ∣ . \epsilon=\sup_{x,y\in M}|d_g(x,y)-d_h(x,y)|. ϵ = x , y ∈ M sup ∣ d g ( x , y ) − d h ( x , y ) ∣. Then ∣ R E ( g ) − R E ( h ) ∣ ≤ ϵ |R_E(g)-R_E(h)|\le\epsilon ∣ R E ( g ) − R E ( h ) ∣ ≤ ϵ For every a > 0 a>0 a > 0 one also has R E ( a 2 g ) = a R E ( g ) R_E(a^2g)=aR_E(g) R E ( a 2 g ) = a R E ( g ) allowing the value ∞ \infty ∞
Proof. For r > ϵ r>\epsilon r > ϵ U r − ϵ ( h ) ⊂ U r ( g ) U_{r-\epsilon}(h)\subset U_r(g) U r − ϵ ( h ) ⊂ U r ( g ) R E ( h ) ≥ R E ( g ) − ϵ R_E(h)\ge R_E(g)-\epsilon R E ( h ) ≥ R E ( g ) − ϵ g , h g,h g , h d a 2 g = a d g d_{a^2g}=a\,d_g d a 2 g = a d g
Proposition 7.4 (Pullback and products). If f : ( M , g ) → ( N , h ) f:(M,g)\to(N,h) f : ( M , g ) → ( N , h ) is smooth and 1 1 1 and F → N F\to N F → N is a bundle, then
R f ∗ F ( g ) ≥ R F ( h ) . R_{f^*F}(g)\ge R_F(h). R f ∗ F ( g ) ≥ R F ( h ) . In particular, for a product metric on X × Y X\times Y X × Y
R pr X ∗ F ( g X ⊕ g Y ) = R F ( g X ) . R_{\operatorname{pr}_X^*F}(g_X\oplus g_Y)=R_F(g_X). R pr X ∗ F ( g X ⊕ g Y ) = R F ( g X ) . Proof. The map f × f f\times f f × f U r ( g ) U_r(g) U r ( g ) U r ( h ) U_r(h) U r ( h ) X X X X × { y 0 } X\times\{y_0\} X × { y 0 }
Proposition 7.5 (Relative gauge description). If P 0 ∈ C r ( E , g ) P^0\in\mathcal{C}_r(E,g) P 0 ∈ C r ( E , g ) the map A ↦ P 0 A A\mapsto P^0A A ↦ P 0 A is a homeomorphism
Gau Δ ( p 1 ∗ E ∣ U r ) ≅ C r ( E , g ) , \operatorname{Gau}_{\Delta}(p_1^*E|_{U_r})\cong\mathcal{C}_r(E,g), Gau Δ ( p 1 ∗ E ∣ U r ) ≅ C r ( E , g ) , where the group on the left consists of smooth bundle automorphisms equal to the identity on Δ \Delta Δ
Proof. The inverse sends P P P A x y = ( P x y 0 ) − 1 P x y A_{xy}=(P^{0}_{xy})^{-1}P_{xy} A x y = ( P x y 0 ) − 1 P x y
Thus a nonempty comparison space is a torsor: choosing one comparison identifies it with a group. Without choosing an origin, it has the intrinsic ternary operation
[ P , Q , R ] x y = P x y Q x y − 1 R x y . [P,Q,R]_{xy}=P_{xy}Q_{xy}^{-1}R_{xy}. [ P , Q , R ] x y = P x y Q x y − 1 R x y . which is preserved by restriction. Both the spaces and the restriction maps carry information beyond the single number R E ( g ) R_E(g) R E ( g )
Characteristic obstructions and line bundles
For a natural characteristic class c c c
δ r c ( E ) : = ( p 2 ∗ c ( E ) − p 1 ∗ c ( E ) ) ∣ U r ( g ) . \delta_r c(E):=(p_2^*c(E)-p_1^*c(E))|_{U_r(g)}. δ r c ( E ) := ( p 2 ∗ c ( E ) − p 1 ∗ c ( E )) ∣ U r ( g ) . An isomorphism of the two pullbacks implies δ r c ( E ) = 0 \delta_r c(E)=0 δ r c ( E ) = 0
Theorem 7.6 (Line-bundle obstruction and topology). For a complex line bundle L L L
C r ( L , g ) ≠ ∅ ⟺ δ r c 1 ( L ) = 0 in H 2 ( U r ( g ) ; Z ) . \mathcal{C}_r(L,g)\ne\varnothing\quad\Longleftrightarrow\quad\delta_r c_1(L)=0\text{ in }H^2(U_r(g);\mathbb{Z}). C r ( L , g ) = ∅ ⟺ δ r c 1 ( L ) = 0 in H 2 ( U r ( g ) ; Z ) . For a real line bundle, the corresponding criterion is δ r w 1 ( L ) = 0 \delta_r w_1(L)=0 δ r w 1 ( L ) = 0 H 1 ( U r ( g ) ; F 2 ) H^1(U_r(g);\mathbb{F}_2) H 1 ( U r ( g ) ; F 2 ) π 0 C r ( L , g ) \pi_0\mathcal{C}_r(L,g) π 0 C r ( L , g ) H 1 ( U r ( g ) , Δ ; Z ) H^1(U_r(g),\Delta;\mathbb{Z}) H 1 ( U r ( g ) , Δ ; Z ) C r ( L , g ) \mathcal{C}_r(L,g) C r ( L , g )
Proof. A comparison is a nowhere-zero section of the line bundle p 1 ∗ L ∨ ⊗ p 2 ∗ L p_1^*L^\vee\otimes p_2^*L p 1 ∗ L ∨ ⊗ p 2 ∗ L c 1 c_1 c 1 w 1 w_1 w 1 [5 Proposition 3.10] .
Fix P 0 P^0 P 0 P 0 f P^0f P 0 f f : U r → C × f:U_r\to\mathbb{C}^{\times} f : U r → C × f ∣ Δ = 1 f|_\Delta=1 f ∣ Δ = 1 H 1 ( U r , Δ ; Z ) H^1(U_r,\Delta;\mathbb{Z}) H 1 ( U r , Δ ; Z ) C × ≃ K ( Z , 1 ) \mathbb{C}^{\times}\simeq K(\mathbb{Z},1) C × ≃ K ( Z , 1 ) Δ ⊂ U r \Delta\subset U_r Δ ⊂ U r
The space U r U_r U r ( x , y ) (x,y) ( x , y ) ( x , x ) (x,x) ( x , x ) Δ \Delta Δ f = 1 f=1 f = 1 f f f u : U r → C u:U_r\to\mathbb{C} u : U r → C u ∣ Δ = 0 u|_\Delta=0 u ∣ Δ = 0 u u u U r U_r U r
The real-line radius
Let L L L α = w 1 ( L ) ∈ H 1 ( M ; F 2 ) \alpha=w_1(L)\in H^1(M;\mathbb{F}_2) α = w 1 ( L ) ∈ H 1 ( M ; F 2 )
σ g ( α ) = inf c closed and piecewise smooth ⟨ α , [ c ] ⟩ = 1 Length g ( c ) . \sigma_g(\alpha)=\inf_{\substack{c\text{ closed and piecewise smooth}\\ \langle\alpha,[c]\rangle=1}}\operatorname{Length}_g(c). σ g ( α ) = c closed and piecewise smooth ⟨ α , [ c ]⟩ = 1 inf Length g ( c ) . It is positive, since sufficiently short loops lie in a contractible metric ball, and finite, since α ≠ 0 \alpha\ne0 α = 0
Theorem 7.8 (Half-systole formula). For every nontrivial real line bundle,
R L ( g ) = 1 2 σ g ( w 1 ( L ) ) . R_L(g)=\frac{1}{2}\sigma_g(w_1(L)). R L ( g ) = 2 1 σ g ( w 1 ( L )) . Moreover, a comparison exists at r = σ g ( w 1 ( L ) ) / 2 r=\sigma_g(w_1(L))/2 r = σ g ( w 1 ( L )) /2
Proof. Choose a fiber metric. The two unit vectors in each fiber form a double cover of M M M L L L
Suppose 0 < r ≤ σ g ( α ) / 2 0<r\leq\sigma_g(\alpha)/2 0 < r ≤ σ g ( α ) /2 d g ( x , y ) < r d_g(x,y)<r d g ( x , y ) < r x x x y y y r r r α \alpha α 2 r ≤ σ g ( α ) 2r\leq\sigma_g(\alpha) 2 r ≤ σ g ( α ) P x y P_{xy} P x y r r r P x x = i d P_{xx}=\mathrm{id} P xx = id
Conversely, if r > σ g ( α ) / 2 r>\sigma_g(\alpha)/2 r > σ g ( α ) /2 α \alpha α c c c x x x ℓ < 2 r \ell<2r ℓ < 2 r c c c ℓ / 2 < r \ell/2<r ℓ /2 < r x x x P x , c ( t ) v P_{x,c(t)}v P x , c ( t ) v 0 ≠ v ∈ L x 0\neq v\in L_x 0 = v ∈ L x c ∗ L c^*L c ∗ L ⟨ α , [ c ] ⟩ = 1 \langle\alpha,[c]\rangle=1 ⟨ α , [ c ]⟩ = 1
Example 7.9 (A circle and products). The Möbius line over a circle of circumference a a a Proposition 7.4 . In particular, shrinking the extra factor can send the product volume to zero while keeping this radius fixed.
Flat tori
Let T = V / Λ T=V/\Lambda T = V /Λ V V V Λ \Lambda Λ H 2 ( T ; Z ) ≅ Alt 2 ( Λ , Z ) H^2(T;\mathbb{Z})\cong\operatorname{Alt}^2(\Lambda,\mathbb{Z}) H 2 ( T ; Z ) ≅ Alt 2 ( Λ , Z ) L L L B = c 1 ( L ) B=c_1(L) B = c 1 ( L )
Λ 0 = { λ ∈ Λ : B ( λ , μ ) = 0 for all μ ∈ Λ } . \Lambda_0=\{\lambda\in\Lambda:B(\lambda,\mu)=0\text{ for all }\mu\in\Lambda\}. Λ 0 = { λ ∈ Λ : B ( λ , μ ) = 0 for all μ ∈ Λ } . Theorem 7.10 (The radical-lattice formula). With the convention min ∅ = ∞ \min\varnothing=\infty min ∅ = ∞
R L ( g ) = 1 2 min λ ∈ Λ ∖ Λ 0 ∥ λ ∥ . R_L(g)=\frac{1}{2}\min_{\lambda\in\Lambda\setminus\Lambda_0}\|\lambda\|. R L ( g ) = 2 1 λ ∈ Λ ∖ Λ 0 min ∥ λ ∥. If the value is finite, a comparison exists at that radius.
Proof. If B = 0 B=0 B = 0 ℓ \ell ℓ
Upper bound. Choose λ ∉ Λ 0 \lambda\notin\Lambda_0 λ ∈ / Λ 0 ℓ \ell ℓ μ ∈ Λ \mu\in\Lambda μ ∈ Λ B ( μ , λ ) ≠ 0 B(\mu,\lambda)\neq0 B ( μ , λ ) = 0 r > ℓ / 2 r>\ell/2 r > ℓ /2
F : ( R / Z ) 2 ⟶ U r ( g ) , F ( t , s ) = ( t μ , t μ + s λ ) F:(\mathbb{R}/\mathbb{Z})^2\longrightarrow U_r(g),\qquad F(t,s)=(t\mu,t\mu+s\lambda) F : ( R / Z ) 2 ⟶ U r ( g ) , F ( t , s ) = ( t μ , t μ + s λ ) is well defined: the two points are at distance at most min ( s , 1 − s ) ℓ ≤ ℓ / 2 \min(s,1-s)\ell\leq\ell/2 min ( s , 1 − s ) ℓ ≤ ℓ /2 0 ≤ s ≤ 1 0\leq s\leq1 0 ≤ s ≤ 1 F F F
⟨ F ∗ δ r c 1 ( L ) , [ ( R / Z ) 2 ] ⟩ = B ( μ , λ ) ≠ 0. \langle F^*\delta_r c_1(L),[(\mathbb{R}/\mathbb{Z})^2]\rangle=B(\mu,\lambda)\neq0. ⟨ F ∗ δ r c 1 ( L ) , [( R / Z ) 2 ]⟩ = B ( μ , λ ) = 0. Theorem 7.6 rules out a comparison.
Lower bound. Put W = span R Λ 0 W=\operatorname{span}_{\mathbb{R}}\Lambda_0 W = span R Λ 0 B B B W W W Λ \Lambda Λ q ( Λ ) q(\Lambda) q ( Λ ) V / W V/W V / W
q : T ⟶ S : = ( V / W ) / q ( Λ ) q:T\longrightarrow S:=(V/W)/q(\Lambda) q : T ⟶ S := ( V / W ) / q ( Λ ) is a quotient torus. The alternating form descends to Λ / Λ 0 \Lambda/\Lambda_0 Λ/ Λ 0 L ≅ q ∗ L ′ L\cong q^*L' L ≅ q ∗ L ′ L ′ → S L'\to S L ′ → S r ≤ ℓ / 2 r \le\ell/2 r ≤ ℓ /2 B r T ( 0 ) B_r^T(0) B r T ( 0 ) T T T z z z v ∈ V v \in V v ∈ V ∥ v ∥ < r \lVert v\rVert< r ∥ v ∥ < r
a r : B r T ( 0 ) ⟶ V / W , a r ( z ) = v + W , a_r:B_r^T(0)\longrightarrow V/W,\qquad a_r(z)=v+W, a r : B r T ( 0 ) ⟶ V / W , a r ( z ) = v + W , is well defined: two such representatives differ by ν ∈ Λ \nu\in\Lambda ν ∈ Λ ∥ ν ∥ < 2 r ≤ ℓ \lVert\nu\rVert< 2r \le\ell ∥ ν ∥ < 2 r ≤ ℓ ν ∈ Λ 0 \nu\in\Lambda_0 ν ∈ Λ 0 a r ( 0 ) = 0 a_r(0)=0 a r ( 0 ) = 0 ( x , y ) ∈ U r ( g ) (x,y)\in U_r(g) ( x , y ) ∈ U r ( g )
t ⟼ q ( x ) + t a r ( y − x ) in S , 0 ≤ t ≤ 1. t\longmapsto q(x)+t\,a_r(y-x)\quad\text{in }S,\qquad0\le t\le1. t ⟼ q ( x ) + t a r ( y − x ) in S , 0 ≤ t ≤ 1. Its endpoints are q ( x ) , q ( y ) q(x),q(y) q ( x ) , q ( y ) L ′ L' L ′ L ≅ q ∗ L ′ L\cong q^*L' L ≅ q ∗ L ′
Example 7.11 (Trivial on every ball does not suffice). Let T 2 = R 2 / ( a Z ⊕ b Z ) T^2=\mathbb{R}^2/(a\mathbb{Z}\oplus b\mathbb{Z}) T 2 = R 2 / ( a Z ⊕ b Z ) 0 < a < b 0<a<b 0 < a < b L L L R L = a / 2 R_L=a/2 R L = a /2 a / 2 < r < b / 2 a/2<r<b/2 a /2 < r < b /2 L L L
Examples beyond the extension radius
Rectangular tangent tori
For the same rectangular T 2 T^2 T 2 R T T 2 = ∞ R_{TT^2}=\infty R T T 2 = ∞
( U r , Δ ) ≅ ( T 2 × B r T ( 0 ) , T 2 × { 0 } ) . (U_r,\Delta)\cong(T^2\times B_r^T(0),T^2\times\{0\}). ( U r , Δ ) ≅ ( T 2 × B r T ( 0 ) , T 2 × { 0 }) . Using the translation frame, comparisons become smooth maps to GL 2 ( R ) \operatorname{GL}_2(\mathbb{R}) GL 2 ( R ) I I I U r U_r U r GL 2 + ( R ) \operatorname{GL}_2^+(\mathbb{R}) GL 2 + ( R ) SO ( 2 ) ≅ S 1 \operatorname{SO}(2)\cong S^1 SO ( 2 ) ≅ S 1
π 0 C r ( T T 2 , g ) ≅ H 1 ( U r , Δ ; Z ) ≅ H 1 ( B r T ( 0 ) ; Z ) ≅ { 0 , 0 < r ≤ a / 2 , Z , a / 2 < r ≤ b / 2 , Z 2 , r > b / 2. \pi_0\mathcal{C}_r(TT^2,g)\cong H^1(U_r,\Delta;\mathbb{Z})\cong H^1(B_r^T(0);\mathbb{Z})\cong
\begin{cases}
0, & 0<r\le a/2,\\
\mathbb{Z}, & a/2<r\le b/2,\\
\mathbb{Z}^2, & r>b/2.
\end{cases} π 0 C r ( T T 2 , g ) ≅ H 1 ( U r , Δ ; Z ) ≅ H 1 ( B r T ( 0 ) ; Z ) ≅ ⎩ ⎨ ⎧ 0 , Z , Z 2 , 0 < r ≤ a /2 , a /2 < r ≤ b /2 , r > b /2. Each path component is contractible: the positive-definite factor in polar decomposition is contractible, and the normalized S 1 S^1 S 1 Theorem 7.6 .
For completeness, the ball is a disk for r ≤ a / 2 r\le a/2 r ≤ a /2 a / 2 < r ≤ b / 2 a/2<r\le b/2 a /2 < r ≤ b /2 b / 2 b/2 b /2 a 2 + b 2 / 2 \sqrt{a^2+b^2}/2 a 2 + b 2 /2
The identifications as groups use the translation frame as origin. Across the second transition, restriction induces the projection Z 2 → Z \mathbb{Z}^2\to\mathbb{Z} Z 2 → Z Z \mathbb{Z} Z Z 2 \mathbb{Z}^2 Z 2
An unstable sphere obstruction
On the unit round S 5 S^5 S 5 inj = diam = π \operatorname{inj}=\operatorname{diam}=\pi inj = diam = π [41 ] , while
T S 5 ⊕ R ≅ R 6 . TS^5\oplus\mathbb{R}\cong\mathbb{R}^6. T S 5 ⊕ R ≅ R 6 . Proposition 7.2 therefore gives
R T S 5 = π , R T S 5 ⊕ R = ∞ . R_{TS^5}=\pi,\qquad R_{TS^5\oplus\mathbb{R}}=\infty. R T S 5 = π , R T S 5 ⊕ R = ∞. All positive-degree Stiefel–Whitney and Pontryagin classes of T S 5 TS^5 T S 5 S 5 ⊂ C 3 S^5 \subset\mathbb{C}^3 S 5 ⊂ C 3 x ↦ i x x \mapsto ix x ↦ i x
Surface reconstruction from finite covers
Let S S S g g g Γ = π 1 ( S ) \Gamma= \pi_1(S) Γ = π 1 ( S ) H ≤ Γ H \leq\Gamma H ≤ Γ p H : S H → S p_H:S_H \to S p H : S H → S 0 ≠ α ∈ H 1 ( S H ; F 2 ) = Hom ( H , F 2 ) 0 \ne\alpha\in H^1(S_H;\mathbb{F}_2) = \operatorname{Hom}(H,\mathbb{F}_2) 0 = α ∈ H 1 ( S H ; F 2 ) = Hom ( H , F 2 ) L α L_\alpha L α w 1 ( L α ) = α w_1(L_\alpha)=\alpha w 1 ( L α ) = α
R g ( H , α ) = R L α ( p H ∗ g ) . R_g(H,\alpha)=R_{L_\alpha}(p_H^*g). R g ( H , α ) = R L α ( p H ∗ g ) . These positive finite numbers are labelled by ( H , α ) (H,\alpha) ( H , α ) ℓ g ( γ ) \ell_g(\gamma) ℓ g ( γ ) 1 ≠ γ ∈ Γ 1 \ne\gamma\in\Gamma 1 = γ ∈ Γ
The reconstruction uses two covering results. Scott’s theorem, in the form quantified by Patel [42 ] , gives a finite cover in which a primitive closed geodesic has a degree-one simple lift. Theorem C of Alvarez–Brum–Martínez–Potrie, with Wolff [43 ] , states that a simple closed geodesic on a closed hyperbolic surface has, for every K > 0 K>0 K > 0 K K K
Theorem 7.12 (Reconstruction from labelled radii). For a primitive γ ≠ 1 \gamma\ne1 γ = 1
I γ = { ( H , α ) : [ Γ : H ] < ∞ , γ ∈ H , α ( γ ) = 1 } . I_\gamma=\{(H,\alpha):[\Gamma:H]<\infty,\ \gamma\in H,\ \alpha(\gamma)=1\}. I γ = {( H , α ) : [ Γ : H ] < ∞ , γ ∈ H , α ( γ ) = 1 } . Then
ℓ g ( γ ) = 2 sup ( H , α ) ∈ I γ R g ( H , α ) . \ell_g(\gamma)=2\sup_{(H,\alpha)\in I_\gamma} R_g(H,\alpha). ℓ g ( γ ) = 2 ( H , α ) ∈ I γ sup R g ( H , α ) . The supremum is attained. Thus all the labelled radii determine the marked length spectrum. In particular, if g , h g,h g , h ϕ \phi ϕ ϕ ∗ h = g \phi^*h=g ϕ ∗ h = g
Proof. If ( H , α ) ∈ I γ (H,\alpha)\in I_\gamma ( H , α ) ∈ I γ γ \gamma γ α \alpha α ℓ g ( γ ) \ell_g(\gamma) ℓ g ( γ ) Theorem 7.8 ,
2 R g ( H , α ) = σ p H ∗ g ( α ) ≤ ℓ g ( γ ) . 2R_g(H,\alpha)=\sigma_{p_H^*g}(\alpha)\leq\ell_g(\gamma). 2 R g ( H , α ) = σ p H ∗ g ( α ) ≤ ℓ g ( γ ) . For the opposite inequality, first pass to a finite cover S 1 S_1 S 1 γ \gamma γ q q q S 1 S_1 S 1 C C C Length q ( c ) ≤ C Length g ( c ) \operatorname{Length}_q(c)\leq C\operatorname{Length}_g(c) Length q ( c ) ≤ C Length g ( c ) c c c g g g [43 ] to the q q q K > C ℓ g ( γ ) K>C\ell_g(\gamma) K > C ℓ g ( γ )
In the resulting cover there is a nonseparating degree-one lift γ ^ \hat{\gamma} γ ^ q q q K K K g g g K / C > ℓ g ( γ ) K/C>\ell_g(\gamma) K / C > ℓ g ( γ ) γ ^ \hat{\gamma} γ ^ g g g α \alpha α α ( γ ^ ) = 1 \alpha(\hat{\gamma})=1 α ( γ ^ ) = 1 γ ^ \hat{\gamma} γ ^ ℓ g ( γ ) \ell_g(\gamma) ℓ g ( γ ) γ \gamma γ I γ I_\gamma I γ
Every nontrivial surface-group element is a power of a primitive element, and ℓ g ( η k ) = ∣ k ∣ ℓ g ( η ) \ell_g(\eta^k)=|k|\ell_g(\eta) ℓ g ( η k ) = ∣ k ∣ ℓ g ( η ) [44 ] ; see also the stronger Anosov-surface result [45 ] . ∎
Corollary 7.13 (Exact multiplicative comparison). For two such metrics on the same marked surface,
sup H , α ≠ 0 ∣ log R g ( H , α ) R h ( H , α ) ∣ = sup γ ≠ 1 ∣ log ℓ g ( γ ) ℓ h ( γ ) ∣ . \sup_{H,\ \alpha\ne0}\left|\log\frac{R_g(H,\alpha)}{R_h(H,\alpha)}\right|=\sup_{\gamma\ne1}\left|\log\frac{\ell_g(\gamma)}{\ell_h(\gamma)}\right|. H , α = 0 sup log R h ( H , α ) R g ( H , α ) = γ = 1 sup log ℓ h ( γ ) ℓ g ( γ ) . Both suprema are finite.
Proof. A uniform multiplicative bound for marked lengths passes to the infimum over α \alpha α Theorem 7.8 . Conversely, the same bound for every labelled radius passes to the supremum over I γ I_\gamma I γ Theorem 7.12 . The metrics are bi-Lipschitz on compact S S S
Related work
The individual maps P x y P_{xy} P x y [46 ] , formerly called quasi-connections. Discrete principal-bundle connections provide a related framework [47 ] . The organizing emphasis here is the metric family of domains U r U_r U r
The line-bundle criteria use classical classification [5 Proposition 3.10] . Persistent characteristic classes of line bundles have been studied by Tinarrage [48 ] . Homotopic distance between maps [49 ] is adjacent to the comparison of the two projections, although it measures a different invariant.
The real-line formula translates extension into the length of loops detected by a mod-2 class. Subsection 7.7 combines that formula with finite-cover isolation and marked-length rigidity results. Further questions concern higher-rank unstable obstructions, behavior at critical radii, and examples where the restriction system detects geometric features missed by ordinary characteristic classes.
Harmonic closure of differential forms
We study the real algebra generated by harmonic forms under wedge product, exterior differentiation and Hodge star. All closure operations are algebraic, not topological.
Definitions and basic properties
Throughout this section, M n M^n M n n ≥ 2 n\ge2 n ≥ 2 g g g Ω ∙ ( M ) \Omega^\bullet(M) Ω ∙ ( M ) δ g \delta_g δ g L 2 L^2 L 2 d d d
Δ g = d δ g + δ g d , H g k = ker ( Δ g | Ω k ( M ) ) . \Delta_g=d\delta_g+\delta_gd,\qquad\mathcal{H}_g^k=\ker\left(\Delta_g\middle|_{\Omega^k(M)}\right). Δ g = d δ g + δ g d , H g k = ker ( Δ g Ω k ( M ) ) . Hodge theory gives H g k ≅ H d R k ( M ) \mathcal{H}_g^k\cong H_{\mathrm{dR}}^k(M) H g k ≅ H dR k ( M ) b k = dim H d R k ( M ) b_k=\dim H_{\mathrm{dR}}^k(M) b k = dim H dR k ( M ) B = ∑ k = 0 n b k B=\sum_{k=0}^{n}b_k B = ∑ k = 0 n b k
Definition 8.1 (Harmonic closure). The harmonic closure of g g g
C g = ⟨ H g ∙ ⟩ ∧ , d , ∗ g ⊆ Ω ∙ ( M ) \mathcal{C}_g=\langle\mathcal{H}_g^\bullet\rangle_{\wedge,d,*_g}\subseteq\Omega^\bullet(M) C g = ⟨ H g ∙ ⟩ ∧ , d , ∗ g ⊆ Ω ∙ ( M ) containing every harmonic form and preserved by d d d ∗ g \ast_g ∗ g finite harmonic closure if dim R C g < ∞ \dim_{\mathbb{R}} C_g < \infty dim R C g < ∞
Only real scalar coefficients are allowed: C g C_g C g C ∞ ( M ) C^\infty(M) C ∞ ( M )
V 0 = H g ∙ V_0=\mathcal{H}_g^\bullet V 0 = H g ∙ and
V j + 1 = span R ( V j ∪ d V j ∪ ∗ g V j ∪ { α ∧ β : α , β ∈ V j } ) . V_{j+1}=\operatorname{span}_{\mathbb{R}}\left(V_j\cup dV_j\cup\ast_gV_j\cup\{\alpha\wedge\beta:\alpha,\beta\in V_j\}\right). V j + 1 = span R ( V j ∪ d V j ∪ ∗ g V j ∪ { α ∧ β : α , β ∈ V j } ) . Then each V j V_j V j C g = ⋃ j ≥ 0 V j C_g=\bigcup_{j\geq0}V_j C g = ⋃ j ≥ 0 V j 1 , vol g ∈ C g 1,\operatorname{vol}_g\in C_g 1 , vol g ∈ C g δ g = ± ∗ g d ∗ g \delta_g=\pm\ast_gd\ast_g δ g = ± ∗ g d ∗ g δ g \delta_g δ g Δ g \Delta_g Δ g C g C_g C g
For any diffeomorphism ϕ : N → M \phi:N\to M ϕ : N → M C ϕ ∗ g = ϕ ∗ C g C_{\phi^\ast g}=\phi^\ast C_g C ϕ ∗ g = ϕ ∗ C g C c 2 g = C g C_{c^2g}=C_g C c 2 g = C g c > 0 c>0 c > 0 ∗ c 2 g ∣ Ω k = c n − 2 k ∗ g \left.\ast_{c^2g}\right|_{\Omega^k}=c^{n-2k}\ast_g ∗ c 2 g Ω k = c n − 2 k ∗ g
Proposition 8.2 (Scalar criterion and pointwise bounds). The following conditions are equivalent:
dim C g < ∞ ⟺ C g 0 = R ⋅ 1. \dim C_g<\infty\quad\Longleftrightarrow\quad C_g^0=\mathbb{R}\cdot1. dim C g < ∞ ⟺ C g 0 = R ⋅ 1. If they hold, same-degree elements of C g C_g C g x ∈ M x\in M x ∈ M
ev x : C g k ↪ Λ k T x ∗ M . \operatorname{ev}_x:C_g^k\hookrightarrow\Lambda^kT_x^\ast M. ev x : C g k ↪ Λ k T x ∗ M . Consequently, with c k = dim C g k c_k=\dim C_g^k c k = dim C g k
b k ≤ c k ≤ ( n k ) , c k = c n − k , dim C g ≤ 2 n . b_k\leq c_k\leq\binom{n}{k},\qquad c_k=c_{n-k},\qquad\dim C_g\leq2^n. b k ≤ c k ≤ ( k n ) , c k = c n − k , dim C g ≤ 2 n . Every nonzero element of C g k C_g^k C g k C g C_g C g f f f 1 , f , f 2 , … 1,f,f^2,\ldots 1 , f , f 2 , …
Proof. In a finite-dimensional algebra of functions, the powers of any f f f P ( f ) = 0 P(f)=0 P ( f ) = 0 f f f C g 0 = R C_g^0=\mathbb{R} C g 0 = R
⟨ α , β ⟩ g = ∗ g ( α ∧ ∗ g β ) ∈ R ( α , β ∈ C g k ) . \langle\alpha,\beta\rangle_g=\ast_g(\alpha\wedge\ast_g\beta)\in\mathbb{R}\qquad(\alpha,\beta\in C_g^k). ⟨ α , β ⟩ g = ∗ g ( α ∧ ∗ g β ) ∈ R ( α , β ∈ C g k ) . A form vanishing at one point therefore vanishes everywhere. This proves evaluation injectivity and the bounds, without initially assuming finite dimensionality. Hodge star gives c k = c n − k c_k=c_{n-k} c k = c n − k M M M
Proposition 8.3 (Finite Hodge model). If C g C_g C g Ω ∙ ( M ) \Omega^\bullet(M) Ω ∙ ( M )
C g = H g ∙ ⊕ d C g ⊕ δ g C g . C_g=\mathcal{H}_g^\bullet\oplus dC_g\oplus\delta_gC_g. C g = H g ∙ ⊕ d C g ⊕ δ g C g . Writing r k = rank ( d : C g k → C g k + 1 ) r_k=\operatorname{rank}(d:C_g^k\to C_g^{k+1}) r k = rank ( d : C g k → C g k + 1 ) r − 1 = r n = 0 r_{-1}=r_n=0 r − 1 = r n = 0
c k = b k + r k − 1 + r k , ε ( g ) : = dim C g − B 2 = ∑ k = 0 n − 1 r k ∈ Z ≥ 0 . c_k=b_k+r_{k-1}+r_k,\qquad\varepsilon(g):=\frac{\dim C_g-B}{2}=\sum_{k=0}^{n-1}r_k\in\mathbb{Z}_{\geq0}. c k = b k + r k − 1 + r k , ε ( g ) := 2 dim C g − B = k = 0 ∑ n − 1 r k ∈ Z ≥ 0 . Moreover, ε ( g ) = 0 \varepsilon(g)=0 ε ( g ) = 0 g g g
Proof. The restrictions of d d d δ g \delta_g δ g H g ∙ \mathcal{H}_g^\bullet H g ∙ d α = δ g α = 0 d\alpha=\delta_g\alpha=0 d α = δ g α = 0 ε ( g ) = 0 \varepsilon(g)=0 ε ( g ) = 0 C g = H g ∙ C_g=\mathcal{H}_g^\bullet C g = H g ∙ [50 ] .
The profile ( c 0 , … , c n ) (c_0,\ldots,c_n) ( c 0 , … , c n ) ε ( g ) \varepsilon(g) ε ( g ) ε ( g ) = ∞ \varepsilon(g)=\infty ε ( g ) = ∞
Examples
Example 8.4 (Cohomology spheres and flat tori). If M M M
C g = H g ∙ = span R { 1 , vol g } , dim C g = 2. \mathcal{C}_g=\mathcal{H}^{\bullet}_g=\operatorname{span}_{\mathbb{R}}\{1,\operatorname{vol}_g\},\qquad\dim\mathcal{C}_g=2. C g = H g ∙ = span R { 1 , vol g } , dim C g = 2. On a flat n n n C g = H g ∙ \mathcal{C}_g=\mathcal{H}^{\bullet}_g C g = H g ∙ 2 n 2^n 2 n ε ( g ) = 0 \varepsilon(g)=0 ε ( g ) = 0
Example 8.5 (The Heisenberg three-manifold). Let M M M
d α = d β = 0 , d γ = α ∧ β . d\alpha=d\beta=0,\qquad d\gamma=\alpha\wedge\beta. d α = d β = 0 , d γ = α ∧ β . Make this coframe orthonormal and orient by α ∧ β ∧ γ \alpha\wedge\beta\wedge\gamma α ∧ β ∧ γ
H g ∙ = span R { 1 , α , β , α ∧ γ , β ∧ γ , α ∧ β ∧ γ } . \mathcal{H}^{\bullet}_g=\operatorname{span}_{\mathbb{R}}\{1,\alpha,\beta,\alpha\wedge\gamma,\beta\wedge\gamma,\alpha\wedge\beta\wedge\gamma\}. H g ∙ = span R { 1 , α , β , α ∧ γ , β ∧ γ , α ∧ β ∧ γ } . Indeed, these forms are closed and coclosed; invariant cohomology and Nomizu’s theorem [51 ] show that they account for all cohomology classes. Since ∗ g ( α ∧ β ) = γ \ast_g(\alpha\wedge\beta)=\gamma ∗ g ( α ∧ β ) = γ
C g = Λ ∙ span R { α , β , γ } , dim C g = 8 , ε ( g ) = 1. \mathcal{C}_g=\Lambda^{\bullet}\operatorname{span}_{\mathbb{R}}\{\alpha,\beta,\gamma\},\qquad\dim\mathcal{C}_g=8,\qquad\varepsilon(g)=1. C g = Λ ∙ span R { α , β , γ } , dim C g = 8 , ε ( g ) = 1. This metric is not geometrically formal: α ∧ β = d γ \alpha\wedge\beta=d\gamma α ∧ β = d γ ⟨ [ α ] , [ α ] , [ β ] ⟩ \langle[\alpha],[\alpha],[\beta]\rangle ⟨[ α ] , [ α ] , [ β ]⟩ ± α ∧ γ \pm\alpha\wedge\gamma ± α ∧ γ
Example 8.6 (An infinite closure on the two-torus). On T 2 = R 2 / Z 2 T^2=\mathbb{R}^2/\mathbb{Z}^2 T 2 = R 2 / Z 2 g = e 2 u ( x , y ) ( d x 2 + d y 2 ) g=e^{2u(x,y)}(dx^2+dy^2) g = e 2 u ( x , y ) ( d x 2 + d y 2 ) u u u d x , d y dx,dy d x , d y
∗ g ( d x ∧ d y ) = e − 2 u ∈ C g 0 . \ast_g(dx\wedge dy)=e^{-2u}\in\mathcal{C}^0_g. ∗ g ( d x ∧ d y ) = e − 2 u ∈ C g 0 . Proposition 8.2 gives dim C g = ∞ \dim\mathcal{C}_g=\infty dim C g = ∞ q ≥ 2 q\ge2 q ≥ 2 b 1 = 2 q > 2 b_1=2q>2 b 1 = 2 q > 2
Proposition 8.7 (Homogeneous examples). Every compact oriented Riemannian manifold admitting a connected transitive group of isometries has finite harmonic closure.
Proof. Each group element is isotopic to the identity and hence acts trivially on cohomology. It preserves harmonic forms, so uniqueness of harmonic representatives makes them invariant. Invariant forms constitute a finite-dimensional algebra preserved by d d d ∗ g \ast_g ∗ g C g \mathcal{C}_g C g
Lie algebra and reconstruction
Theorem 8.8 (Canonical infinitesimal action). Suppose C g \mathcal{C}_g C g
g g = { α ♯ : α ∈ C g 1 } , r = dim C g 1 . \mathfrak{g}_g=\{\alpha^\sharp:\alpha\in\mathcal{C}^1_g\},\qquad r=\dim\mathcal{C}^1_g. g g = { α ♯ : α ∈ C g 1 } , r = dim C g 1 . Then g g \mathfrak{g}_g g g r r r r r r C g \mathcal{C}_g C g
Proof. For α ∈ C g 1 \alpha\in C_g^1 α ∈ C g 1 δ g α \delta_g\alpha δ g α ⟨ δ g α , 1 ⟩ L 2 = ⟨ α , d 1 ⟩ L 2 = 0 \langle\delta_g\alpha,1\rangle_{L^2}=\langle\alpha,d1\rangle_{L^2}=0 ⟨ δ g α , 1 ⟩ L 2 = ⟨ α , d 1 ⟩ L 2 = 0 δ g α = 0 \delta_g\alpha=0 δ g α = 0
δ g ( α ∧ β ) = ( δ g α ) β − ( δ g β ) α − [ α ♯ , β ♯ ] . \delta_g(\alpha\wedge\beta)=(\delta_g\alpha)\beta-(\delta_g\beta)\alpha-[\alpha^\sharp,\beta^\sharp]. δ g ( α ∧ β ) = ( δ g α ) β − ( δ g β ) α − [ α ♯ , β ♯ ] . Hence l g \mathfrak{l}_g l g ι α ♯ ω = ± ∗ g ( α ∧ ∗ g ω ) \iota_{\alpha^\sharp}\omega=\pm*_g(\alpha\wedge*_g\omega) ι α ♯ ω = ± ∗ g ( α ∧ ∗ g ω ) C g C_g C g
Corollary 8.9 (Full rank). If r = n r=n r = n C g = Λ ∙ ( C g 1 ) C_g=\Lambda^\bullet(C_g^1) C g = Λ ∙ ( C g 1 ) ( M , g ) (M,g) ( M , g ) Γ \ G \Gamma\backslash G Γ\ G G G G Γ \Gamma Γ r = n − 1 r=n-1 r = n − 1
Proof. Constant inner products give a global orthonormal coframe θ 1 , … , θ n \theta^1,\ldots,\theta^n θ 1 , … , θ n C g 1 C_g^1 C g 1 C g C_g C g
d θ i = ∑ j < k c j k i θ j ∧ θ k with constant c j k i . d\theta^i=\sum_{j<k}c^i_{jk}\theta^j\wedge\theta^k \qquad\text{with constant }c^i_{jk}. d θ i = j < k ∑ c j k i θ j ∧ θ k with constant c j k i . These are Maurer–Cartan equations. Equivalently, the complete action of Theorem 8.8, taken as a right action, is transitive with discrete stabilizer Γ \Gamma Γ M = Γ \ G M=\Gamma\backslash G M = Γ\ G r = n − 1 r=n-1 r = n − 1 C g 1 C_g^1 C g 1 C g 1 C_g^1 C g 1
Theorem 8.10 (Nilmanifold reconstruction). Let G G G Γ < G \Gamma<G Γ < G M = Γ \ G M=\Gamma\backslash G M = Γ\ G
C g = Λ ∙ g ∗ , dim C g = 2 n . C_g=\Lambda^\bullet\mathfrak{g}^*,\qquad\dim C_g=2^n. C g = Λ ∙ g ∗ , dim C g = 2 n . Thus ( C g , ∧ , d , ∗ g ) (C_g,\wedge,d,*_g) ( C g , ∧ , d , ∗ g ) Γ \Gamma Γ
Proof. The invariant algebra A = Λ ∙ g ∗ A=\Lambda^\bullet\mathfrak{g}^* A = Λ ∙ g ∗ d d d ∗ g *_g ∗ g [51 ] makes A ↪ Ω ∙ ( M ) A\hookrightarrow\Omega^\bullet(M) A ↪ Ω ∙ ( M ) A A A C g ⊂ A C_g\subset A C g ⊂ A C g C_g C g
Choose a coframe adapted to the nilpotent structure, with d θ i ∈ Λ 2 span R { θ 1 , … , θ i − 1 } d\theta^i\in\Lambda^2\operatorname{span}_{\mathbb{R}}\{\theta^1,\ldots,\theta^{i-1}\} d θ i ∈ Λ 2 span R { θ 1 , … , θ i − 1 } C g C_g C g d θ i ∈ C g d\theta^i\in C_g d θ i ∈ C g η ∈ C g 1 \eta\in C_g^1 η ∈ C g 1 d η = d θ i d\eta=d\theta^i d η = d θ i θ i − η \theta^i-\eta θ i − η θ i ∈ C g \theta^i\in C_g θ i ∈ C g d d d ∗ g ( α ∧ ∗ g β ) *_g(\alpha\wedge*_g\beta) ∗ g ( α ∧ ∗ g β )
Conformal rigidity
Theorem 8.11 (Rigidity from one cohomology class). Let 0 < k < n 0<k<n 0 < k < n 0 ≠ α ∈ H d R k ( M ) 0\ne\alpha\in H^k_{\mathrm{dR}}(M) 0 = α ∈ H dR k ( M ) α \alpha α g g g h = e 2 u g h=e^{2u}g h = e 2 u g u u u
Proof. Put p = n / k > 1 p=n/k>1 p = n / k > 1 α \alpha α
E g ( ω ) = ∫ M ∣ ω ∣ g p d V g . (63) E_g(\omega)=\int_M|\omega|_g^p\,dV_g. \tag*{(63)} E g ( ω ) = ∫ M ∣ ω ∣ g p d V g . (63) These quotient metrics are locally homogeneous; they need not be globally homogeneous as Riemannian manifolds. If α \alpha α ∣ α ∣ g ≡ c > 0 |\alpha|_g \equiv c > 0 ∣ α ∣ g ≡ c > 0 ω = α + d η \omega= \alpha+ d\eta ω = α + d η
E g ( ω ) ≥ E g ( α ) + p c p − 2 ∫ M ⟨ α , d η ⟩ g d V g = E g ( α ) , E_g(\omega) \ge E_g(\alpha) + p c^{p-2} \int_M \langle\alpha, d\eta\rangle_g\,dV_g = E_g(\alpha), E g ( ω ) ≥ E g ( α ) + p c p − 2 ∫ M ⟨ α , d η ⟩ g d V g = E g ( α ) , with equality only for ω = α \omega= \alpha ω = α α \alpha α E g E_g E g ∣ ω ∣ h = e − k u ∣ ω ∣ g |\omega|_h = e^{-ku}|\omega|_g ∣ ω ∣ h = e − k u ∣ ω ∣ g d V h = e n u d V g dV_h = e^{nu}dV_g d V h = e n u d V g E h = E g E_h = E_g E h = E g c h = e − k u c g c_h = e^{-ku}c_g c h = e − k u c g u u u
Corollary 8.12 (Conformal uniqueness and symmetries). Suppose M M M
Proof. Some intermediate cohomology class is nonzero. Proposition 8.2 and Theorem 8.11 apply to its harmonic representatives. For a conformal self-map ϕ \phi ϕ ϕ ∗ g \phi^*g ϕ ∗ g ϕ ∗ g = c 2 g \phi^*g = c^2g ϕ ∗ g = c 2 g c = 1 c = 1 c = 1
Theorem 8.13 (Closedness and generic infinitude). The finite-closure locus
F ( M ) = { g ∈ Met ∞ ( M ) : dim C g < ∞ } \mathcal{F}(M) = \{g \in\operatorname{Met}^{\infty}(M) : \dim C_g < \infty\} F ( M ) = { g ∈ Met ∞ ( M ) : dim C g < ∞ } is closed in the C ∞ C^\infty C ∞ M M M F ( M ) \mathcal{F}(M) F ( M ) F ( M ) = Met ∞ ( M ) \mathcal{F}(M) = \operatorname{Met}^{\infty}(M) F ( M ) = Met ∞ ( M )
Proof. Suppose g j → g g_j \to g g j → g g j ∈ F ( M ) g_j \in\mathcal{F}(M) g j ∈ F ( M ) g j g_j g j g g g d d d ∧ \wedge ∧ ∗ g j \ast_{g_j} ∗ g j C g C_g C g 2 n + 1 2^n + 1 2 n + 1 L 2 L^2 L 2 j j j dim C g j ≤ 2 n \dim C_{g_j} \le2^n dim C g j ≤ 2 n C ∞ C^\infty C ∞ F ( M ) \mathcal{F}(M) F ( M )
Low-dimensional consequences
Proposition 8.14 (Dimension three). For a closed, connected, oriented three-manifold, the finite-closure metrics are exactly as follows.
Table 3.
Proof. The bound gives b 1 ≤ 3 b_1 \le3 b 1 ≤ 3 C g 2 = ∗ C g 1 C_g^2 = \ast C_g^1 C g 2 = ∗ C g 1 n − 1 n - 1 n − 1 r = dim C g 1 ∈ { 0 , 1 , 3 } r = \dim C_g^1 \in\{0,1,3\} r = dim C g 1 ∈ { 0 , 1 , 3 } b 1 = 0 b_1 = 0 b 1 = 0 b 1 = 1 b_1 = 1 b 1 = 1 α \alpha α span R { 1 , α , ∗ α , vol g } \operatorname{span}_{\mathbb{R}}\{1,\alpha,\ast\alpha,\operatorname{vol}_g\} span R { 1 , α , ∗ α , vol g }
If b 1 = 2 b_1 = 2 b 1 = 2 α , β \alpha,\beta α , β γ = ∗ ( α ∧ β ) \gamma= \ast(\alpha\wedge\beta) γ = ∗ ( α ∧ β ) d α = d β = 0 d\alpha= d\beta= 0 d α = d β = 0 d ∗ α = d ∗ β = 0 d * \alpha= d * \beta= 0 d ∗ α = d ∗ β = 0 d γ = c α ∧ β d\gamma= c\alpha\wedge\beta d γ = c α ∧ β c ≠ 0 c \ne0 c = 0 Theorem 8.10 . If b 1 = 3 b_1 = 3 b 1 = 3
Proposition 8.15 (A four-dimensional reduction). If n = 4 n = 4 n = 4 χ ( M ) ≠ 0 \chi(M) \ne0 χ ( M ) = 0 g g g
Proof. A nonzero element of C g 1 C_g^1 C g 1 χ ( M ) ≠ 0 \chi(M) \ne0 χ ( M ) = 0 C g 1 = C g 3 = 0 C_g^1 = C_g^3 = 0 C g 1 = C g 3 = 0 C g C_g C g χ ( M ) = 2 + b 2 ( M ) > 0 \chi(M) = 2 + b_2(M) > 0 χ ( M ) = 2 + b 2 ( M ) > 0
Related work
Geometric formality [50 ] is exactly C g = H g C_g = \mathcal{H}_g C g = H g [52 ] ; the high-b 1 b_1 b 1 [51 ] supplies the invariant cohomology model. The conformal argument is closely related to Ornea–Pilca’s middle-degree uniqueness theorem [53 Proposition 2.4] and the nonlinear conformal n / k n/k n / k [54 ] . Hájek’s small subalgebras are generated by harmonic data and preserved by a special propagator [55 Definition 3.19] ; ABC-geometric formality admits a Hodge-star-stable bidifferential algebra formulation with additional complex hypotheses [56 Definition 4.4] . The construction considered here is the real canonical ( ∧ , d , ∗ ) (\wedge, d, *) ( ∧ , d , ∗ )
