HHOMOLOGY OF RELATIVE TRISECTION AND ITSAPPLICATION
HOKUTO TANIMOTO
Abstract.
Feller, Klug, Schirmer and Zemke showed the homology andthe intersection form of a closed trisected 4-manifold are described in termsof trisection diagram. In this paper, it is confirmed that we are able tocalculate those of a trisected 4-manifold with boundary in a similar way.Moreover, we describe a representative of the second Stiefel-Whitney classby the relative trisection diagram. Introduction
Gay and Kirby [1] introduced a trisection as a decomposition of a 4-manifoldinto three 4-dimensional handlebodies. They mainly dealt with closed man-ifolds. Since Castro, Gay and Pinz´on-Caicedo defined a relative trisectionclearly in [2] and [3], we are able to deal with the case of 4-manifolds withboundary in a similar way to the closed case such as a trisection diagramgiven by three families of curves on the central surface.Feller, Klug, Schirmer and Zemke [4] expressd the homology and the inter-section form of a closed 4-manifold in terms of trisection diagram in the wayone of three families of curves plays a key role. On the other hand, Florensand Moussard [5] described the homology such that roles of three families aresymmetric.In this paper, we show that the homology and the intersection form of a tri-sected 4-manifold with boundary can be calculated in a similar way to [4] and[5]. Our approaches derive from handle decompositions of the manifold associ-ated with the relative trisection in both cases. Using a chain complex given bythis approach, we express a representative of the second Stiefel-Whitney classin terms of relative trisection diagram. As a corollary, we obtain a necessaryand sufficient condition for the existence of spin structures on a 4-manifold,which is described by curves on the central surface.This paper is organized as follows. In Section 2, we recall briefly definitionsregarding relative trisections and their diagrams, and then state the mainresults of this paper. In Section 3, we consider two handle decompositionsassociated with the relative trisection: one’s union of a 0-handle and 1-handlesis X and the other’s union is D × Σ. In Section 4, we review several factsof 3-dimensional topology. In Section 5, we compute the homology and theintersection form of X . As an application, we give a description of the secondStiefel-Whitney class in Section 6. a r X i v : . [ m a t h . G T ] J a n HOKUTO TANIMOTO
The author would like to thank Hisaaki Endo for many discussions andencouragement. 2.
Main results
Let X be a compact, connected, oriented, smooth 4-manifold with con-nected boundary. Relative trisections are more complicated than usual onesfor closed 4-manifolds. We employ the same definition as [2] and [3]. Let g, k = ( k , k , k ) , p, b be integers with g ≥ p ≥ b ≥ g + p + b − ≥ k i ≥ p + b −
1, and put l = 2 p + b − D , ∂ D and ∂ ± D denote a third of aunit 2-dimensional disk, its arc and ∂D \ Int ∂ D , respectively. Moreover, let ∂ − D and ∂ + D be each radius. Definition 2.1.
A ( g, k ; p, b )- relative trisection of X is a decomposition X = X ∪ X ∪ X such that:i) X i ∩ X ∩ X is diffeomorphic to Σ, a genus g surface with b boundarycomponents;ii) X i ∩ X j = ∂X i ∩ ∂X j for i (cid:54) = j , and each of them is diffeomorphic to a3-dimensional compression body from Σ to a genus p surface P ;iii) X i is diffeomorphic to (cid:92) k i S × D ;iv) X i ∩ ∂X is diffeomorphic to P × ∂ D ∪ ∂P × D .v) ( X i − ∩ X i ) ∪ ( X i ∩ X i +1 ) is a sutured Heegaard splitting of P × ∂ ± D (cid:93) ( (cid:93) k i − l S × S ).We orient each compression body X i ∩ X i − as a submanifold of ∂X i , surfaces X ∩ X ∩ X and ( X i ∩ X i − ) ∩ ∂X as submanifolds of ∂ ( X i ∩ X i − ).The above definition can be extended to a 4-manifold with more than oneboundary components. In that case, our main results are also true. However,we assume that the boundary is connected for simplicity.Same as Heegaard splittings and trisections, there is a diagram that deter-mines the relative trisection. Let µ , ν be a family of g − p disjoint simpleclosed curves on Σ. For (Σ; µ, ν ) and (Σ; µ (cid:48) , ν (cid:48) ), we call that (Σ; µ, ν ) is dif-feomorphism and handle slide equivalent to (Σ; µ (cid:48) , ν (cid:48) ) if they are related by adiffeomorphism between Σ and a sequence of handle slides within each µ and ν . Definition 2.2.
A ( g, k ; p, b ) -relative trisection diagram (Σ; α, β, γ ) is a 4-tuplesuch that:i) Σ is a genus g surface with b boundary components;ii) each α, β and γ is a family of g − p disjoint simple closed curves on Σ;iii) each triple (Σ; α, β ) , (Σ; β, γ ) , (Σ; γ, α ) is diffeomorphism and handleslide equivalent to (Σ; δ k i , (cid:15) k i ) shown in Figure 1, which is a standardsutured Heegaard diagram of P × ∂ ± D (cid:93) ( (cid:93) k i − l S × S ). OMOLOGY OF RELATIVE TRISECTION AND ITS APPLICATION 3 . . . . . . . . . ...
Figure 1.
A standard diagram (Σ; δ k , (cid:15) k ), where the red curvesare δ k , and blue curves are (cid:15) k .For ν ∈ { α, β, γ } , let C ν be a compression body given by attaching 3-dimensional 2-handles to Σ × I along a family of curves ν × { } . Furthermore,Σ ν denotes a surface given by performing surgeries to Σ × { } along ν × { } .A relative trisection is determined by the spine C α ∪ C β ∪ C γ , that is, a relativetrisection diagram describes the relative trisection such that X ∩ X ∩ X = Σ, X ∩ X = C α , X ∩ X = C β and X ∩ X = C γ , respectively.For each homomorphism induced by inclusions ι ν : H (Σ) → H ( C ν ) and ι ∂ν : H (Σ , ∂ Σ) → H ( C ν , ∂C ν \ Σ), let L ν be ker ι ν , and L ∂ν ker ι ∂ν . Using thissubgroups, we calculate the homology in two ways. Theorem 1.
The homology of X can be obtained from the following chaincomplex C Y : L α ∩ L γ ) ⊕ ( L β ∩ L γ ) L γ Hom( L ∂α ∩ L ∂β , Z ) Z , π ρ where π ( x, y ) = x + y and ρ ( x ) = (cid:104)− , x (cid:105) Σ . Theorem 2.
The homology of X can also be obtained from the followingchain complex C Z : L α ∩ L β ) ⊕ ( L β ∩ L γ ) ⊕ ( L γ ∩ L α ) L α ⊕ L β ⊕ L γ H (Σ) Z , ζ ι where ζ ( x, y, z ) = ( x − z, y − x, z − y ) and ι is a homomorphsim induced bythe inclusions ι ν .In both cases, H ( X ) is isomorphic to the group described by L ν in the sameway as [4]. Moreover, the intersection form of X is also described by this groupand the intersection form of Σ.Let µ , ν be families of oriented simple closed curves on Σ or families oforiented arcs proper embedded in Σ. Definition 2.3.
For µ and ν , an intersection matrix µ Q ν of µ , ν is the matrixwhich consists of the intersection numbers (cid:104) [ µ i ] , [ ν j ] (cid:105) Σ .For a relative trisection diagram (Σ; α, β, γ ), performing handleslides on α and β , we can suppose by Definition 2.2.iii) that ( α, β ) is diffeomorphismequivalent to ( δ k , (cid:15) k ). Therefore, there exists a family of arcs a such that HOKUTO TANIMOTO { [ α ] , [ a ] } and { [ β ] , [ a ] } are bases of L ∂α and L ∂β , respectively. We define severalmatrices to describe the second Stiefel-Whitney class as follows; γ Q β,∂ = (cid:0) γ Q β γ Q a (cid:1) , α∂ Q γ = (cid:18) α Q γa Q γ (cid:19) ,R gp,b = I g − p ⊕ p (cid:18) (cid:19) ⊕ O b − . Theorem 3.
Suppose that ( α, β ) is diffeomorphism equivalent to ( δ k , (cid:15) k ).Then, as a representative of the second Stiefel-Whitney class w , we can obtain c : L γ → Z defined by the following formula with respect to the γ -basis c ( x ) = g − p (cid:88) i =1 (cid:0) γ Q β,∂ R gp,b α∂ Q γ (cid:1) i i x i (mod 2) . Next, let a be a family of arcs such that { [ a ] } is a basis of H (Σ , ∂ Σ).Regarding the complex C Z , we can describe a representative by using thefollowing matrix; ν Q a = α Q aβ Q aγ Q a , a Q ν = − t ( ν Q a ) ,S g,b = ⊕ g (cid:18) (cid:19) ⊕ O b − . Theorem 4.
As a representative of the second Stiefel-Whitney class, we canobtain c (cid:48) : L α ⊕ L β ⊕ L γ → Z defined by the following formula with respect tothe α -, β -, γ -basis c (cid:48) ( x ) = g − p ) (cid:88) i =1 ( ν Q a S g,b a Q ν ) i i x i (mod 2) . Relative trisection and handle decomposition
According to [1], non-relative trisections induce handle decompositions ofa 4-manifold. Therefore, trisection is considered as a variant of handle de-composition. We can calculate the homology and the intersection form of a4-manifold by using a trisection. In this section, we show that relative trisec-tion has similar properties.First, we introduce a handle decomposition constructed from X . Lemma 3.1.
For a trisected 4-manifold X with connected boundary, there isa decomposition X = Y ∪ V ∪ V such that:(1) each V , V is diffeomorphic to (cid:92) l S × D , and those intersection V ∩ V is an empty set;(2) Y ∩ V i is included in ∂V i , and is diffeomorphic to P × I ; OMOLOGY OF RELATIVE TRISECTION AND ITS APPLICATION 5 (3) Y is a handlebody that consists of one 0-handle, k g − p k + k − l γ are the attaching circles of 2-handles for Y , and theirframings are induced by Σ;(5) Y = X . Proof.
Since X is diffeomorphic to (cid:92) k S × D , we regard it as Y . X ∩ X is obtained by attaching 3-dimensional 2-handles to X ∩ X ∩ X ∼ = Σ along γ with framing induced by Σ. We get 4-dimensional 2-handlesby thickening them. In other words, for the 3-dimensional attaching region S × D ⊂ Σ and an embedded Σ × D in ( X ∩ X ) ∪ ( X ∩ X ) such thatΣ × D ∩ ( X ∩ X ) = Σ × [ − ,
0] and Σ × D ∩ ( X ∩ X ) = Σ × [0 , S × D × D . A surgery derived from thisattaching 2-handles changes C α ∪ C β into ( C α ∪ C γ ) ∪ P × D ∪ ( C γ ∪ C β ). X is obtained by attaching (cid:92) k S × D , (cid:92) k S × D on it along C α ∪ C γ , C γ ∪ C β which are diffeomorphic to P × ∂ ± D (cid:93) ( (cid:93) k i − l S × S ).We separate (cid:92) k i S × D into (cid:92) k i − l S × D included in Int X and (cid:92) l S × D . Weregard the former as a 4-dimensional handlebody which consists of one 0-handleand k i − l k i − l + 1 3-handles and one4-handle. It is clear that the pair of a 3-handle of boundary sum and the 4-handle is a canceling pair. In this way, we obtain the handlebosy Y satisfying(3), (4) and (5). Each V , V should be (cid:92) l S × D attached along C γ ∪ C β , C α ∪ C γ . (cid:3) On the other hand, there is another handle decomposition constructed fromΣ × D . Lemma 3.2.
For a trisected 4-manifold X with connected boundary, there isa decomposition X = Z ∪ W ∪ W ∪ W such that:(1) W i is diffeomorphic to (cid:92) l S × D , and those intersection W i ∩ W j ( i (cid:54) = j )is an empty set;(2) Z ∩ W i is included in ∂W i , and is diffeomorphic to P × I ;(3) Z is a handlebody that consists of one 0-handle, 2 g + b − g − p ) 2-handles, (cid:80) i k i − l α , β , γ are the attaching circles of 2-handles for Z . Further-more, their framings are induced by Σ;(5) Z ∼ = Σ × D . Proof.
Since Σ × D is diffeomorphic to (cid:92) g + b − S × D , we regard it as Z .Considering how to reconstruct a trisection from a diagram, we can obtain thedecomposition in a similar way to the proof of Lemma 3.1. (cid:3) HOKUTO TANIMOTO
Figure 2.
Two decompositions of X : Y ∪ V ∪ V and Z ∪ W ∪ W ∪ W .According to the following lemma, X is homotopy equivalent to each of Y and Z . Therefore, the homology and the intersection form of X can becalculated from the handlebodies Y and Z . Lemma 3.3.
There are deformation retracts r : ( X, ∂X ) → ( Y, ∂Y ) and r (cid:48) :( X, ∂X ) → ( Z, ∂Z ). Consequently, H i ( X ) is isomorphic to H i ( Y ) and H i ( Z ),and H ∗ ( X, ∂X ) is isomorphic to each of H ∗ ( Y, ∂Y ) and H ∗ ( Z, ∂Z ) as rings.
Proof.
Since V i ∼ = (cid:92) l S × D and ( Y ∩ V i ) ∪ ∂V i \ ( Y ∩ V i ) is a standard Heegaardsplitting of ∂V i ∼ = (cid:93) l S × S , we can construct deformation retracts r i : V i → Y ∩ V i . Using this r i and id Y , we obtain the desired map r . The constructionof r (cid:48) is the same. (cid:3) Figure 3. r : ( X, ∂X ) → ( Y, ∂Y ).4.
Facts of 3-dimensional topology
To consider the homology and the intersection of Y and Z , we review topol-ogy of 3-manifolds. We modify some lemmas in [4] to obtain the lemmas inthis section, which can be applied to sutured Heegaard splitting. We omit theproofs of several lemmas in this paper since they are similar to those of [4]. Lemma 4.1.
Let C ν be a relative compression body with surface Σ. Then:(1) { [ ν ] , . . . , [ ν g − p ] } forms a basis of L ν ;(2) there is a family of arcs a such that { [ ν ] , . . . , [ ν g − p ] , [ a ]; . . . , [ a l ] } formsa basis of L ∂ν ,(3) { x ∈ H (Σ) | ∀ y ∈ L ∂ν , (cid:104) y, x (cid:105) Σ = 0 } = L ν . OMOLOGY OF RELATIVE TRISECTION AND ITS APPLICATION 7
Proof.
We consider the exact sequence H ( C ν ) H ( C ν , Σ) H (Σ) H ( C ν ) ∂ ι ν of ( C ν , Σ) to show Claim (1). H ( C ν ) = 0 since C ν is diffeomorphic to (cid:92) g + p + b − S × D . The core disks of 2-handles form a basis of H ( C ν , Σ), and theimage of each core disk is its attaching circle. These prove Claim (1) becauseof the exactness.To consider exact sequences of triples ( C ν , Σ , ∂ Σ) and ( C ν , ∂C ν \ Σ , ∂ Σ), wedeal with the sequence of ( C ν , ∂ Σ): H ( C ν , ∂ Σ) H ( ∂ Σ) H ( C ν ) H ( C ν , ∂ Σ) H ( ∂ Σ) H ( C ν ) . σ σ Let c , . . . , c b − be boundary components of ∂ Σ, and then the homomorphism σ satisfies σ ([ c ]) = − [ c ] − · · · − [ c b − ], σ ([ c i ]) = [ c i ] ( i = 1 , . . . , b − σ is the free submodule which has one generator [ c ] + · · · + [ c b − ],[Σ] forms a basis of H ( C ν , ∂ Σ). Each c i ⊂ (cid:92) g − p + b − S × D can be regardedas S × {∗ i } , and those homology classes form a basis of Im σ . Therefore,coker σ is a free module. It is easy to see that ker σ is free. These prove that H ( C ν , ∂ Σ) is also free.Since the image of [Σ] by the homomorphism H ( C ν , ∂ Σ) → H ( C ν , Σ) iszero, we obtain the following diagram: H ( C ν , Σ) H (Σ , ∂ Σ) H ( C ν , ∂ Σ) 00 H ( ∂C ν \ Σ , ∂ Σ) H ( C ν , ∂ Σ) H ( C ν , ∂C ν \ Σ) 0 . ∂ η ι ∂ν η θθ The upper row is the exact sequence of ( C ν , Σ , ∂ Σ), and the lower row isthe exact sequence of ( C ν , ∂C ν \ Σ , ∂ Σ). The upper sequence splits such that H (Σ , ∂ Σ) (cid:39) Im ∂ ⊕ H ( C ν , ∂ Σ), and ι ∂ν ( x, y ) = θ ( y ). Let a be a family of l proper arcs in Σ \ ν such that the surgery along a changes Σ ν into a disk. { [ a ] , . . . , [ a l ] } forms a basis of H ( ∂C ν \ Σ , ∂ Σ). Since we can regard a ⊂ Σ,so ker ι ∂ν = Im ∂ ⊕ ker θ . This proves Claim (2).We separate Σ into Σ g − p (cid:93) Σ ν such that ν ⊂ Σ g − p and a ⊂ Σ ν . Let ν ∗ bea family of g − p curves in Σ g − p such that { [ ν ] , [ ν ∗ ] } is a symplectic basis of H (Σ g − p ) regarding the intersection form of Σ g − p . Let a ∗ be a family of l curves in Σ ν as the following diagram. HOKUTO TANIMOTO ......
Figure 4.
The curves a and a ∗ . { [ a ∗ ] } is a basis of H (Σ ν ), and then { [ ν ] , [ ν ∗ ] , [ a ∗ ] } is a basis of H (Σ). For x = ( x , . . . , x g + b − ) ∈ H (Σ), we have (cid:104) ν i , x (cid:105) Σ = x i + g − p , (cid:104) a i , x (cid:105) Σ = x i +2( g − p ) . (1), (2) and this show that the left-hand side of (3) is L ν . (cid:3) Let ( L ∂ν ) ⊥ be the left-hand side of Lemma 4.1 (3), then ( L ∂ν ) ⊥ = L ν accordingto (3). We can regard L ν as a submodule in L ∂ν .Let (Σ; C µ , − C ν ) be a sutured Heegaard splitting of a 3-manifold M . We de-fine ι M and ι ∂M in the same way as ι ν and ι ∂ν . We consider the Mayer-Vietoris se-quences of (Σ; C µ , − C ν ) and (cid:16) (Σ , ∂ Σ) ; ( C µ , ∂C µ \ Σ) , − ( C ν , ∂C ν \ Σ) (cid:17) . Sup-pose Q ⊂ Int M is an oriented embedded surface which is transverse to Σand represents x ∈ H ( M ). The image of the boundary homomorphism ∂x isrepresented by the family of curves Q ∩ Σ. We orient them as the boundaryof the oriented surface Q ∩ C µ . Note that the orientation induced from Q ∩ C ν is the opposite. The boundary homomorphism of H ( M, ∂M ) is also similar.
Lemma 4.2.
Let ∂ : H ( M ) → H (Σ) and ∂ (cid:48) : H ( M, ∂M ) → H (Σ , ∂ Σ) bethe boundary homomorphisms coming from the Mayer-Vietoris sequences forthe triple (Σ; C µ , − C ν ) and (cid:16) (Σ , ∂ Σ) ; ( C µ , ∂C µ \ Σ) , − ( C ν , ∂C ν \ Σ) (cid:17) , respec-tively. Then:(1) the maps ∂ and ∂ (cid:48) are isomorphisms between H ( M ) and L µ ∩ L ν , and H ( M, ∂M ) and L ∂µ ∩ L ∂ν , respectively;(2) for any z ∈ H ( M, ∂M ) and x ∈ H (Σ), (cid:104) z, ι M x (cid:105) M = (cid:104) ∂ (cid:48) z, x (cid:105) Σ ;(3) if H ( M, ∂M ) is torsion free and x ∈ H (Σ), then x ∈ ker ι M if andonly if x ∈ ( L ∂µ ∩ L ∂ν ) ⊥ . Proof.
The proof of this lemma is similar to the proof of Lemma 2.2 [4]. (cid:3)
The intersection form of a 4-manifold with handle decomposition can becalculated by using a linking matrix of the framed link obtained from attachingcircles of the 2-handles. We can also define the linking number in the case of 3-manifolds with boundaries by using Seifert surfaces. In regard to a relationship
OMOLOGY OF RELATIVE TRISECTION AND ITS APPLICATION 9 between the linking number and the intersection on Σ, we obtain the followinglemma.
Lemma 4.3.
Suppose J and K are families of embedded oriented curves inInt Σ such that ι M ([ J ]) = ι M ([ K ]) = 0, J ∩ K = ∅ . Then we have:(1) there is j ∈ L µ such that (cid:104) y, j (cid:105) Σ = (cid:104) y, [ J ] (cid:105) Σ for all y ∈ L ∂ν ;(2) for j ∈ L µ satisfying Claim (1), lk( J, K ) M = (cid:104) j, [ K ] (cid:105) Σ . Proof.
The proof of this lemma is the same as the proof of Lemma 2.4 [4]. (cid:3)
We call the closed version of this lemma also Lemma 4.3.Let N and M be (cid:93) k S × S and P × ∂ ± D (cid:93) ( (cid:93) k − l S × S ), respectively. Consid-ering a genus l Heegaard splitting of (cid:93) l S × S constructed from P × ∂ ± D and P × ∂ D ∪ ∂P × D , we can regard M as a submanifold of N . H ( N ) is a free mod-ule which has a basis (cid:8) [ {∗ i } × S ] (cid:9) . Suppose that a is the family of arcs on P as in Figure 4, then (cid:8) [ a × ∂ ± D ] , . . . , [ a l × ∂ ± D ] , (cid:2) {∗ l +1 }× S (cid:3) , . . . , (cid:2) {∗ k }× S (cid:3)(cid:9) is a basis of H ( M, ∂M ). We define an isomorphism b : H ( N ) → H ( M, ∂M )as follows: b ([ {∗ i } × S ]) = (cid:40) [ a i × ∂ ± D ] ( i ≤ l ) (cid:2) {∗ i } × S (cid:3) ( l + 1 ≤ i ) . In regard to the intersections in N and M , we have: Lemma 4.4.
For all z ∈ H ( N ) and x ∈ H (Σ), (cid:104) z, ι N x (cid:105) N = (cid:104) b ( z ) , ι M x (cid:105) M . Proof.
It is sufficient to show this regarding the basis { [ {∗ i } × S ] } . Thecase l + 1 ≤ i is clear. Considering the above canonical Heegaard splitting of (cid:93) l S × S , we obtain N by attaching a handlebody H l to M along this splitting.Therefore, {∗ i } × S is the sphere which is made of a i × ∂ ± D and a core diskof H l , and then a i × ∂ ± D = ( {∗ i } × S ) ∩ M . Let γ be a representative of x such that it is transverse to {∗ i } × S . Since γ is also in M , ( {∗ i } × S ) ∩ γ =( a i × ∂ ± D ) ∩ γ . (cid:3) Calculating the homology and the intersection form
Let (Σ; X , X , X ) be a ( g, k ; p, b )-relative trisection of a 4-manifold X . M µν denotes the manifold which admits a sutured Heegaard splitting (Σ; C µ , − C ν ),and let ∂ µν , ∂ (cid:48) µν be the maps in Lemma 4.2 for this M µν . In this section, weshow how the homology and the intersection form of X are calculated in termsof the intersection form of Σ and the subgroups L ν . Theorem 1.
The homology of X can be obtained from the following chaincomplex C Y : L α ∩ L γ ) ⊕ ( L β ∩ L γ ) L γ Hom( L ∂α ∩ L ∂β , Z ) Z , π ρ where π ( x, y ) = x + y and ρ ( x ) = (cid:104)− , x (cid:105) Σ . Proof.
We calculate the homology of the handlebody Y . Let ( C ∗ , ∂ ∗ ) be thechain complex of Y . Since our manifold is oriented and connected, we willfocus on the following nontrivial part of ( C ∗ , ∂ ∗ ):0 C C C . ∂ ∂ Let c : C → H ( ∂X ) be the isomorphism which sends each 1-handle gener-ator of C to the element of H ( ∂X ) represented by its belt sphere, and let b : C → L γ be the isomorphism which sends each 2-handle to the element of L γ ⊂ H (Σ) represented by its attaching circle. The boundary homomorphism ∂ is expressed as follows: ∂ ( h ) = (cid:88) j (cid:104) c ( h j ) , ι ∂X b ( h ) (cid:105) ∂X h j . We fix a diffeomorphism φ of ∂X ∼ = (cid:93) k S × S such that ( ∂X , M αβ , ∂ Σ × [ − , , Σ α , Σ β ) correspond to ( (cid:93) k S × S , P × ∂ ± D (cid:93) ( (cid:93) k − l S × S ) , ∂P × ∂ ± D,P − , P + ). We define b = ( φ ∗ ) − ◦ b ◦ φ ∗ by using this φ and the homomorphism b in Lemma 4.4. The above equality can be replaced by ∂ ( h ) = (cid:88) j (cid:104) b c ( h j ) , ι M αβ b ( h ) (cid:105) M αβ h j . Furthermore, by Lemma 4.2 (2), this equality can also be replaced by ∂ ( h ) = (cid:88) j (cid:104) ∂ (cid:48) αβ b c ( h j ) , b ( h ) (cid:105) Σ h j . Let D : L ∂α ∩ L ∂β → Hom( L ∂α ∩ L ∂β , Z ) be the dual map induced by the basis { [ ∂ (cid:48) αβ b c ( h j )] } . If we define f = D ◦ ∂ (cid:48) αβ ◦ b ◦ c and f = b , then we have(5.1) ρ ◦ f = f ◦ ∂ . We separate C into C ⊕ C to define f , where C i is the free modulegenerated by the 3-handles derived from X i . Using this decomposition, wedefine an isomorphism b : C → H ( M γα ) ⊕ H ( M βγ ) which maps each 3-handle generator of C to the element of H ( M γα ) represented by its attachingsphere in M γα , and likewise for the generator of C and H ( M βγ ). Let f : C → ( L α ∩ L γ ) ⊕ ( L β ∩ L γ ) be the isomorphism obtained by composing this b withthe sum of ∂ γα : H ( M γα ) → L α ∩ L γ and − ∂ βγ : H ( M βγ ) → L β ∩ L γ fromLemma 4.2 (1). We prove that π ◦ f = f ◦ ∂ . (5.2)These equations (5.1), (5.2) will show that f ∗ is a chain isomorphism between C and C Y , that is, the theorem holds.It is sufficient to show (5.2) in regard to the generators of C . M γα consistsof 3-dimensional 2-handles whose attaching circles are γ , Σ × D and 2-handleswhose attaching circles are α . Every embedded sphere S ⊂ M γα is isotopic toa sphere which intersects 2-handles whose attaching circles are γ only in disks OMOLOGY OF RELATIVE TRISECTION AND ITS APPLICATION 11 that are parallel to the core of a 2-handle. We focus on each 2-handle D × D to show this. S ∩ D × D consists of some Σ ,t j ( t j ≥
1) whose boundary ∂ Σ ,t j areincluded in ∂D × D . If there is Σ ,t j which is not a disk and has a boundarycomponent that is not null-homologous in ∂D × D , we move S as in thefollowing figure so that it produces a disk which is parallel to the core andΣ ,t j whose non null-homologous boundary components decrease. . . .. . .. . .. . . . . .. . . . . . . . . . . . . . . . . . . . . Figure 5.
Isotopy of S .If necessary, by performing this operation several times, S ∩ D × D becomessome disks which are parallel to the core and some Σ ,t j whose boundary com-ponents are null-homologous. Since we can regard the latter as the boundaryof a tubular neighborhood of an embedded graph whose boundary componentsare included in ∂D × D , these can be disjoint from { } × D . Therefore, forsufficiently small (cid:15) >
0, every Σ ,t j is disjoint from D (cid:15) × D . If we replace a2-handle D × D with D (cid:15) × D , S intersects the 2-handle only in disks that areparallel to the core. This shows that S is isotopic to a sphere which satisfiesthe condition.Performing the same operation regarding α -side, we can assume that S intersects the both sides only in the parallel disks. Furthermore, S is isotopicto S γ which intersects C γ in the parallel disks. ( S is also isotopic to S α in thesame way.)Let S be the attaching sphere of a 3-handle h . Since f ( h ) = ∂ γα ([ S ]) and π is an inclusion, we have π ◦ f ( h ) = [ S γ ∩ Σ] . Let D ij be a disk component of S γ ∩ C γ which is parallel to the core diskof 2-handle h j , and then [ S γ ∩ Σ] = (cid:80) i,j [ ∂D ij ]. From the way to define theorientation of ∂D ij , [ ∂D ij ] is equivalent to (cid:104) D ij , ( { } × D ) j (cid:105) M γα f ( h j ). Since S γ ∩ h j = (cid:96) i D ij , we have (cid:88) i,j [ ∂D ij ] = (cid:88) j (cid:104) S γ , ( { } × D ) j (cid:105) ∂Y f ( h j ) . The belt sphere of h j appears as ( { } × D ) j in M γα . Therefore, the right-handside of the formula is equal to f ◦ ∂ ( h ). (cid:3) Theorem 2.
The homology of X can also be obtain from the following chaincomplex C Z : L α ∩ L β ) ⊕ ( L β ∩ L γ ) ⊕ ( L γ ∩ L α ) L α ⊕ L β ⊕ L γ H (Σ) Z , ζ ι where ζ ( x, y, z ) = ( x − z, y − x, z − y ) and ι is a homomorphism induced bythe inclusions ι ν . Proof.
Let ( C (cid:48)∗ , ∂ (cid:48)∗ ) be the chain complex of Z . Its nontrivial part is0 C (cid:48) C (cid:48) C (cid:48) . ∂ (cid:48) ∂ (cid:48) Note that Z is diffeomorphic to Σ × D . Applying p = g, k = 2 g + b − b (cid:48) : H ( ∂Z ) → H (Σ × ∂ ± D, ∂ (Σ × ∂ ± D )) and ∂ (cid:48) : H (Σ × ∂ ± D, ∂ (Σ × ∂ ± D )) → H (Σ , ∂ Σ). If wedefine f (cid:48) and f (cid:48) in the same way as in the proof of Theorem 1, the followingformula also holds: ι ◦ f (cid:48) = f (cid:48) ◦ ∂ (cid:48) . Similarly, we obtain f (cid:48) by composing b (cid:48) : C (cid:48) → H ( M αβ ) ⊕ H ( M βγ ) ⊕ H ( M γα ) and ∂ αβ ⊕ ∂ βγ ⊕ ∂ γα . Let S be the attaching sphere of a 3-handle h ∈ C (cid:48) , and then we have ζ ◦ f (cid:48) ( h ) = ( ∂ αβ ([ S ]) , − ∂ αβ ([ S ]) , .S is isotopic to S α and S β as in the proof of Theorem 1. Let D αij and D βij bea disk component of S α ∩ C α and S β ∩ − C β , respectively. From the way toorient ∂ αβ , the right-hand side is equal to ( (cid:80) i,j [ ∂D αij ] , (cid:80) i,j [ ∂D βij ] , h βj appears as − ( { } × D ) βj in M αβ , theintersection number of the core disk of h βj with it in − C β is +1. Therefore, weobtain ζ ◦ f (cid:48) = f (cid:48) ◦ ∂ (cid:48) . (cid:3) Using Theorem 1 and 2, we can describe each H ( X ), H ( X ) and H ( X ) by L ν in the same way as [4] and [5]. We give a precise proof of the descriptionof H ( X ) by using Theorem 1. Corollary 5.1. H ( X ) (cid:39) L γ ∩ ( L α + L β )( L γ ∩ L α ) + ( L γ ∩ L β ) . Proof.
We should calculate ker ρ and Im π . It is easy to show Im π = ( L γ ∩ L α ) + ( L γ ∩ L β ) and ker ρ = L γ ∩ ( L ∂α ∩ L ∂β ) ⊥ . Therefore, it is sufficient to show( L ∂α ∩ L ∂β ) ⊥ = L α + L β .( L ∂α ∩ L ∂β ) ⊥ ⊃ L α + L β is clear because of Lemma 4.1 (3). Since (Σ; α, β )is equivalent to (Σ; δ k , (cid:15) k ), performing handle slides on each α and β if nec-essary, we obtain a family of proper arcs a which is disjoint from α ∪ β , andcuts each Σ α and Σ β into a disk. Moreover, we can assume [ α i ] = [ β i ] ∈ L α ∩ L β ( i ≤ k − l ), (cid:104) [ α i ] , [ β j ] (cid:105) Σ = δ ij ( i, j ≥ k − l + 1). For this α , let OMOLOGY OF RELATIVE TRISECTION AND ITS APPLICATION 13 { [ α ] , [ α ∗ ] , [ a ∗ ] } be the basis as in the proof of Lemma 4.1 (3). We can also as-sume [ α ∗ i ] = [ β i ] ( i ≥ k − l + 1). Since { [ α ] , . . . , [ α k − l ] , [ a ] , . . . , [ a l ] } becomesa basis of L ∂α ∩ L ∂β , x ∈ ( L ∂α ∩ L ∂β ) ⊥ must be represented as a liner combinationof [ α ] , . . . , [ α g − p ] , [ α ∗ k − l +1 ] , . . . , [ α ∗ g − p ]. (cid:3) Since the proof of the following corollary is the same as [4] and [5], we onlyintroduce the forms of H , H . Corollary 5.2. H ( X ) (cid:39) H (Σ) L α + L β + L γ , H ( X ) (cid:39) L α ∩ L β ∩ L γ . To consider the intersection form of X , we define a bilinear form Φ. Definition 5.3.
Let Φ : L γ ∩ ( L α + L β ) × L γ ∩ ( L α + L β ) → Z be the bilinearform given as follows; Φ( x, y ) = −(cid:104) x (cid:48) , y (cid:105) Σ , where x (cid:48) is any element of L α which satisfies x − x (cid:48) ∈ L β .There is an element x (cid:48) satisfying the above condition since x ∈ L α + L β ,but it is not unique in general.. For another x (cid:48)(cid:48) , x (cid:48) − x (cid:48)(cid:48) ∈ L α ∩ L β . Since y ∈ ( L ∂α ∩ L ∂β ) ⊥ , (cid:104) x (cid:48) − x (cid:48)(cid:48) , y (cid:105) Σ = 0. Therefore, the value of Φ does not dependon a choice of x (cid:48) . We describe the bilinear form induced by Φ on any quotientgroup as also Φ.The intersection form of X is equivalent to Φ. We can prove the followingproposition by using Lemmas in Section 4 in the same way as [4]. Furthermore,we can also prove it in the same way as [5] since this intersection is includedin Int Z . Proposition 5.4.
There is an isomorphism( H ( X ) , (cid:104)− , −(cid:105) X ) (cid:39) (cid:18) L γ ∩ ( L α + L β )( L γ ∩ L α ) + ( L γ ∩ L β ) , Φ (cid:19) . Representative of the Stiefel-Whitney class
As an application of chain complexes C Y and C Z , we describe representativesof the second Stiefel-Whitney class by using the matrices µ Q ν . Since we canconvert a trisection diagram to a Kirby diagram via handlebodies, the followingtheorem shown by Gompf and Stipsicz [6] plays an important role to describethe representative. Theorem. (Gompf-Stipsicz [6]) For an oriented handlebody X given by aKirby digram in dotted circle notation, w ( X ) ∈ H ( X ; Z ) is representedby the cocycle c ∈ C ( X ; Z ) whose value on each 2-handle is its framingcoefficient modulo 2.In regard to the trisection, we have: Theorem 3.
Suppose that ( α, β ) is diffeomorphism equivalent to ( δ k , (cid:15) k ).Then, as a representative of the second Stiefel-Whitney class w , we can obtain c : L γ → Z defined by the following formula with respect to the γ -basis c ( x ) = g − p (cid:88) i =1 (cid:0) γ Q β,∂ R gp,b α∂ Q γ (cid:1) i i x i (mod 2) . Proof.
Let a be a family of arcs in Σ as in the proof of Corollary 5.1. Fromthe assumption and Lemma 3.1, we can draw a Kirby diagram based on thetrisection diagram as follows: Σ × D is embedded in R ⊂ S , where S isthe boundary of the 0-handle. Dotted circles for 1-handles are ∂ ( a i × D )and α i which is parallel to β i . Attaching circles of 2-handles are γ , and itsframings are induced by Σ. ∂X is a union of C α , C β and ∂ ( P × D ) \ P × ∂ ± D .Since P × ∂ − D ∪ ( P × ∂ + D ∪ ∂ ( P × D ) \ P × ∂ ± D ) is a Heegaard splitting,we can construct ∂X from C α by attaching a 3-dimensional 1-handlebodywhose system consists of β and ∂ ( a i × [ − , ∂C α . Note that ∂C α is diffeomorphic to Σ g + p + b − , and C α itself is a 1-handlebody whose systemconsists of α and ∂ ( a i × [ − , c ∈ C ( X ; Z ). When we calculate a framing (or a linking number), wecan ignore dotted circles for 1-handles to regard ∂X ⊂ S . This is equivalentto changing the system { ∂C α ; β , . . . , β g − p , ∂ ( a × [ − , , . . . , ∂ ( a l × [ − , } into { ∂C α ; β l , . . . , β lg − p , a ∗ × { } , a ∗ × {− } , . . . , a ∗ p − × { } , a ∗ p × {− } , a ∗ p +1 ×{ } , . . . , a ∗ l × { }} , where β l is a family of curves which maps to (cid:15) l by the dif-feomorphism. There is no need to change C α .If we apply Lemma 4.3 to this decomposition of S , there is j γ for an at-taching circle γ such that c ( γ ) = −(cid:104) j γ , γ (cid:105) ∂C α . Since { [ α ] , [ β l ] , [ a ∗ ] } is a basisof H (Σ), γ is expressed as (cid:80) i ( x i [ α i ] + y i [ β li ]) + (cid:80) j z j [ a ∗ j ]. In ∂C α , [ a ∗ k − ]corresponds to [ a ∗ k − × { } ], and [ a ∗ k ] does to [ a ∗ k × {− } ] + [ ∂ ( a k − × [ − , ≤ k ≤ p . Since we can obtain j γ as an element which sat-isfies γ − j γ ∈ (cid:104) [ β l ] , [ a ∗ k − × { } ] , [ a ∗ k × {− } ] , a ∗ i × { }(cid:105) , j γ is expressed as (cid:80) i x i [ α i ] + (cid:80) k z k [ ∂ ( a k − × [ − , j γ as an element of L ∂α , wehave j γ = (cid:88) i x i [ α i ] + (cid:88) k z k [ a k − ] . For this j γ , we have c ( γ ) = −(cid:104) j γ , γ (cid:105) Σ because of j γ ∩ γ ⊂ Σ.Let α ∂ be a basis { [ α ] , [ a ] } of L ∂α . Note that we orient them to satisfy (cid:104) [ β li ] , [ α j ] (cid:105) Σ = (cid:104) [ a i ] , [ a ∗ j ] (cid:105) Σ = δ ij . In regard to this basis α ∂ , we obtain j γ i = (cid:80) j ( t R gp,b β∂ Q γ ) ji [ α ∂j ]. Since the coefficient of [ α ∂j ] is equal to − ( γ Q β,∂ R gp,b ) ij ,this equality can be replaced by j γ i = − (cid:88) j ( γ Q β,∂ R gp,b ) ij [ α ∂j ] . OMOLOGY OF RELATIVE TRISECTION AND ITS APPLICATION 15
For x = (cid:80) i x i [ γ i ] ∈ L γ , c ( x ) is − (cid:80) i (cid:104) j γ i , [ γ i ] (cid:105) Σ x i . Therefore, we can calculate c ( x ) as follows: c ( x ) = (cid:88) i (cid:88) j ( γ Q β,∂ R gp,b ) ij (cid:104) [ α ∂j ] , [ γ i ] (cid:105) Σ x i , = (cid:88) i (cid:0) γ Q β,∂ R gp,b α∂ Q γ (cid:1) ii x i . (cid:3) Theorem 4.
As a representative of the second Stiefel-Whitney class, we canobtain c (cid:48) : L α ⊕ L β ⊕ L γ → Z defined by the following formula with respect tothe α -, β -, γ -basis c (cid:48) ( x ) = g − p ) (cid:88) i =1 ( ν Q a S g,b a Q ν ) i i x i (mod 2) . Proof.
Let a and a ∗ be families of curves in Σ as in the Figure 4. We candraw a Kirby diagram as in the proof of Theorem 3. The main difference isthat dotted circles for 1-handles are only ∂ ( a i × D ). Therefore, when we split ∂Z to two 1-handlebodies, their systems are the same { ∂ (Σ × D ); ∂ ( a × D ) , . . . , ∂ ( a g + b − × D ) } . We should change the one of the two systemsinto { ∂ (Σ × D ); a ∗ × { } , a ∗ × {− } , . . . , a ∗ g − × { } , a ∗ g × {− } , a ∗ g +1 ×{ } , . . . , a ∗ g + b − × { }} to calculate a framing.Note that each { [ a ] } and { [ a ∗ ] } is a basis of H (Σ , ∂ Σ) and H (Σ), respec-tively, and these satisfy (cid:104) [ a i ] , [ a ∗ j ] (cid:105) Σ = δ ij . For ν i , we obtain j ν i = (cid:80) j ( t S g,b a Q ν ) ji [ a j ]regarding a basis { [ a ] } . Moreover, by simple calculation, this equality is equiv-alent to j ν i = − (cid:88) j ( ν Q a S g,b ) ij [ a j ] . For x = (cid:80) i x i [ ν i ] ∈ L α ⊕ L β ⊕ L γ , we can calculate c ( x ): c ( x ) = − (cid:88) i (cid:104) j ν i , [ ν i ] (cid:105) Σ x i , = (cid:88) i ( ν Q a S g,b a Q ν ) ii x i . (cid:3) The second Stiefel-Whitney class is strongly related to the existence of spinstructure of a 4-manifold X . Therefore, as corollaries, we have: Corollary 6.1.
Suppose that ( α, β ) is diffeomorphism equivalent to ( δ k , (cid:15) k ).Then, a trisected 4-manifold X admits a spin structure if and only if thereexists d ∈ L ∂α ∩ L ∂β such that (cid:104) d, x (cid:105) Σ = c ( x ) (mod 2) for all x ∈ L γ . Proof.
The first cochain group ( C Y ) is canonically isomorphic to L ∂α ∩ L ∂β .Let φ : L ∂α ∩ L ∂β → ( C Y ) be this isomorphism. We define another coboundarymap d (cid:48) : L ∂α ∩ L ∂β → Hom( L γ , Z ) by d (cid:48) ( d ) = (cid:104) d, −(cid:105) Σ . It is clear that this is acoboundary map. Moreover, for x ∈ L γ , we have( d φ ( d ))( x ) = (cid:104) d, x (cid:105) Σ . Therefore (( C Y ) , d ) is isomorphic to ( L ∂α ∩ L ∂β , d (cid:48) ). (cid:3) Corollary 6.2.
A trisected 4-manifold X admits a spin structure if and onlyif there exists d ∈ H (Σ , ∂ Σ) such that (cid:104) d, x (cid:105) Σ = c (cid:48) ( x ) (mod 2) for all x ∈ L α ⊕ L β ⊕ L γ . Proof. ( C Z ) is isomorphic to H (Σ , ∂ Σ) by the universal coefficient theoremand the Poincar´e duality. Let φ : ( C Z ) → H (Σ , ∂ Σ) be this isomorphism,and then we can show this corollary in the same way as Corollary 6.1. (cid:3)
Calculating −(cid:104) j ν i , ν j (cid:105) Σ in the same way as the proofs of Theorem 3 and 4,we obtain the linking matrices regarding to framed links of 2-handles. Proposition 6.3.
Each linking matrix of X regarding to handle decomposi-tions Y and Z is γ Q β,∂ R gp,b α∂ Q γ and ν Q a S g,b a Q ν , respectively.Finally, we compute the second Stiefel-Whitney class of the D bundle over S with Euler number − Figure 6.
A (2 ,
1; 0 , D bundle over S with Euler number −
1, where the central greencurve is γ .We obtain the following intersection matrix; γ Q β = (cid:18) − (cid:19) , α Q γ = (cid:18) −
11 1 (cid:19) , β Q α = I , a Q γ = (cid:0) (cid:1) . Using Proposition 6.3, we can compute a linking matrix: γ Q β,∂ R gp,b α∂ Q γ = (cid:18) − − − (cid:19) . Therefore, a representation matrix of c regarding to the basis { [ γ ] , [ γ ] } is(0 − L ∂α ∩ L ∂β has a basis { [ a ] } ,and a Q γ is equal to (1 0),this disk bundle does not admit a spin structure. OMOLOGY OF RELATIVE TRISECTION AND ITS APPLICATION 17
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