Trisections, intersection forms and the Torelli group
Peter Lambert-Cole

TL;DR
This paper explores the relationship between trisections of 4-manifolds, intersection forms, and the Torelli group, demonstrating how certain 4-manifold invariants can be used to understand their topology and classify them.
Contribution
It establishes an analogy between 3-manifold regluing via the Johnson kernel and 4-manifold trisections, linking invariants to intersection form classification.
Findings
Trisections of 4-manifolds can be related through the Johnson kernel.
Invariants of homology 3-spheres can obstruct intersection forms of 4-manifolds.
Casson invariant recovers Rohlin's theorem on spin 4-manifolds.
Abstract
We apply mapping class group techniques and trisections to study intersection forms of smooth 4-manifolds. Johnson defined a well-known homomorphism from the Torelli group of a compact surface. Morita later showed that every homology 3-sphere can be obtained from the standard Heegaard decomposition of by regluing according to a map in the kernel of this homomorphism. We prove an analogous result for trisections of 4-manifolds. Specifically, if and admit handle decompositions without 1- or 3-handles and have isomorphic intersection forms, then a trisection of can be obtained from a trisection of by cutting and regluing by an element of the Johnson kernel. We also describe how invariants of homology 3-spheres can be applied, via this result, to obstruct intersection forms of smooth 4-manifolds. As an application, we use the Casson invariant to recover Rohlin's…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Trisections, intersection forms and the Torelli group
Peter Lambert-Cole
School of Mathematics
Georgia Institute of Technology
[email protected] https://www.math.gatech.edu/~plambertcole3
Abstract.
We apply mapping class group techniques and trisections to study intersection forms of smooth 4-manifolds. Johnson defined a well-known homomorphism from the Torelli group of a compact surface. Morita later showed that every homology 3-sphere can be obtained from the standard Heegaard decomposition of by regluing according to a map in the kernel of this homomorphism. We prove an analogous result for trisections of 4-manifolds. Specifically, if and admit handle decompositions without 1- or 3-handles and have isomorphic intersection forms, then a trisection of can be obtained from a trisection of by cutting and regluing by an element of the Johnson kernel. We also describe how invariants of homology 3-spheres can be applied, via this result, to obstruct intersection forms of smooth 4-manifolds. As an application, we use the Casson invariant to recover Rohlin’s Theorem on the signature of spin 4-manifolds.
Key words and phrases:
4-manifolds
2010 Mathematics Subject Classification:
57M27
1. Introduction
Every closed, oriented 3-manifold admits a Heegaard decomposition into the union of two genus-handlebodies. Such a decomposition can be obtained from the standard Heegaard splitting of by cutting and regluing by some element of the mapping class group of a closed, genus- surface. When is an integral homology 3-sphere, the map can be chosen in the Torelli group , which consists of mapping classes that act trivially on homology. This connection has proven enormously useful in understanding both the mapping class group and integral homology 3-spheres.
Trisections of smooth 4-manifolds are analogous to Heegaard splittings in dimension 3. Specifically, a -trisection of a smooth 4-manifold is a decomposition such that
- (1)
the sector is diffeomorphic to the 4-dimensional 1-handlebody , 2. (2)
each double intersection is diffeomorphic to the 3-dimensional 1-handlebody , 3. (3)
the triple intersection is a closed, oriented surface of genus ,
It follows immediately from the definition that is a genus Heegaard splitting of the boundary . Conversely, take a genus- handlebody and three homeomorphisms . Suppose the union
[TABLE]
is homeomorphic to for . Then a result of Laudenbach and Poénaru [LP72] states that we can build a unique smooth 4-manifold, up to diffeomorphism, by gluing three handlebodies to via the homeomorphisms and then capping off with three 4-dimensional 1-handlebodies.
Given the analogy between Heegaard splittings and trisections, it is natural to ask:
- (1)
Do results about Heegaard splittings and 3-manifolds extend to 4-dimensions? 2. (2)
Can we apply 3-dimensional techniques to answer questions in 4-manifold topology?
Of specific interest in this paper is the work of Birman and Craggs [BC78], Johnson [Joh80, Joh85a, Joh85b], Casson, and Morita [Mor89, Mor91] that relates the Torelli group to invariants of integral homology 3-spheres. Birman and Craggs used the Rohlin invariant of homology 3-spheres to define a collection of homomorphisms from the Torelli group to . Johnson later completely classified these homomorphisms and subsequently used these maps, along with his namesake homomorphism, to determine the abelianization of Torelli group. Casson lifted the Rohlin invariant and defined a famous -valued invariant of homology 3-spheres. Morita then applied Johnson’s work to reinterpret the Casson invariant in terms of the mapping class group. In particular, every integral homology 3-sphere can be obtained by taking the standard Heegaard decomposition of , cutting along the Heegaard surface, and regluing by an element of , the kernel of the Johnson homomorphism. As Johnson showed, this subgroup is precisely the subgroup generated by separating twists. The change in the Casson invariant can then be computed using the surgery formula.
The main result of this paper is the following 4-dimensional analogue of Morita’s result.
Theorem 1.1**.**
Let and be homeomorphic, closed smooth 4-manifolds. Suppose that and admit -trisections and , respectively. Then can be obtained from by cutting and regluing by an element of the Johnson kernel .
Note that the condition that and admit -trisections is equivalent to the condition that they are simply-connected and admit perfect Morse functions (see Theorem 6.2).
1.1. Intersection forms
A second goal is this paper is to describe a way in which, via trisections, mapping class group techniques may be used to obstruct intersection forms. Let be a 4-manifold. Multiplication in the cohomology ring determines a symmetric, bilinear pairing . When is simply-connected, it is a result of Wall that determines the homotopy-type of and a result of Freedman that determines the homeomorphism type of . Every unimodular quadratic form over is the intersection form of a topological 4-manifold. However, this is not true for smooth 4-manifolds and determining which quadratic forms can be realized is an important open problem in 4-manifold theory.
There is an extra subtlety that distinguishes trisections from Heegaard splittings. It is not true that an arbitrary choice of gluing maps determines a closed 4-manifold. The pairwise union may not be homeomorphic to , let alone have the same integral homology. This gluing data only determines a more general Heegaard triple and only when each has the correct homeomorphism type does this construction describe a unique, closed 4-manifold. When is homeomorphic to and and are integral homology 3-spheres, we refer to the triple as a pseudotrisection.
While this appears unfortunate, in fact it is an opportunity to show that some quadratic forms cannot be realized as the intersection form of any closed, smooth, oriented 4-manifold. Let be a triple of gluing maps. Each map induces a map on homology . The kernel of this map is a -dimensional subspace that is Lagrangian with respect to the intersection form on . As shown by Feller, Klug, Schirmer and Zemke [FKSZ18], the triple of Lagrangian subspaces determines the cohomology ring of the induced 4-manifold. Moreover, even if the triple is only a pseudotrisection, we can formally compute the ‘intersection form’ using the same algebraic formula.
We then have a strategy to obstruct as the intersection form of smooth 4-manifold. Take the set of all pseudotrisections whose ‘intersection form’ is . As in Theorem 1.1, any two pseudotrisections with the same ‘intersection form’ are related by an element of the Johnson kernel. To exclude , it now suffices to show that after cutting and regluing by any element in , at least one of the resulting 3-manifolds is always a nontrivial homology 3-sphere.
1.2. Rohlin’s Theorem
A standard fact in 4-manifold topology is Rohlin’s Theorem.
Theorem 1.2** (Rohlin [Roh52]).**
Let be a closed, spin 4-manifold. Then .
There exist several alternate proofs of this result in the literature (see [KM61, FK78, Kir89, LM89, KM91]). We present a new proof here using the Casson invariant of homology 3-spheres, whose reduction mod 2 is better known as the Rohlin invariant . A standard approach uses Theorem 1.2 to show that is well-defined. However, using trisections we can reverse the logical causality and apply the mod 2 Casson invariant to exclude spin 4-manifolds whose signature is .
By standard surgery theory techniques, every spin 4-manifold is spin-cobordant to a simply-connected 4-manifold that has indefinite intersection form and a handle decomposition without and -handles (Lemma 6.1). Since is spin, the Stiefel–Whitney class vanishes and so the intersection form is even. By the classification of unimodular, indefinite and even symmetric bilinear forms, the intersection form is isomorphic to , where a negative value of corresponds to summands of . Rohlin’s Theorem is then equivalent to the statement that must be even. Thus, it suffices to prove the following theorem.
Theorem 1.3**.**
Suppose that be a spin 4-manifold that admits a -trisection. Then the intersection form of has the form
[TABLE]
for some integer .
The first step is find a triple of maps that determine pseudotrisection with intersection form (Figure 1). The resulting 3-manifold can be immediately identified as the Poincaré homology sphere, whose Casson invariant is . In particular, since is homeomorphic to , we see that
[TABLE]
Then, we use the surgery formulas for the Casson invariant to prove that
[TABLE]
for all . More generally, if we start with a pseudotrisection with intersection form , we show that
[TABLE]
for any . This suffices to prove Theorem 1.3 and therefore Rohlin’s Theorem.
It would be interesting to apply a stronger invariant of homology 3-spheres, in order to obstruct other intersection forms. A naive hope is that, since the Casson invariant counts representations of the fundamental group and Donaldson invariants count instantons, one might recover Donaldson’s Diagonalization theorem [Don83]. However, the full Casson invariant does not appear to add more information than its mod 2 version. For example, it does not obstruct the intersection forms or , which are definite but not diagonal and therefore excluded by Donaldson.
1.3. Acknowledgements
I would like to thank Dan Margalit for his excellent class on the Torelli group. I would also like to thank Justin Lanier, Agniva Roy and Trent Schirmer for helpful discussions.
2. Torelli group and the Johnson homomorphism
2.1. Torelli group
Let be a closed, oriented surface of genus and let denote a compact, oriented surface of genus and 1 boundary component. Let denote the mapping class group of and let denote the Torelli group of . It is the subgroup of consisting of classes of homeomorphisms of to itself that induce the trivial homomorphism on homology. Let be the mapping class group of , consisting of isotopy classes of homeomorphisms of that fix the boundary pointwise. Let denote the Torelli group of . For technical reasons, it is often convenient to work with instead of . The groups are related by the exact sequence
[TABLE]
where denote the unit tangent bundle of . Let and be the subgroups of and generated by Dehn twists on separating curves.
The following is a basic fact in the theory of mapping class groups.
Proposition 2.1**.**
Let \mbox{\boldmath\alpha}=(\alpha_{1},\dots,\alpha_{g}) be a collection of disjoint, simple closed curves representing linearly independent classes in . Let \mbox{\boldmath\alpha}^{\prime}=(\alpha^{\prime}_{1},\dots,\alpha^{\prime}_{g}) be a second collection of disjoint, simple closed curves satisfying for all . There exists an element such that for all .
Proof.
We sketch the proof; more details can be found in [FM12, Chapter 6.3.2]. The cut systems \mbox{\boldmath\alpha},\mbox{\boldmath\alpha}^{\prime} can be extended to geometric symplectic bases \{\mbox{\boldmath\alpha},\mbox{\boldmath\beta}\} and \{\mbox{\boldmath\alpha}^{\prime},\mbox{\boldmath\beta}^{\prime}\} such that . Each complement \Sigma\smallsetminus(\mbox{\boldmath\alpha}\cup\mbox{\boldmath\beta}) and \Sigma\smallsetminus(\mbox{\boldmath\alpha}^{\prime}\cup\mbox{\boldmath\beta}^{\prime}) is a sphere with boundary components. We can choose a homeomorphism between the spheres that identifies with and with and extends to a homeomorphism of . The induced map on homology sends to and to . In other words, acts trivially on homology and is therefore in the Torelli group. ∎
2.2. Johnson homomorphism
Let denote the fundamental group of , with a basepoint chosen in the boundary, let denote the term in the lower central series for and let be the nilpotent quotient. Note that . There is an exact sequence of groups
[TABLE]
where is the center of . There is a natural action of on and the canonical morphism is injective. The subgroups are characteristic and therefore preserved by any automorphism of . Thus, we obtain a map and denote its kernel by . The Torelli group is precisely . There are sequence of homomorphisms
[TABLE]
For , the homomorphism
[TABLE]
is known as the the Johnson homomorphism.
Theorem 2.2** ([Joh85a]).**
The kernel of is .
Recall that a bounding pair consists of two disjoint, homologous curves and the corresponding bounding pair map is is an element of the Torelli group. The genus of a bounding pair is the genus of the subsurface cut off by . A -chain is a collection of simple closed curves such that and for . If is odd, then the boundary of a neighborhood of is a bounding pair of genus . If is even, then the boundary of a neighborhood of is a separating curve that cuts off a subsurface of genus .
Lemma 2.3**.**
Let be a 3-chain and let be the corresponding bounding pair. Then
[TABLE]
.
To understand equivalences between Heegaard splittings and between trisections, we are interested in bounding pair maps that extend across a handlebody .
Lemma 2.4**.**
Let be a bounding pair that bounds an annulus in a handlebody . Then the bounding pair map extends across the handlebody.
Proof.
Cut along the annulus and reglue via a Dehn twist. This gives a homeomorphism of the handlebody to itself that restricts to the bounding pair map on boundary surface. ∎
Lemma 2.5**.**
Let be a 3-chain and suppose that one of the three curves bounds in the handlebody . Then the corresponding boundary pair extends across the handlebody.
Proof.
Without loss of generality, assume that either or bound in the handlebody. Then the separating curve , which is the boundary of a neighborhood of the 2-chain , bounds in the handlebody. Consequently, the curve is obtained from by a band sum with this curve . It is now clear that and bound an annulus in the handlebody. Thus by Lemma 2.4, the bounding pair extends across the handlebody. ∎
3. Heegaard splittings and Trisections
Throughout this section, let denote a 3-dimensional, genus handlebody and let be an abstract, closed surface of genus . Let denote the mapping class group of and let denote the mapping class group of . Let denote the subgroup of consisting of classes that can be represented by a homeomorphism that extends across .
A cut system of disks for is a collection of disjoint, properly embedded disks in whose complement is homeomorphic to . The boundaries of the disks are a collection \mbox{\boldmath\alpha}=\{\alpha_{1},\dots,\alpha_{g}\} of disjoint, simple closed curves in that generate a -dimensional subspace in . If , then the cut system can be connected to by a sequence of handleslides. A handleslide of over consists of replacing by , which is obtained by joining to a second copy of by an embedded arc in .
A collection of disjoint, simple closed curves in whose complement has genus 0 is known as a cut system of curves. A homeomorphism sends the boundaries of a cut system of disks in to a cut system of curves .
We assume, once and for all, that we have fixed a cut system of disks with boundary for .
3.1. Heegaard splittings
Let denote the set of pair where each is an isotopy class of homeomorphisms. Given any element , we can build a closed 3-manifold
[TABLE]
This decomposition of into two handlebodies is a Heegaard decomposition.
A collection of simple closed curves on is a geometric symplectic basis if the geometric intersection numbers satisfy
[TABLE]
Definition 3.1**.**
A Heegaard decomposition , with , is standard if is a geometric symplectic basis for .
Definition 3.2**.**
A Heegaard decomposition , with , is standard if there exists a geometric symplectic basis such that
[TABLE]
Many pairs specify the same Heegaard decomposition up to homeomorphism. Recall that denotes the subgroup of consisting of the classes of homeomorphisms that extend across the handlebody . Then acts on on the left as
[TABLE]
and the mapping class group acts on on the right as
[TABLE]
Let .
Lemma 3.3**.**
The -orbits of are precisely the equivalence classes of Heegaard splittings.
An important and well-known fact, due to Waldhausen, is that the standard Heegaard splitting of is essentially unique.
Theorem 3.4** ([Wal68]).**
Suppose that the pair determines a Heegaard splitting of . Then
[TABLE]
where is standard.
By inductively applying Haken’s Lemma, a similar statement holds for connected sums of .
Corollary 3.5**.**
Suppose that the pair determines a Heegaard splitting of . Then
[TABLE]
where is standard.
The homology of can be computed using the following chain complex
[TABLE]
All of the maps are nonzero except for . Let denote a basis for and denote a second basis. The linear map is defined by the formula
[TABLE]
where denotes the intersection pairing on .
Proposition 3.6**.**
Let and the associated 3-manifold.
- (1)
* is an integral homology 3-sphere if and only if is unimodular.* 2. (2)
If and is a standard Heegaard decomposition of , then the pair determines an integral homology 3-sphere . 3. (3)
If is an integral homology, then there exists some standard Heegaard decomposition of and an element such that
[TABLE]
Morita proved a stronger version of Proposition 3.6, which we state in the current formalism.
Theorem 3.7** ([Mor89]).**
If is an integral homology 3-sphere, then there exists some standard Heegaard decomposition of and an element such that
[TABLE]
3.2. Trisections of closed, smooth 4-manifolds
Recall that a trisection of a closed, oriented, smooth 4-manifold is a decomposition , where each double intersection is a genus handlebody and is a closed, oriented genus surface. We can identify each double intersection with a fixed, abstract and the central surface with a fixed, abstract . These identification induce a triple of maps
[TABLE]
for . Thus, every trisection determines a triple of homeomorphisms. This triple uniquely describes up to diffeomorphism. Furthermore, the pair determines a Heegaard splitting of .
The triple depends on choices. For any choices and , the triple determines a diffeomorphic 4-manifold. Let denote the set of triples where each is an isotopy class of homeomorphisms. The group acts on on the left by the rule
[TABLE]
and the group acts on on the right by the rule
[TABLE]
Let . We can combine the above actions into a single action on and each diffeomorphism class of trisection corresponds with a unique -orbits. We refer to the action of as global reparametrization and the action of as a handlebody diffeomorphism.
Remark 3.8**.**
If is a triple arising from a trisection of a closed 4-manifold, then by Theorem 3.4 and Corollary 3.5, each triple is -equivalent to a standard . However, it is not true in general that is -equivalent to some where every pair . It is known that only connected sums of and admit this property.
It is often convenient to work just in and interpret as a subgroup of . Let be a fixed triple. Define subgroups of
[TABLE]
Consequently, the orbits of the action of on the left are precisely the orbits of the action of on the right. Similarly for the orbits of and . Furthermore, let .
Lemma 3.9**.**
Fix a triple . If , then
[TABLE]
Proof.
Since , there exist such that
[TABLE]
Thus
[TABLE]
∎
3.3. Heegaard diagrams
For many purposes, such as computing the algebraic topology of , it is often easier and sufficient to encode by its action on a cut system.
Given a collection of maps that determine a trisection of a 4-manifold , we obtain a trisection diagram (\Sigma,\mbox{\boldmath\alpha}_{1},\mbox{\boldmath\alpha}_{2},\mbox{\boldmath\alpha}_{3}), where \alpha_{\lambda}=\phi_{\lambda}(\mbox{\boldmath\alpha}). To agree with conventions in Heegaard Floer theory, we often denote the trisection diagram as (\Sigma,\mbox{\boldmath\alpha},\mbox{\boldmath\beta},\mbox{\boldmath\gamma}) instead. A collection of the form (\Sigma,\mbox{\boldmath\alpha},\mbox{\boldmath\beta},\mbox{\boldmath\gamma}), where each of \mbox{\boldmath\alpha},\mbox{\boldmath\beta} and is a cut system of curves, is known as a Heegaard triple. Conversely, tiven a Heegaard triple (\Sigma,\mbox{\boldmath\alpha}_{1},\mbox{\boldmath\alpha}_{2},\mbox{\boldmath\alpha}_{3}), we can choose a (nonunique) triple of homeomorphisms such that \mbox{\boldmath\alpha}_{\lambda} is the image under of a fixed cut system of curves.
Lemma 3.10**.**
A Heegaard triple (\Sigma,\mbox{\boldmath\alpha},\mbox{\boldmath\beta},\mbox{\boldmath\gamma}) determines a unique -orbit in .
The cohomology ring of can be computed purely from the Heegaard triple (\Sigma,\mbox{\boldmath\alpha},\mbox{\boldmath\beta},\mbox{\boldmath\gamma}). In particular, the cut systems \mbox{\boldmath\alpha},\mbox{\boldmath\beta},\mbox{\boldmath\gamma} determines a -dimensional subspace in . As detailed in [FKSZ18], the cohomology ring of is determined purely by the triple , up to a symplectic automorphism of .
Let (\Sigma,\mbox{\boldmath\alpha},\mbox{\boldmath\beta},\mbox{\boldmath\gamma}) be a trisection diagram for a -trisection. Let be the -matrix of intersection numbers and define and similarly. Then the intersection form is given in matrix-form by the formula ([FKSZ18, Theorem 4.3])
[TABLE]
Theorem 3.11** ([FKSZ18]).**
Let be a -trisection of . Then admits a diagram (\Sigma,\mbox{\boldmath\alpha},\mbox{\boldmath\beta},\mbox{\boldmath\gamma}) such that
- (1)
(\Sigma,\alpha,\mbox{\boldmath\beta})* is a standard Heegaard diagram for .* 2. (2)
In , we have
[TABLE]
where is the intersection form of and .
Combining the above theorem with Proposition 2.1, we obtain
Corollary 3.12**.**
Let be closed, smooth, oriented 4-manifolds such that . Suppose that admit -trisections and . Then there exists a diagram (\Sigma,\mbox{\boldmath\alpha},\mbox{\boldmath\beta},\mbox{\boldmath\gamma}) for and an element such that (\Sigma,\mbox{\boldmath\alpha},\mbox{\boldmath\beta},\rho(\mbox{\boldmath\gamma})) is a diagram for .
3.4. Pseudotrisections
We introduce the more general notion of a pseudotrisection. Recall from the previous subsection that a trisection of a closed, smooth, oriented 4-manifold can be encoded by a triple of maps or a Heegaard triple (\Sigma,\mbox{\boldmath\alpha},\mbox{\boldmath\beta},\mbox{\boldmath\gamma}). In an honest trisection, the 3-manifolds obtained as the union of any pair of handlebodies must by homeomorphic to or for some . In a pseudotrisection, we relax the condition and allow two of the 3-manifolds to merely have the same -homology as one of those manifolds.
Definition 3.13**.**
Let (\Sigma,\mbox{\boldmath\alpha},\mbox{\boldmath\beta},\mbox{\boldmath\gamma}) be a diagram that defines a -pseudotrisection . The intersection form of the pseudotrisection is the quadratic form determines by the formula in Equation 1.
For each unimodular intersection form , we can build a pseudotrisection with intersection form as follows:
Proposition 3.14**.**
Let be a unimodular, symmetric bilinear form of rank . Then there exists a pseudotrisection (\Sigma,\mbox{\boldmath\alpha}_{Q},\mbox{\boldmath\beta}_{Q},\mbox{\boldmath\gamma}_{Q}) such that
- (1)
* are standard,* 2. (2)
* are standard, and* 3. (3)
**
where .
Proof.
Start with the standard trisection (\Sigma,\mbox{\boldmath\alpha},\mbox{\boldmath\beta},\mbox{\boldmath\gamma}^{\prime}) of . The standard -trisection of is given by the diagram where intersect geometrically once and represents the class . Taking the connected sum of copies gives the standard trisection of .
Let be curves homotopic to and , respectively, and disjoint from . In addition, let be a band sum of and disjoint from . Now set
[TABLE]
and let \mbox{\boldmath\gamma}=\phi(\mbox{\boldmath\gamma}^{\prime}). We can assume that each Dehn twist is along a curve disjoint from a fixed collection of curves . Thus, the geometric intersection number between and never changes. In particular, and still form a geometric symplectic basis for the surface. The induced map on satisfies
[TABLE]
Therefore, (\Sigma,\mbox{\boldmath\alpha},\mbox{\boldmath\beta},\mbox{\boldmath\gamma}) gives a pseudotrisection with intersection form .
To obtain a pseudotrisection, we can stabilize times by connected sum with the standard trisection of . A diagram for this trisection is where is a standard Heegaard diagram for the genus 1 splitting of . ∎
We refer to the pseudotrisection \mathcal{T}_{Q}=(\Sigma,\mbox{\boldmath\alpha}_{Q},\mbox{\boldmath\beta}_{Q},\mbox{\boldmath\gamma}_{Q}) constructed in Proposition 3.14 as the standard pseudotrisection for .
Proposition 3.15**.**
Let be an intersection form.
- (1)
Let \mathcal{T}=(\Sigma,\mbox{\boldmath\alpha}^{\prime},\mbox{\boldmath\beta}^{\prime},\mbox{\boldmath\gamma}^{\prime}) a -pseudotrisection with intersection form . There exists an equivalent diagram (\Sigma,\mbox{\boldmath\alpha},\mbox{\boldmath\beta},\mbox{\boldmath\gamma}) for such that is a symplectic basis for and
[TABLE]
where . 2. (2)
Let \mathcal{T}=(\Sigma,\mbox{\boldmath\alpha}^{\prime},\mbox{\boldmath\beta}^{\prime},\mbox{\boldmath\gamma}^{\prime}) a -pseudotrisection diagram satisfying the conclusions of part (1). Then there exists some such that \mathcal{T}_{Q,\phi}\coloneqq(\Sigma,\mbox{\boldmath\alpha}_{Q},\mbox{\boldmath\beta}_{Q},\phi(\mbox{\boldmath\gamma}_{Q})) is equivalent to . 3. (3)
Let \mathcal{T}_{1}=(\Sigma_{g},\mbox{\boldmath\alpha}_{1},\mbox{\boldmath\beta}_{1},\mbox{\boldmath\gamma}_{1}) and \mathcal{T}_{2}=(\Sigma_{g},\mbox{\boldmath\alpha}_{2},\mbox{\boldmath\beta}_{2},\mbox{\boldmath\gamma}_{2}) be -pseudotrisections that satisfy the conclusions of part (1) and have the same intersection form . Then there exists some such that \phi(\mbox{\boldmath\gamma}_{1})=\mbox{\boldmath\gamma}_{2}.
Proof.
The proof of (1) is essentially the same as the proof of Theorem 3.11, which is a restatement of [FKSZ18, Theorem 4.4]. The only difference is that we cannot assume (\Sigma,\mbox{\boldmath\alpha},\mbox{\boldmath\beta}) is geometrically standard. However, the proof only requires that form a symplectic basis for . This follows since the 3-manifold constructed from the diagram (\Sigma,\mbox{\boldmath\alpha},\mbox{\boldmath\beta}) is an integral homology 3-sphere.
Part (2) follows from part (1) by Proposition 2.1.
To prove (3), note that by part (2) we can assume has a diagram (\Sigma,\mbox{\boldmath\alpha}_{Q},\mbox{\boldmath\beta}_{Q},\phi_{1}(\mbox{\boldmath\gamma}_{Q})) and has a diagram (\Sigma,\mbox{\boldmath\alpha}_{Q},\mbox{\boldmath\beta}_{Q},\phi_{2}(\mbox{\boldmath\gamma}_{Q})). Now set . ∎
4. Trisections and Torelli group
The main result of this section is an analogue of Morita’s result (Theorem 3.7) for trisections of closed, smooth 4-manifolds.
Theorem 4.1**.**
Let be a unimodular, symmetric bilinear form of rank over . Let be a closed, smooth 4-manifold with intersection form and which admits a trisection . Let (\Sigma,\mbox{\boldmath\alpha}_{Q},\mbox{\boldmath\beta}_{Q},\mbox{\boldmath\gamma}_{Q}) be the pseudotrisection from Part (1). Then there exists a map such that (\mbox{\boldmath\alpha}_{Q},\mbox{\boldmath\beta}_{Q},\phi(\mbox{\boldmath\gamma}_{Q})) is a trisection diagram for the trisection .
Fix an intersection form and let be the standard -pseudotrisection with intersection form . For concreteness, let be a triple of maps encoding this pseudotrisection. We then obtain subgroups of homeomorphisms that extend across the and -handlebodies, respectively. From now own, we supress the subscript , although it is important that these groups depend on .
Let be the intersection of with the Torelli group . Define and similarly, where denotes the intersection of and . Finally, let denote the Johnson homomorphism.
Proposition 4.2**.**
Let be a unimodular, symmetric bilinear form and let (\Sigma,\mbox{\boldmath\alpha}_{Q},\mbox{\boldmath\beta}_{Q},\mbox{\boldmath\gamma}_{Q}) be the standard -pseudotrisection for . Then
[TABLE]
Proof.
We just need to find explicit bounding pairs that map onto a basis for .
To do this, we decompose into subspaces and then check that each subspace is in . Let for be the left generators and for be the right generators.
Let be the subspace spanned by elements of the form , where are left generators; let be the subspace spanned by elements of the form , where are left generators; let be their intersection and let their sum. Define similarly using only right generators. Finally, define in the same manner but allowing elements of any combination between left and right.
By assumption, we know that bound in , that bound in and some curves bound in , where
[TABLE]
First, it follows from Morita’s original argument that is in the image . Specifically, the following elements lie in :
[TABLE]
In addition, the following elements
[TABLE]
lie in and the elements
[TABLE]
lie in . These elements suffice to span .
Next, let . The following elements are each the image of a 3-chain, at least one of which bounds in and at least one of which bounds in . Thus, by Lemma 2.5, they lie in the image :
[TABLE]
Thus .
Next, since are geometrically standard, we can find 3-chains that live in mapping to
[TABLE]
Since is unimodular, we can express any as a -linear combination of the and therefore . Finally, it again follows from Morita’s original proof that
[TABLE]
is in . Consequently, .
Finally, we need to ensure all mixed elements lie in the image. From now on, we will let and .
By a similar argument as above, we can obtain the elements
[TABLE]
in the image . This is enough to completely span .
Moreover, the elements
[TABLE]
also lie in and so we can span . And finally, the elements
[TABLE]
are in the image . We have thus constructed a basis for that lies in .
∎
Theorem 4.3**.**
Let be a closed, oriented, smooth 4-manifold that admits a trisection. Then admits a trisection diagram
[TABLE]
where is the intersection form of , \mathcal{T}_{Q}=(\Sigma,\mbox{\boldmath\alpha}_{Q},\mbox{\boldmath\beta}_{Q},\mbox{\boldmath\gamma}_{Q}) is the standard pseudotrisection for constructed in Proposition 3.14, and .
Proof.
Just as in the proof of Corollary 3.12, we can apply Proposition 2.1 and find a trisection diagram (\Sigma,\mbox{\boldmath\alpha}_{Q},\mbox{\boldmath\beta}_{Q},\psi(\mbox{\boldmath\gamma}_{Q})) for , where is an element of the Torelli group . By Proposition 4.2, we can find some elements and such that
[TABLE]
Consequently, . Thus, . It now follows from Lemma 3.9 that the trisection diagrams
[TABLE]
are equivalent. ∎
5. Casson invariant and linking forms
5.1. Casson invariant
If is a knot in a homology 3-sphere , let denote the integral homology 3-sphere obtained by Dehn surgery on with slope .
Let denote the Casson invariant of an integral homology 3-sphere . It has the following properties:
- (1)
. 2. (2)
For any , any knot in and any integer , the difference
[TABLE]
is independent of . 3. (3)
If is a boundary link in , then
[TABLE] 4. (4)
. 5. (5)
. 6. (6)
. 7. (7)
.
Moreover, we have that
[TABLE]
5.2. Linking form
The Arf invariant and Casson invariant of knots can be computed in terms of a linking form on the homology of a Seifert surface.
Let be an oriented 3-manifold and an oriented, embedded surface in . The linking form is a map
[TABLE]
defined in terms of the pairwise linking of curves on . Specifically, let and be simple closed curves in . Then
[TABLE]
where denotes a pushoff of in the positive normal direction to . The linking pairing is well-defined on homology, bilinear, and satisfies the symmetry relation
[TABLE]
Reducing mod 2, we obtain a map defined by setting
[TABLE]
It is a quadratic enhancement of the intersection pairing , meaning that it satisfies the relation
[TABLE]
Now, suppose that is a Seifert surface for a knot , where is an integral homology 3-sphere. Let be a geometric symplectic basis for . The Casson invariant of can be computed in terms of the linking form on according to the formula
[TABLE]
The Arf invariant can be computing using the simpler formula
[TABLE]
5.3. Linking forms on the central surface
Let be the central surface of a pseudotrisection. This surface embeds in each 3-manifold as a Heegaard surface. Each embedding determines a linking form on and a quadratic enhancements . In general, these linking forms are distinct. However, with respect to a particular basis, we can describe and in terms of the quadratic form of the pseudotrisection. In addition, let be a separating simple closed curve in . Then, via the embedding of in , it determines a knot . In general, the Arf invariant, Alexander polynomial and Casson invariant of depends on . However, these also can be described in terms of the quadratic form of the pseudotrisection.
Throughout this subsection, let (\Sigma,\mbox{\boldmath\alpha},\mbox{\boldmath\beta},\mbox{\boldmath\gamma}) be a pseudotrisection with quadratic form . Let and .
First, we describe the difference between the quadratic enhancements and .
Lemma 5.1**.**
Suppose that is even. Then .
Proof.
By Equation 3, it is enought to check on the symplectic basis . Furthermore, we assume that we have performed handeslides so that the diagram satisfies the conclusions of Proposition 3.15.
First, the mod 2 linking form in vanishes on each element of the basis since we can choose representatives that bound disks in one of the two handlebodies and . By the same argument, the mod 2 linking form in vanishes on the basis since each can be represented by the boundary of a compressing disk. To check that the mod 2 linking forms agree, we need to check that always vanishes. By Equation 3, we have
[TABLE]
But since is an even intersection form, every diagonal element is even and vanishes mod . ∎
As a corollary, we see that the Arf invariants of and agree when the intersection form is even.
Lemma 5.2**.**
Suppose that is even. Let be a separating curve in and let and be the knots obtained as the image of in and . Then .
Proof.
Since is separating on , it cuts off a subsurface . The images of this surface in and are Seifert surfaces for and , respectively. Choose a symplectic basis for . By Equation 5 and Lemma 5.1,
[TABLE]
∎
Now we describe the linking forms and in terms of the quadratic form of the pseudotrisection. Recall that we have a symplectic basis for such that is a basis for and is a basis for Furthermore. To simplify notation, let . Then is a basis for .
Proposition 5.3**.**
The linking form satisfies
[TABLE]
Proof.
For any , we have that
[TABLE]
Consequently, we easily see that
[TABLE]
Using the relation in Equation 2, we then see that
[TABLE]
Furthermore, we have that . Consequently
[TABLE]
Using the bilinearity relation, we obtain
[TABLE]
Using the above formulas, we see that
[TABLE]
Next, we have
[TABLE]
so for all . This implies that and consequently . ∎
Proposition 5.4**.**
The linking form satisfies
[TABLE]
Proof.
As in the proof of Proposition 5.3, we have that
[TABLE]
for any . We then easily obtain
[TABLE]
It then follows that
[TABLE]
Now,
[TABLE]
which implies that . Finally,
[TABLE]
Thus . Since is the matrix for the linking form in the basis , we see that is the matrix for the linking form in the basis . ∎
6. Rohlin’s Theorem
Lemma 6.1**.**
Let be a closed, spin 4-manifold. Then is spin-cobordant to a 4-manifold such that
- (1)
* has a handle decomposition without 1- or 3-handles, and* 2. (2)
the intersection form of is even and indefinite.
In particular, is spin and satisfies .
Proof.
Following the standard surgery theory trick, we can attach 5-dimensional 2-handles to kill the 1-handles of . Moreover, we can extend the spin structure across these 2-handles. Turning the manifold upside down, we can also kill all of the 3-handles. This results in some spin that has no 1- or 3-handles. By attaching one more 2-handle along a contractible loop, we get a spin cobordism to , which has an even, indefinite intersection form. ∎
The connection between handle decompositions and trisections is given by the following result.
Theorem 6.2** ([GK16]).**
Let be a closed, oriented 4-manifold that admits a handle decomposition with a single [math]-handle, 1-handles, 2-handles, and 3-handles. Then admits a trisection for some .
Corollary 6.3**.**
Suppose that a closed, smooth oriented 4-manifold admits a handle decomposition without 1- or 3-handles. Then
- (1)
* admits a trisection for some , and* 2. (2)
* admits a trisection for some .*
Proof.
The first statement follows directly from Theorem 6.2, while the second follows from the first by cyclically permuting the sectors of the trisection. ∎
A pseudotrisection with intersection form is guaranteed by Proposition 3.14. An example of such a pseudotrisection (which is not obtained by the method of the proof of Proposition 3.14, however), is given in Figure 1.
Proposition 6.4**.**
Let (\Sigma,\mbox{\boldmath\alpha},\mbox{\boldmath\beta},\mbox{\boldmath\gamma}) be the Heegaard triple in Figure 1. Then and is the Poincaré homology sphere.
Proof.
Each intersects a unique -curve in a single point and a unique curve in a single point. This implies that the pairs and are the standard Heegaard diagram for .
Viewing as a Heegaard surface in , the surface framing of each curve is and the curves are a collection of unknots that link according to the Dynkin diagram. It is well-known (e.g. [KS79]) that the result of surgery on this framed link results in the Poincaré homology sphere. ∎
We now have sufficient tools to prove Rohlin’s Theorem
Proof of Theorem 1.3.
Fix and set . Then we can obtain a pseudotrisection (\Sigma,\mbox{\boldmath\alpha},\mbox{\boldmath\beta},\mbox{\boldmath\gamma}) with intersection form by taking a connected sum of copies of the counterfeit , copies of the standard trisection of , and copies of the standard trisection of .
Let be any element of the Johnson kernel and let (\Sigma,\mbox{\boldmath\alpha},\mbox{\boldmath\beta},\phi(\mbox{\boldmath\gamma})) be the resulting pseudotrisection. We obtain 3-manifolds and as the union of handlebodies. Since , it is the product of separating twists. Consequently, is obtained from by surgery on a boundary link and is obtained from by surgery on a boundary link . The surgery formula for the Casson invariant implies that
[TABLE]
But by Lemma 5.2, we have that for all . Consequently
[TABLE]
Now, suppose that is any closed, smooth, oriented 4-manifold with intersection and a trisection. By Theorem 4.1, we can assume there is some element such that (\Sigma_{g},\mbox{\boldmath\alpha},\mbox{\boldmath\beta},\phi(\mbox{\boldmath\gamma})) is a trisection diagram for . Moreover, since this is an honest trisection decomposition, we have that . Therefore
[TABLE]
∎
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[BC 78] Joan S. Birman and R. Craggs. The μ 𝜇 \mu -invariant of 3 3 3 -manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented 2 2 2 -manifold. Trans. Amer. Math. Soc. , 237:283–309, 1978.
- 2[Don 83] S. K. Donaldson. An application of gauge theory to four-dimensional topology. J. Differential Geom. , 18(2):279–315, 1983.
- 3[FK 78] Michael Freedman and Robion Kirby. A geometric proof of Rochlin’s theorem. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2 , Proc. Sympos. Pure Math., XXXII, pages 85–97. Amer. Math. Soc., Providence, R.I., 1978.
- 4[FKSZ 18] Peter Feller, Michael Klug, Trenton Schirmer, and Drew Zemke. Calculating the homology and intersection form of a 4-manifold from a trisection diagram. Proceedings of the National Academy of Sciences , 115(43):10869–10874, 2018.
- 5[FM 12] Benson Farb and Dan Margalit. A primer on mapping class groups , volume 49 of Princeton Mathematical Series . Princeton University Press, Princeton, NJ, 2012.
- 6[GK 16] David Gay and Robion Kirby. Trisecting 4-manifolds. Geom. Topol. , 20(6):3097–3132, 2016.
- 7[Joh 80] Dennis Johnson. Quadratic forms and the Birman-Craggs homomorphisms. Trans. Amer. Math. Soc. , 261(1):235–254, 1980.
- 8[Joh 83a] Dennis Johnson. The structure of the Torelli group. I. A finite set of generators for ℐ ℐ \mathcal{I} . Ann. of Math. (2) , 118(3):423–442, 1983.
