Trisections of 4-manifolds via Lefschetz fibrations
We develop a technique for gluing relative trisection diagrams of -manifolds with nonempty connected boundary to obtain trisection diagrams for closed -manifolds. As an application, we describe a trisection of any closed -manifold which admits a Lefschetz fibration over equipped with a section of square , by an explicit diagram determined by the vanishing cycles of the Lefschetz fibration. In particular, we obtain a trisection diagram for some simply connected minimal complex surface of general type. As a consequence, we obtain explicit trisection diagrams for a pair of closed -manifolds which are homeomorphic but not diffeomorphic. Moreover, we describe a trisection for any oriented -bundle over any closed surface and in particular we draw the corresponding diagrams for and using our gluing technique. Furthermore, we provide an alternate proof of a recent result of Gay and Kirby which says that every closed -manifold admits a trisection. The key feature of our proof is that Cerf theory takes a back seat to contact geometry.
2000 Mathematics Subject Classification:
Recently, Gay and Kirby  proved that every smooth, closed, oriented, connected -manifold admits a trisection, meaning that for every , there exist non-negative integers such that is diffeomorphic to the union of three copies of the -dimensional -handlebody , intersecting pairwise in -dimensional handlebodies, with triple intersection a closed, oriented, connected -dimensional surface of genus . Such a decomposition of is called a -trisection or simply a genus trisection, since is determined by using the fact that , where denotes the Euler characteristic of .
Moreover, they showed that the trisection data can be encoded as a -tuple , which is called a -trisection diagram, such that each triple , , and is a genus Heegaard diagram for . Furthermore, they proved that trisection of (and its diagram) is unique up to a natural stabilization operation.
On the other hand, various flavors of Lefschetz fibrations have been studied extensively in the last two decades to understand the topology of smooth -manifolds. Suppose that a closed -manifold admits a Lefschetz fibration over , whose regular fiber is a smooth, closed, oriented, connected surface of genus . The fibration induces a handle decomposition of , where the essential data can be encoded by a finite set of ordered simple closed curves (the vanishing cycles) on a surface diffeomorphic to . The only condition imposed on the set curves is that the product of right-handed Dehn twists along these curves is isotopic to the identity diffeomorphism of .
In addition, every -manifold with nonempty boundary has a relative trisection and under favorable circumstances also admits an achiral Lefschetz fibration over with bounded fibers. The common feature shared by these structures is that each induces a natural open book on . To exploit this feature in the present paper, we develop a technique to obtain trisection diagrams for closed -manifolds by gluing relative trisection diagrams of -manifolds with nonempty connected boundary. The precise result is stated in Proposition 2.12, which is too technical to include in the introduction. Nevertheless, our gluing technique has several applications — one of which is the following result.
Theorem 3.7. Suppose that is a smooth, closed, oriented, connected -manifold which admits a genus Lefschetz fibration over with singular fibers, equipped with a section of square . Then, an explicit -trisection of can be described by a corresponding trisection diagram, which is determined by the vanishing cycles of the Lefschetz fibration. Moreover, if denotes the -manifold obtained from by blowing down the section of square , then we also obtain a -trisection of along with a corresponding diagram.
In particular, Theorem 3.7 provides a description of a -trisection diagram of the Horikawa surface (see [14, page 269] for its definition and properties), a simply connected complex surface of general type which admits a genus Lefschetz fibration over with singular fibers, equipped with a section of square . This section is the unique sphere in with self-intersection so that by blowing it down, we obtain a trisection diagram for the simply connected minimal complex surface of general type.
To the best of our knowledge, none of the existing methods in the literature can be effectively utilized to obtain explicit trisection diagrams for complex surfaces of general type. For example, Gay and Kirby describe trisections of , , , closed -manifolds admitting locally trivial fibrations over or (including of course , and ) and arbitrary connected sums of these in .
Note that, by Freedman’s celebrated theorem, the Horikawa surface is homeomorphic to (and also to the elliptic surface ), since it is simply connected, nonspin and its Euler characteristic is , while its signature is . On the other hand, since is a simply connected complex surface (hence Kähler) with , it has non-vanishing Seiberg-Witten invariants, while has vanishing Seiberg-Witten invariants which follows from the fact that . Hence, we conclude that is certainly not diffeomorphic to .
As a consequence, we obtain explicit -trisection diagrams for a pair of closed -manifolds, the Horikawa surface and , which are homeomorphic but not diffeomorphic. Note that has a natural -trisection diagram (obtained by the connected sum of the standard -trisection diagrams of and ), which can be stabilized four times to yield a -trisection diagram.
More generally, Theorem 3.7 can be applied to a large class of -manifolds. A fundamental result of Donaldson  says that every closed symplectic -manifold admits a Lefschetz pencil over . By blowing up its base locus the Lefschetz pencil can be turned into a Lefschetz fibration over , so that each exceptional sphere becomes a (symplectic) section of square . Conversely, any -manifold satisfying the hypothesis of Theorem 3.7 must carry a symplectic structure where the section of square can be assumed to be symplectically embedded. Therefore, is necessarily a nonminimal symplectic -manifold.
In [11, Theorem 3], Gay describes a trisection for any closed -manifold admitting a Lefschetz pencil, although he does not formulate the trisection of in terms of the vanishing cycles of the pencil (see [11, Remark 9]). He also points out that his technique does not extend to cover the case of Lefschetz fibrations on closed -manifolds [11, Remark 8].
We would like to point out that Theorem 3.7 holds true for any achiral Lefschetz fibration equipped with a section of square . In this case, is not necessarily symplectic. We opted to state our result only for Lefschetz fibrations to emphasize their connection to symplectic geometry.
Next we turn our attention to another natural application of our gluing technique where we find trisections of doubles of -manifolds with nonempty connected boundary. It is well-known (see, for example, [14, Example 4.6.5]) that there are two oriented -bundles over a closed, oriented, connected surface of genus : the trivial bundle and the twisted bundle . The former is the double of any -bundle over with even Euler number, while the latter is the double of any -bundle over with odd Euler number. We obtain trisections of these -bundles by doubling the relative trisections of the appropriate -bundles. In particular, we draw the corresponding -trisection diagram for and the -trisection diagram for using our gluing technique.
For any , the twisted bundle is not covered by the examples in , while our trisection for has smaller genus compared to that of given in . We discuss the case of oriented -bundles over nonorientable surfaces in Section 4.5.
Finally, we provide a simple alternate proof of the following result due to Gay and Kirby.
Theorem 5.1. Every smooth, closed, oriented, connected -manifold admits a trisection.
Our proof is genuinely different from the two original proofs due to Gay and Kirby , one with Morse -functions and one with ordinary Morse functions, since not only contact geometry plays a crucial role in our proof, but we also employ a technique for gluing relative trisections.
After the completion of our work, we learned that Baykur and Saeki  gave yet another proof of Theorem 5.1, setting up a correspondence between broken Lefschetz fibrations and trisections on -manifolds, using a method which is very different from ours. In particular, they prove the existence of a -trisection on a 4-manifold which admits a genus Lefschetz fibration over with Lefschetz singularities — generalizing the first assertion in our Theorem 3.7, but without providing the corresponding explicit diagram for the trisection. They also give examples of trisections (without diagrams) on a pair of closed -manifolds (different from ours) which are homeomorphic but not diffeomorphic. In addition, for any , they give small genus trisections (again without the diagrams) for .
Conventions: All -manifolds are assumed to be smooth, compact, oriented and connected throughout the paper. The corners which appear in gluing manifolds are smoothed in a canonical way.
2. Gluing relative trisections
We first review some basic results about trisections and their diagrams (cf. ). Let denote the standard genus Heegaard splitting of obtained by stabilizing the standard genus Heegaard splitting times.
A -trisection of a closed -manifold is a decomposition such that for each ,
there is a diffeomorphism and
taking indices mod , and .
It follows that is a closed surface of genus . Also note that and determine each other, since the Euler characteristic is equal to , which can be easily derived by gluing and first and then gluing .
Suppose that each of and is a collection of disjoint simple closed curves on some compact surface . We say that two such triples and are diffeomorphism and handleslide equivalent if there exists a diffeomorphism such that is related to by a sequence of handleslides and is related to by a sequence of handleslides.
A -trisection diagram is an ordered -tuple such that
is a closed genus surface,
each of , and is a non-separating collection of disjoint, simple closed curves on ,
each triple , and is diffeomorphism and handleslide equivalent to the standard genus Heegaard diagram of depicted in Figure 1.
According to Gay and Kirby , every closed -manifold admits a trisection, which in turn, can be encoded by a diagram. Conversely, every trisection diagram determines a trisected closed -manifold, uniquely up to diffeomorphism.
Suppose that is a -manifold with nonempty connected boundary . We would like to find a decomposition , such that each is diffeomorphic to for some fixed . Since , it would be natural to require that part of each contribute to . Hence, we need a particular decomposition of to specify a submanifold of to be embedded in . With this goal in mind, we proceed as follows to develop the language we will use throughout the paper.
Suppose that are non-negative integers satisfying and
Let and for , and , .
We denote by a fixed genus surface with boundary components. Let
be a third of the unit disk in the complex plane whose boundary is decomposed as , where
The somewhat unusual choice of the disk will be justified by the construction below. We set , which is indeed diffeomorphic to . Then, inherits a decomposition , where and .
Let be the standard genus one Heegaard splitting of . For any , let , where the boundary connected sum is taken in neighborhoods of the Heegaard surfaces, inducing the standard genus Heegaard splitting of . Stabilizing this Heegaard splitting times we obtain a genus Heegaard splitting of .
We set and . Note that we can identify , where the boundary connected sum again takes place along the neighborhoods of points in the Heegaard surfaces. We now have a decomposition of as follows:
A -relative trisection of a -manifold with non-empty connected boundary is a decomposition such that for each ,
there is a diffeomorphism and
taking indices mod , and
As a consequence, is diffeomorphic to , which is a genus surface with boundary components. Note that the Euler characteristic is equal to , which can be calculated directly from the definition of a relative trisection. We also give alternate method to calculate in Corollary 2.10.
According to , every -manifold with nonempty connected boundary admits a trisection. Moreover, there is a natural open book induced on , whose page is diffeomorphic to , which is an essential ingredient in our definition of .
Informally, the contribution of each to is one third of an open book. This is because the part of each that contributes to is diffeomorphic to
where is one third of the truncated pages, while is one third of the neighborhood of the binding. In other words, not only we trisect the -manifold , but we also trisect its boundary . Conversely, if an open book is fixed on , then admits a trisection whose induced open book coincides with the given one.
A -relative trisection diagram is an ordered -tuple such that
is a genus surface with boundary components,
each of and is a collection of disjoint, essential, simple closed curves,
each triple , and is diffeomorphism and handleslide equivalent to the diagram depicted in Figure 2.
It was shown in  that every relative trisection diagram determines uniquely, up to diffeomorphism, (i) a relatively trisected -manifold with nonempty connected boundary and (ii) the open book on induced by the trisection. Moreover, the page and the monodromy of the open book on is determined completely by the relative trisection diagram by an explicit algorithm, which we spell out below.
Suppose that is a -relative trisection diagram, which represents a relative trisection of a -manifold with nonempty connected boundary. The page of the induced open book on is given by , which is the genus surface with boundary components obtained from by performing surgery along the curves. This means that to obtain , we cut open along each curve and glue in disks to cap off the resulting boundaries.
Now that we have a fixed identification of the page of as , we use Alexander’s trick to describe the monodromy of . Namely, we cut into a single disk via two distinct ordered collections of arcs, so that for each arc in one collection there is an arc in the other collection with the same endpoints. As a result, we get a self-diffeomorphism of that takes one collection of arcs to the other respecting the ordering of the arcs, and equals to the identity otherwise. This diffeomorphism uniquely extends to a self-diffeomorphism of the disk, up to isotopy. Therefore, we get a self-diffeomorphism of fixing pointwise, which is uniquely determined up to isotopy. Next, we provide some more details (see [5, Theorem 5]) about how to obtain the aforementioned collection of arcs.
Let be any ordered collection of disjoint, properly embedded arcs in disjoint from , such that the image of in cuts into a disk. We choose a collection of arcs , and a collection of simple closed curves disjoint from in such that is handleslide equivalent to , and is handleslide equivalent to . This means that arcs are obtained by sliding arcs over curves, and is obtained by sliding curves over curves. Next we choose a collection of arcs , and a collection of simple closed curves disjoint from in such that is handleslide equivalent to , and is handleslide equivalent to . This means that arcs are obtained by sliding arcs over curves, and is obtained by sliding curves over curves. Finally, we choose a collection of arcs , and a collection of simple closed curves disjoint from in such that is handleslide equivalent to , and is handleslide equivalent to . This means that arcs are obtained by sliding arcs over curves, and is obtained by sliding curves over curves. It follows that is handleslide equivalent to for some collection of arcs disjoint from in .
We call the triple a cut system of arcs associated to the diagram .
Now we have two ordered collections of arcs and in , such that each of their images in cuts into a disk. Then, as we explained above, there is a unique diffeomorphism , up to isotopy, which fixes pointwise such that .
It is shown in  that, up to isotopy, the monodromy of the resulting open book is independent of the choices in the above algorithm.
Next, we give a very simple version of the general gluing theorem  for relatively trisected -manifolds. Here we present a different proof — where we use the definition of a relative trisection as given in  instead of  — for the case of a single boundary component.
Suppose that and are -manifolds such that and are both nonempty and connected. Let and be - and -relative trisections with induced open books and on and , respectively. If is an orientation-reversing diffeomorphism which takes to (and hence and ), then the relative trisections on and can be glued together to yield a -trisection of the closed -manifold , where and .
Since there is an orientation-reversing diffeomorphism which takes the open book to the open book , we have and . Let and be - and -relative trisections, respectively. Then, by Definition 2.3, there are diffeomorphisms and for .
Let and be the bindings of and , where and are the projection maps of these open books, respectively. Since the gluing diffeomorphism takes to by our assumption, we have , for all Moreover, we may assume that
is a diffeomorphism for each . Informally, we identify each third of on with the appropriate third of on via the gluing map . This allows us to define
where if and , where .
We claim that is a -trisection, where
In order to prove our claim, we first need to describe, for each , a diffeomorphism . The diffeomorphism is essentially obtained by gluing the diffeomorphisms and using the diffeomorphism , as we describe below.
To construct the desired diffeomorphism , it suffices to describe how to glue with to obtain by identifying with using the gluing map .
By definition, , and similarly , where and . Note that is diffeomorphic to via . To glue to we identify with via the diffeomorphism . Next, we observe that by gluing and along the aforementioned parts of their boundaries using , we get , where .
To see this, we view as , and similarly as , where . We glue with and then take its product with . To glue with we identify with using . The result of this identification is diffeomorphic to . However, to complete the identification dictated by , we have to take the quotient of by the relation for all . Note that we suppressed here since we have already identified with via . The result is still diffeomorphic to the handlebody . Therefore, the gluing of and is diffeomorphic to , since it is a thickening of by taking its product with .
As a consequence, the result of gluing to is diffeomorphic to
To finish the proof of the lemma, we need to show that taking indices mod ,
where is the standard genus Heegaard splitting of .
We observe that
where if and .
Note that by definition, and similarly . Since the boundary connected sums are taken along the interior of Heegaard surfaces, the identification does not interact with and and hence
But we see that is diffeomorphic to , by exactly the same argument used above when we discussed the gluing of with . Therefore, we have
since . Similarly, we have
where if and . Thus we obtain . Moreover,
Therefore, and gives the standard genus Heegaard splitting of , as desired.
To summarize, we showed that there is a diffeomorphism , for each , and moreover, taking indices mod , and . Therefore, we conclude that is a -trisection. ∎
Here is an immediate corollary of Lemma 2.7.
Suppose that is a -relative trisection of a -manifold with nonempty connected boundary. Let denote the double of obtained by gluing and (meaning with the opposite orientation) by the identity map of the boundary . Then admits a -trisection.
If is a -relative trisection of with the induced open book on , then is a -relative trisection of with the induced open book on , where is obtained from by reversing the orientation of the pages. Since the identity map from to is an orientation-reversing diffeomorphism which takes to , we obtain the desired result about by Lemma 2.7. ∎
The point of Corollary 2.8 is that one does not need to know the monodromy of the open book on to describe a trisection on .
Let denote the -bundle over with Euler number . In , there is a description of a -relative trisection of for , and a -relative trisection of . Since the double of is or depending on modulo , we get a -trisection of (resp. ) for any even (resp. odd) nonzero integer , by Corollary 2.8.
In particular, by doubling the -relative trisection of , we obtain a -trisection of which is smaller compared to the -trisection presented in , provided that . Similarly, by setting , we obtain a -trisection for , which is not covered by the examples in , except for . Note that there is also a -relative trisection of given in  for each . Since the double of is or depending on modulo , we get infinitely many -trisections of and .
If is a -relative trisection of a -manifold with nonempty connected boundary, then the Euler characteristic is equal to
Using Corollary 2.8, we compute
One can of course derive the same formula directly from the definition of a relative trisection. ∎
Since every relatively trisected -manifold with connected boundary is determined by some relative trisection diagram, it would be desirable to have a version of Lemma 2.7, where one “glues” the relative trisection diagrams corresponding to and to get a diagram corresponding to the trisection . This is the content of Proposition 2.12, but first we develop some language to be used in its statement.
Let and be - and -relative trisection diagrams corresponding to the relative trisections and , with induced open books and on and