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# Simplified broken Lefschetz fibrations and trisections of 4-manifolds

Edited by Yakov Eliashberg, Stanford University, Stanford, CA, and approved March 13, 2018 (received for review October 3, 2017)

## Significance

As our world, with time included, is 4D, exploring similarities and differences of 4D spaces via maps and decompositions they admit leads to a better understanding of the universe we are living in. This article studies maps from 4D spaces to 2D ones. We illustrate how to modify such maps algorithmically to derive rather special maps that allow us to break 4D spaces into much simpler, much better-understood pieces. Our general methods are presented with a large variety of explicit constructions.

## Abstract

Shapes of 4D spaces can be studied effectively via maps to standard surfaces. We explain, and illustrate by quintessential examples, how to simplify such generic maps on 4-manifolds topologically, to derive simple decompositions into much better-understood manifold pieces. Our methods not only allow us to produce various interesting families of examples but also to establish a correspondence between simplified broken Lefschetz fibrations and simplified trisections of closed, oriented 4-manifolds.

There is a long and rich history of studying geometry and topology of spaces by looking at maps between them. For a 4D manifold, generic maps to surfaces allow one to foliate it by surfaces, some of which are pinched along embedded loops on them. Two of the most paramount classes of maps, which received tremendous attention in recent years, are (broken) Lefschetz fibrations and trisected Morse 2-functions. Both yield decompositions of the ambient 4-manifold into much simpler pieces, such as symplectic fibrations or thickened handlebodies, allowing one to bring a hefty combination of ideas and techniques from complex and symplectic geometry, classical 3-manifold topology, and geometric group theory.

In this paper we will approach broken Lefschetz fibrations and trisections from the vantage point of singularity theory, focusing on how to construct much-simplified versions of these maps/decompositions through topological modifications of generic maps. Most of the topological beautifications we perform by homotopies of maps, guided and argued via diagrammatic representations of their singular images—which we hope will make our rather combinatorial arguments accessible to a broader audience. The reader is invited to glance over some of the figures below.

Our main goal is to demonstrate, with illuminative examples, how naturally and easily such simplified maps and trisections arise on 4-manifolds. Here we will show how to pass from a simplified broken Lefschetz fibration to a simplified trisection and back, without increasing the fibration/trisection genus much. Further, we will authenticate infinite families of examples for smallest possible genera. Background results and their complete proofs, which are of fairly technical nature, are given in our more extensive work in ref. 1.

## Maps with Elementary Singularities

First, we introduce the classes of maps we are interested in. Herein,

### Generic Maps.

The map f is said to have a fold singularity at y, if there are local coordinates around y and

Fold and cusp points constitute a 1D submanifold

By Thom transversality (2), any smooth map

### Broken Lefschetz Fibrations.

The map f is said to have a Lefschetz singularity at a point

This class of maps was first introduced by Auroux, Donaldson, and Katzarkov in ref. 3, as a generalization of honest Lefschetz fibrations (without folds), which have become central objects in symplectic and contact geometry after the pioneering works of Donaldson, Gompf, Seidel, and Giroux since the mid-1990s. While only symplectic 4-manifolds admit Lefschetz fibrations (with

### Trisections.

A trisection of a 4-manifold X is a decomposition into three 4D 1-handlebodies (thickening of a wedge of circles) meeting pairwise in 3D 1-handlebodies, and all three intersecting along a closed, connected, orientable surface.

Trisections were introduced by Gay and Kirby in ref. 4 as natural analogues of Heegaard splittings of 3-manifolds. Just like how Heegaard splittings correspond to certain Morse functions, which are generic maps to the 1D disk *-*functions. This class of maps is characterized by the following: Up to isotopy, they have a single definite fold circle mapped to the boundary *-*trisection of X, where

Any 4-manifold X admits generic maps that can be homotoped to a trisection. Even more remarkably, like the Reidemeister–Singer theorem for Heegaard splittings of 3-manifolds, trisections of 4-manifolds are unique up to an innate stabilization operation (4).

## Maps with Simplified Topologies

We now describe special subclasses of broken Lefschetz fibrations and trisections, which have simpler topologies.

### Simplified Broken Lefschetz Fibrations.

A broken Lefschetz fibration

This subclass of broken Lefschetz fibrations was introduced by the first author in ref. 5. The underlying topology is simple: either we get a genus-g Lefschetz fibration over *-*handle (

Let us digress a little on the latter aspect. Let

### Simplified Trisections.

A trisection is said to be simplified if the singular image of an associated trisection map f is embedded and cusps only appear in triples—like a triangle—in innermost fold circles. This is in great contrast with a general trisection map, which has the so-called Cerf boxes in between the three sectors of the base disk, where folds can cross each other arbitrarily (and therefore, the images of some indefinite fold circles might wind around the origin multiple times). Compare the singular images given in Fig. 1, where the arrows indicate the index-2 fiberwise handle attachments, and the definite fold is given in red.

Simplified trisections were introduced recently in ref. 1. The main difference between a general trisection and a simplified one is manifested in what one might call the hierarchy of handle slides. The inverse image, under an arbitrary trisection map, of any radial cut of the base disk from the origin to its boundary (say, avoiding the cusps) is a genus-

### The Existence.

With all of the necessary definitions in place, we can now quote our main result from ref. 1 motivating this work:

**Theorem 1 (Existence).** *Given any generic map from a closed*, *connected*, *oriented*, *smooth* 4*-manifold* X *to* *there are explicit algorithms to modify it to a simplified broken Lefschetz fibration*, *as well as to a simplified trisection map. Therefore*, *any* X *admits simplified broken Lefschetz fibrations and simplified trisections.*

There is of course an abundance of generic maps from any X to

We can then derive these two types of simplified maps from one another, and the next theorem aims to do it in the most economical way for the genus of the resulting broken Lefschetz fibration or trisection, that is, by keeping it as small as we can (but not to say one cannot do better for specific examples).

**Theorem 2 (Correspondence).** *If there is a genus-*g *simplified broken Lefschetz fibration* *with* *Lefschetz critical points*, *and* *components of* *then there is an associated simplified* *-trisection of* X, *where*

*Conversely*, *if* X *admits a simplified* *-trisection*, *then there is an associated genus-*g *simplified broken Lefschetz fibration* *with* k *Lefschetz singularities and one* *component*, *where*

## Homotopies of Generic Maps and Base Diagram Moves

The base diagram of a map

We will perform homotopies through a sequence of base diagram moves, viz. local modifications of a base diagram, each one of which can always be realized by a 1-parameter family of smooth maps (which do not change outside of this locality). While the transition happens around one point on Σ, the bifurcation of the map may occur around one point (a monogerm move), or two to three points (a multigerm move) in X. It turns out that only some of the possible local changes that can occur in a base diagram during a generic homotopy are always-realizable. However, the bifurcations we get through always-realizable ones will be enough to obtain the desired topology for the resulting map.

These homotopy moves have been studied in varying levels of detail since the mid-1960s by Levine, Hatcher–Wagoner, Eliashberg–Mishachev, Lekili, Williams, Gay–Kirby, Behrens–Hayano, and the authors of this article. Fig. 2 lists the always-realizable moves we will use in this article, with the names and conventions carried on from ref. 1. The normal orientations for cusped arcs are always in the direction of cusps. In our arguments to follow, we will use only these always-realizable base diagram moves. (So the reader can refer to this figure as a chart of legal moves in a board game of sorts.)

The first two rows of Fig. 2 consist of a monogerm move flip and multigerm moves *-*move, and push. Several of these can be regarded as Reidemeister I and II type moves. There are Reidemeister III type moves as well, which play a vital role in the proof of *Theorem 1* but are not needed for our relatively more straightforward constructions here. The third row contains three monogerm moves cusp merge, unsink, and wrinkle. Note that two cusps can be merged using any path between them in the source 4-manifold, but here we simply use an arc with image embedded in the middle region between the two cusped arcs. When the fibers in this region are connected, one can always find such an arc. Finally, the fourth row of Fig. 2 lists two combination moves: flip-and-slip and definite-to-indefinite. These involve a sequence of base diagram moves suppressed in this presentation. Importantly, almost none of these base diagram moves have always-realizable pseudoinverses.

## Bridging Broken Lefschetz Fibrations and Trisections

Here we outline the proof of *Theorem 2*, in hopes of providing insight to the reader how we can use the always-realizable base diagram moves for rearranging the underlining topology of a map to our liking. More details for the arguments below can be found in ref. 1.

### From Simplified Broken Lefschetz Fibrations to Trisections.

Let

First assume that

We can now apply base diagram moves to turn h into a trisection map. Recall that we use the terminology from ref. 1; the names we call out for the moves can be found in Fig. 2.

First, using an

What remains is to arrange the triple-cusped indefinite fold circles, as shown in Fig. 3. Unsinking one of its four cusps, the innermost fold circle becomes a triple-cusped one. Wrinkle one of the Lefschetz singularities to produce the next triple-cusped circle. We push all remaining Lefschetz singularities into the region bounded by this triple-cusped circle and repeat the same procedure until we exhaust all of the Lefschetz singularities. We end up with

If we had

### From Trisections to Simplified Broken Lefschetz Fibrations.

Let

Embed

Let us view all of the

It is an easy exercise to check that throughout all of the modifications we have made to h, every map we got so far had only connected fibers. When we have connected fibers over a region, going against the normal arrow direction of an indefinite fold, we pass to a neighboring region over which the fiber should be connected as well. This is because tracing this path upstairs we attach an index-1 handle to the original regular fiber, simply increasing the genus by one. (To complete the exercise, one can, for example, begin with observing that after the definite-to-indefinite move the fibers over the southern hemisphere had to be connected.)

So, if we flip each circle twice, we can merge all of them into one immersed circle after

Since a regular fiber over the southern hemisphere is obtained by a 1-handle attachment to a genus

## Small Genera Examples and Infinite Families

We will now look at simplified broken Lefschetz fibrations and trisections of small genera. We will present some new examples for the latter, so as to classify simplified trisections of genera at most two, and show that for each

### Classification of Small Simplified Broken Lefschetz Fibrations.

Simplified genus-g broken Lefschetz fibrations of genus

A genus-0 simplified broken Lefschetz fibration cannot have an indefinite fold, or otherwise the fibers would be disconnected. So, genus-0 simplified broken Lefschetz fibrations are all isomorphic to holomorphic rational Lefschetz fibrations on (possibly trivial) blow-ups of complex surfaces

The first interesting examples we get are the genus-1 simplified broken Lefschetz fibrations with indefinite folds. Let

Surprisingly, perhaps, when μ is trivial, that is, when

The complete list of 4-manifolds admitting genus-1 simplified broken Lefschetz fibration is then exhausted by (possibly trivial) blow-ups of all of the 4-manifolds we mentioned, and of

### Classification of Small Simplified Trisections.

Let us now look at the corresponding picture for simplified trisections. General

The indefinite part of the singular image of a genus-

As for

A few observations first. Given any map from X to a surface, localizing the map over a disk with no singular image, one can always introduce a Lefschetz singularity. Furthermore, suppose we have a disk

Using the above tricks, we can start with a genus-1 simplified trisection on *Theorem 2*, the rational fibration on

What can we say about higher genera trisections? When *Theorem 2*. We will see in the next section that in fact many

For a sharper result, we can instead take the infinite family of genus-1 simplified broken Lefschetz fibrations on rational homology 4-spheres *Theorem 2* in this case is an infinite family of genus-3 simplified trisections. Let us remark that we similarly get a genus-3 simplified trisection on *,* which—as a map—is not isotopic to the standard genus-3 trisection used by Gay and Kirby for their stabilization result in ref. 4. Blow-ups of these infinite families then give infinite families of genus-

In the same fashion as our construction of genus-2 simplified trisections on connected sums, one can produce genus-3 simplified trisections on connected sums of

Although a complete classification of genus-

**Question 1.** *Which* 4*-manifolds admit simplified genus-*3 *trisections*? *Is there any* 4*-manifold*, *other than the ones mentioned above*, *which admits genus-*3 *simplified trisections*? *How about genus-*4?

## More Examples: From 3-Manifolds to 4-Manifolds

Our last family of examples are on 3-manifold bundles over the circle, and on 4-manifolds derived from them by a standard surgery. We will show that one can easily derive a simplified broken Lefschetz fibration or a simplified trisection on these 4-manifolds from any given Heegaard splitting of the 3-manifold that is invariant under the monodromy of the bundle.

### General Constructions.

Let Y be a closed, connected, oriented 3-manifold, and let

Let us first derive a simplified broken Lefschetz fibration from

Alternatively, we can derive a simplified trisection from *Theorem 2*, we can turn them into

The constructions above are variations of those we had in ref. 1 for the particular case of *-*manifolds, that is, 4-manifolds obtained by surgering out

The same trick applies to any 4-manifold

### More Examples of Small Simplified Trisections.

Through the constructions above, we can produce many more simplified trisections of genus 3 or genus 4.

Taking

Taking

Together with the genus-1 Heegaard splitting of

Recall that our second construction produces a simplified trisection on a 4-manifold *Question 1* all the more curious! However, any 4-manifold

We finish with a natural question:

**Question 2.** *Is there any* 4*-manifold which admits a trisection*, *but not a simplified one of the same genus*?

Defining the minimal trisection genus (resp. minimal simplified trisection genus) of a 4-manifold X as the smallest genus of a trisection (resp. simplified trisection) on X, one can equivalently ask if there is a 4-manifold whose trisection genus is smaller than its simplified trisection genus. The two are equal for all of the 4-manifolds with (simplified) trisections of genus

## Acknowledgments

We thank Kenta Hayano for his careful comments on a draft of this manuscript. This work was partially supported by NSF Grant DMS-1510395 (to R.I.B.) and Japan Society for the Promotion of Science KAKENHI Grants JP23244008, JP23654028, JP15K13438, JP16K13754, JP16H03936, JP17H01090, and JP17H06128 (to O.S.).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: baykur{at}math.umass.edu.

Author contributions: R.I.B. and O.S. designed research, performed research, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.

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