#### by David T. Gay

#### 1. Chatty introduction (which you can skip if you want to jump to the math)

Rob Kirby was my Ph.D. supervisor at UC Berkeley; I finished my Ph.D.
there in 1999. During my first postdoc position, at the University of
Arizona, my postdoc advisor
Doug Pickrell
suggested that I invite Rob
to give a colloquium talk; I think that was in the fall of 2001. I
remember Rob gave a survey of the beginning of
Heegaard Floer theory
and in particular I clearly remember him saying something about
organizing the various spin__\( ^\mathbb{C} \)__ structures on a 4-manifold
into “buckets”. However, the key event of that visit for me was a
full-moon night hike that Doug organized at Sabino Canyon. (Part of my
point here in the story is to make sure to give Doug due credit for
his role in all of this. Another point is to emphasize the importance
of hiking in Rob’s mathematical life.) During the hike, we ended up
talking about my thesis work on handle-by-handle symplectic
constructions, and I remember Rob commenting, “Oh, so that’s what your
thesis was about.” The point is not that he didn’t read my thesis but
that the essential idea of my thesis was somehow obscured in the whole
mechanics of writing, reviewing, graduating, and so forth; I do know
that Rob and I both talked about most of the details of my thesis but
somehow we had never stepped back and thought about what it was *really* about. And once Rob understood what it was *really* about,
the ideas started flowing.

In this article I want to tell the (mostly mathematical) story of my
collaboration with Rob Kirby that started during that night hike and
ended up with our
discovery of trisections of
4-manifolds
[5].
I use “discovery” cautiously
because much of what we discovered was already known and in the end a
lot of our work can be thought of as just finding the right way to
organize the ideas and the questions. Our work started hovering around
various topological constructions related to the vision of extending
4-dimensional symplectic tools to a larger class of 4-manifolds
using near-symplectic structures (inspired by
Cliff Taubes).
We then
followed what seemed like the natural mathematical trail to the study
of broken Lefschetz fibrations (inspired by
Auroux,
Donaldson
and
Katzarkov,
Usher
and
Perutz),
then to thinking more generally about
generic maps from 4-manifolds to surfaces (inspired by
Lekili),
which we ended up calling *Morse 2-functions*. (A lot of this
involved rediscovering for ourselves in our own way of understanding
lots of things already known to singularity theorists and others.)
From there we more or less stumbled on trisections of 4-manifolds
as a particularly nice outcome of the theory of Morse 2-functions.
At the end I will give a brief survey of some of the developments
surrounding trisections that have happened since our first paper,
emphasizing my work with Rob and
Aaron Abrams
[6]
on group trisections.

I hope to emphasize throughout the powerful influence of Rob’s unique way of seeing the mathematical world on my own mathematical development and the naturalness of the mathematical flow of our collaboration. When I was a graduate student and asked Rob for some advice about what to think about mathematically, I don’t remember the exact context, but I remember clearly that his advice was simultaneously vague, unhelpful and deeply meaningful: “Follow your nose.”

#### 2. Just the math about what trisections are

For those who have not thought about trisections before, the idea is
almost stupidly simple. A trisection of a smooth, closed, oriented,
connected 4-manifold __\( X \)__ is a decomposition of __\( X \)__ into three
topologically simple pieces __\[ X = X_1 \cup X_2 \cup X_3 \]__ which fit
together like three “slices of a pie” as in
Figure 1: Each piece __\( X_i \)__ is diffeomorphic (after
some mild smoothing of corners) to a regular neighborhood of a bouquet
of (zero or more) circles in __\( \mathbb{R}^4 \)__. (A bouquet of zero
circles is a point, one circle is a circle, two circles is a figure
eight, etc.) Each pairwise intersection __\( X_i \cap X_j \)__ is
diffeomorphic to a regular neighborhood of a bouquet of (zero or more)
circles in __\( \mathbb{R}^3 \)__ (a.k.a. a solid handlebody). The triple
intersection __\( X_1 \cap X_2 \cap X_3 \)__ is diffeomorphic to a closed,
orientable, connected surface.

The surprising facts which we discovered between 2011 and 2013 are
(i) that *every smooth, closed, oriented, connected 4-manifold* has
a trisection (*existence*), and in fact it has lots of them;
(ii) that there is a simple-to-describe stabilization operation that increases
the complexity of a trisection; and
(iii) that
any two trisections of a
given 4-manifold have a common stabilization (*uniqueness*).
Furthermore, this means that every such 4-manifold can be described
up to diffeomorphism by a *trisection diagram* which is a diagram
drawn on a closed, oriented surface of genus __\( g \)__ involving __\( g \)__ red
simple closed curves, __\( g \)__ blue simple closed curves, and __\( g \)__ green
simple closed curves. See Figure 2 for an
interesting example. The surface is the triple intersection and the
curves of each color are curves that bound disks in the pairwise
intersections.

One formally satisfying feature of trisections is that they turn out
to be the right generalization of the notion of a Heegaard splitting
of a 3-manifold to the world of 4-manifolds, with the surprise
being that in order to generalize a Heegaard splitting (a
decomposition of a 3-manifold into two simple pieces) one needs to
consider decompositions into *three* pieces. In fact Heegaard
splittings of a 3-manifold __\( M \)__ are best seen as coming from
particularly nice smooth maps __\( f: M \to [-1,1] \)__ (Morse functions with
well organized critical points) and then
decomposing __\( M \)__ into
__\( f^{-1}([-1,0]) \)__ and __\( f^{-1}([0,1]) \)__. Similarly, trisections of a
4-manifold __\( X \)__ are best seen as coming from particularly nice
smooth maps __\( f: X \to D^2 \)__ to the disk, called
*Morse 2-functions*,
with particularly well organized singularities,
and then decomposing into three pieces by pulling back the
decomposition of the disk into three (fat) pie slices.
Also, Heegaard
splittings allow 3-manifolds to be described by *Heegaard
diagrams*, which are just like trisection diagrams but only involve
curves of two colors. Finally, note that a trisection diagram gives
three Heegaard diagrams, by considering each pair of two colors.

In principle, a trisection diagram is not much different from a framed
link diagram, but it is better organized, and there are algorithms to
turn either one into the other. In particular, trisection diagrams
have an interesting formal property summarized in the slogan
“pairwise boring, triply interesting”. There is a standard move on
a collection of disjoint simple closed curves on a surface, called a
*handle slide*, which involves one curve “sliding over” and
picking up a copy of another curve, as shown in Figure 3.
Handle slides do not change either the 4-manifold associated to a
trisection diagram or the 3-manifold associated to a Heegaard
diagram. A trisection diagram __\( (\Sigma, \alpha.\beta,\gamma) \)__ has the
property that each associated Heegaard diagram of two colors
__\( (\Sigma,\alpha,\beta) \)__, __\( (\Sigma,\beta,\gamma) \)__ and
__\( (\Sigma,\gamma,\alpha) \)__ is equivalent, after some handle slides and a
diffeomorphism, to the boring Heegaard diagram in
Figure 4, which is a *Heegaard diagram* for the
connected sum of some number of __\( S^1 \times S^2 \)__s (a boring
3-manifold). However, all 4-manifolds (and thus a very large
class of very interesting objects) can be described by these triples
of collections of curves, despite the fact that in pairs they are
boring. The interested reader might enjoy verifying this pairwise
standardness for the diagram in Figure 2.

A classical example of the “pairwise boring, triply interesting” slogan, by the way, is the three-component link known as the Borromean rings, shown in Figure 5. This link of three circles has the property than any pair of the three are unlinked but all together they are nontrivially linked. Here is a meta-question for the mathematical community at large: Where else in mathematics does this slogan apply?

#### 3. The quixotic quest to make gauge theory mean something topologically

My entire topological career has been dominated by the incredible twin
results of
Freedman
[e5]
that two closed simply connected
smooth 4-manifolds are homeomorphic if and only if they are
homotopy equivalent and of Donaldson
[e6]
that there
exist invariants that can distinguish pairs (known as “exotic
pairs”) of closed, simply connected smooth 4-manifolds that are
homotopy equivalent but not diffeomorphic. Rob was closely connected
with Freedman’s work, but Donaldson’s work used gauge theory (PDEs
coming from physics), and Rob was never a gauge theorist. Ever since I
have known Rob an important part of his mathematical quest has been to
understand in some “genuinely topological” how to see that exotic
pairs are not diffeomorphic. The Donaldson invariants were later
replaced to some extent with the simpler Seiberg–Witten
invariants
[e9]
and eventually many gauge theoretic
results could be proved using
Ozsváth
and
Szabó’s
Heegaard Floer
invariants
[e19],
but, still, underlying it all
is the problem of counting solutions to PDEs. Rob always wanted a way
to *see* what we were counting explicitly and combinatorially, and
to see from a geometric topologist’s perspective, in a direct way, why
these counts were invariants of smooth structures.

We would like to be able to stare at some diagrammatic, combinatorial description of a 4-manifold, “count” something in the diagram, perhaps count smooth isotopy classes of some objects in the diagram, and then understand that this is an invariant because one understands the moves needed to relate any two such diagrammatic descriptions of the same 4-manifold. I would argue that we are still far from such a perspective, but that for formal reasons such a perspective should be out there. The closest we have come that I am aware of are the handful of 4-dimensional results coming out of Khovanov homology, beginning with Rasmussen’s computations of slice genus [e22].1 In what follows, one thread of the story of my collaboration with Rob is that we were constantly trying to catch up with gauge and Floer theory and reinterpret it in ways we nonanalytically minded “mere geometric topologists” could understand and really see.

Generally, however, almost all invariants proceed by first imposing
some additional structure, then counting something, and then showing
that the answer does not depend on the choice of that additional
structure. In the gauge theoretic context this additional structure is
a collection of differential geometric data such as Riemannian
metrics, connections on various bundles, and so forth. In the Floer
theoretic setting the additional structure is generally an almost
complex structure, usually with a symplectic structure lurking in the
background. In our hypothetically more “combinatorial”, more
“topologically understandable” visions, there would still be extra
structure, but this would come in the form of a diagram describing a
construction of the manifold or simply a decomposition of the manifold
into elementary pieces. For us, inspired heavily by Morse theory, an
intermediate structure has typically been some kind of smooth map from
the manifold in question to another simpler manifold, e.g.,
__\( \mathbb{R} \)__ or __\( \mathbb{R}^2 \)__ or __\( S^2 \)__. Symplectic topology has
always lurked in the background as “borderline understandable” to
simple minded topologists, and thus you will find the following
storyline progresses through symplectic topology to understanding
various classes of maps from dimension 4 to dimension 2 and
finally arriving at decompositions of manifolds and associated
diagrams, with an addendum in which the “diagrams” are eventually
replaced with group theory.

There is a great deal also to be said about the extent to which complex algebraic geometry has inspired smooth 4-dimensional topology, with smooth algebraic surfaces being in some sense the prototypical 4-manifolds. This is too much for me to competently discuss here but suffice it to say that this thread very much lurks in the background, and I believe that much of Rob’s insight into the problems he and I have thought about can be traced back to his early work with Harer and Kas on the smooth topology of complex surfaces [1]. (To a complex algebraic geometer, a quick introduction to trisections is to say that they are the natural generalization of the decomposition of the complex projective plane into three coordinate charts.)

Perhaps a better reason for this quest than just the fact that we couldn’t understand gauge theory is that all gauge theoretic methods have failed to say anything about homotopy 4-spheres, and thus shed no light on the holy grail, the smooth 4-dimensional Poincaré conjecture (or its sister, the smooth 4-dimensional Schoenflies problem). Thus, like any good 4-dimensional topologist should, we have always held out hope that some day we might stumble upon some invariant, almost certainly not defined using anything like gauge theory, which could distinguish smooth homotopy 4-spheres and hence disprove the Poincaré conjecture. Owning up to such a dream in public is perhaps not good form, but in our collaboration we have had moments of (probably misguided) optimism and enthusiasm in this direction. The truth is that we have been frequently quite naive about the potential to make hard things easy to understand, but that our naïveté seems to have served us pretty well.

In what follows I will try to give a fairly thorough account of the path Rob and I took that brought us to thinking about trisections of 4-manifolds. Of course we never knew where we were going in the long run, but we looked at a sequence of related problems that each raised new questions that, in hindsight, seemed to lead inexorably towards trisections. All of the questions we thought about along the way are, in my opinion, still very important and serve to set trisections in the right context. At each step we were working on 4-manifolds equipped with certain auxiliary structures, and I will describe those auxiliary structures as they arise.

#### 4. Near symplectic constructions

The relevant auxiliary structures to discuss here are symplectic and
almost complex structures on 4-manifolds, building on the
foundational work of
Gromov
[e7].
Symplectic structures are
closed, nondegenerate 2-forms, which means one can always find
local coordinates in which they have (in dimension 4) the form __\( dx_1
\wedge dy_1 + dx_2 \wedge dy_2 \)__. (A 2-form at a point is an
antisymmetric bilinear form on the tangent space at that point that
should be thought of measuring the “signed area” of the
parallelogram spanned by two vectors, and in this case the area of the
__\( 1 \times 1 \)__ squares in the __\( x_1,y_1 \)__ and __\( x_2,y_2 \)__ planes are 1
while the __\( 1 \times 1 \)__ squares in any of the __\( x_1,x_2 \)__, __\( x_1,y_2 \)__,
__\( y_1,x_2 \)__ and __\( y_1,y_2 \)__ planes have “area” 0.) An almost complex
structure on a 4-manifold is a way to “multiply by __\( i \)__” in the
tangent spaces, in other words a smoothly varying linear automorphism
__\( J \)__ of each tangent space such that __\( J^2 = -\operatorname{id} \)__. The
structure of a 2-dimensional complex manifold on a 4-manifold
induces an almost complex structure on the tangent spaces, but not
every almost complex structure comes from a complex manifold
structure.

A classical way to probe the structure of a complex algebraic variety
is to study enumerative problems for curves in that variety.
Remembering that “curves” over __\( \mathbb{C} \)__ are actually real
2-dimensional surfaces, one can push these enumerative methods to
the softer setting of almost complex manifolds and study
*pseudoholomorphic curves*, which are surfaces in the 4-manifold
whose tangent spaces are fixed by the almost complex structure, in
other words, their real 2-dimensional tangent spaces are actually
complex lines with respect to __\( J \)__. Gromov’s key contribution was to
show that, when an almost complex structure is dominated by a
symplectic structure (meaning that the 2-form assigns positive area
to a parallelogram spanned by __\( V \)__ and __\( JV \)__ for any nonzero tangent
vector __\( V \)__), then one can control families of pseudoholomorphic curves
and get compactness results for their moduli spaces. This then allows,
in special cases, for one to have well-defined counts of
pseudoholomorphic curves and prove that these counts are invariant
under various choices. Note that the pseudoholomorphic condition on an
embedding of a surface is basically a partial differential equation,
so looking for pseudoholomorphic curves is basically counting
solutions to a PDE, but somehow it feels a little more concrete than
the PDEs involved in gauge theory.

Returning now to the project of finding a more “concrete” or
“geometric topologist friendly” understanding of gauge theoretic
invariants of smooth 4-manifolds, Taubes
[e11]
showed
that, when a 4-manifold supports a symplectic structure, its
Seiberg–Witten (SW) invariants can be calculated by counting
pseudoholomorphic curves with respect to a generic almost complex
structure dominated by that symplectic structure, in other words, a
Gromov invariant (Gr). This result is generally summarized as SW=Gr.
This has a range of spectacular implications, but when handed a random
4-manifold, it might not be so clear whether it has a symplectic
structure or not. On the other hand, Taubes,
Honda
[e16]
and others had observed that every
4-manifold with __\( b_2^+ > 0 \)__, which means that it contains surfaces
which have positive signed intersection number with any wiggled
versions of themselves, supports a “near symplectic” structure. This
is a closed 2-form which is symplectic away from a 1-dimensional
locus where it is zero, and along this locus it vanishes transversely
in the appropriate sense, so that its zero locus is a 1-manifold in
a neighborhood of which the 2-form has a standard model. Taubes
floated the idea that perhaps suitably defined Gromov invariants in
this setting would recover Seiberg–Witten invariants for arbitrary
4-manifolds with __\( b_2^+ \)__ positive, and initiated a
study
[e13]
of the behavior of pseudoholomorphic curves
in the standard local model near the vanishing locus. This line of
reasoning seems to have finally reached fruition with
Chris Gerig’s
work
[e26],
which is also a better resource than this paragraph
for a proper account of the history and the motivation for the idea.

Still, the fact that every 4-manifold with positive __\( b_2^+ \)__
supports a near symplectic structure was not constructive, however, so
if the SW=Gr plan panned out in the near symplectic setting it was
still not clear how useful that would be if one did not know how to
explicitly construct a near symplectic form on a particular
4-manifold. After our Sabino Canyon hike, Rob explained these ideas
to me and then Rob and I set out to solve this
problem
[2],
where “explicitly” meant to
us “starting from a framed link diagram describing a handle
decomposition of the 4-manifold” and then proceeding in something
that might loosely be called an algorithm.

It is important to point out here that in the end it is a bit
disingenuous to describe our construction as explicit because of one
key point; at a critical stage in our construction we had two
near-symplectic symplectic structures, one on one half of the
4-manifold and one on the other, and we needed to glue them
together along the separating 3-manifold. There is a standard way
to do this, using a *contact structure* on the 3-manifold as
the appropriate gluing boundary data. (If you don’t know what a
contact structure is, just know that it is the correct boundary data
for symplectic structures.) In order to glue, we needed to know that
the contact structures coming from the constructions on the two halves
were equal (or could be deformed to be equal). We knew this thanks to
Eliashberg’s
result
[e8]
that a certain
class of contact structures, known as *overtwisted contact
structures*, could be classified using algebraic topological
invariants, so we adjusted our construction appropriately to make sure
that the invariants coming from the two sides matched. However it is
not at all clear that Eliashberg’s machinery to go from knowing that
the invariants match to deforming (isotoping) the contact structures
to be equal is “explicit”.

#### 5. Broken Lefschetz fibrations

The auxiliary structures of concern here are Lefschetz fibrations; for
a good overview see
Gompf’s
exposition in the *Notices of the
AMS*
[e18].
In short, a Lefschetz fibration on an
oriented 4-manifold __\( X \)__ is a smooth map from __\( X \)__ to the 2-sphere
__\( S^2 \)__ which has finitely many isolated singularities, each of which is
locally modeled (respecting orientations) on the simplest singularity
that can arise in a holomorphic map from __\( \mathbb{C}^2 \)__ to
__\( \mathbb{C} \)__: __\( (z_1,z_2) \mapsto z_1^2+z_2^2 \)__. Away from these
singularities, the map thus looks like a surface bundle over a
surface, so locally like __\( \Sigma \times B^2 \to B^2 \)__ for some surface
__\( \Sigma \)__. To get a more topological understanding of what it means to
support a symplectic structure, Donaldson
[e10]
showed
that, after blowing up enough times, all symplectic 4-manifolds
support Lefschetz fibrations. (“Blowing up”, despite its name, is a
mild operation that changes a manifold in a very controlled way so as
to allow two surfaces that intersect to become disjoint, at the
expense of introducing a new surface, called the *exceptional
divisor*, that they both intersect; as long as one keeps track of the
exceptional divisor one still has access to the original manifold and
all of its topology.) With this in mind, Usher
[e15]
showed
that one could recover the full Gromov pseudoholomorphic curve count
by a certain count of pseudoholomorphic “multisections” of
symplectic Lefschetz fibrations, arguably making Taubes’ __\( \text{SW}=\text{Gr} \)__ result
slightly more meaningful to the average geometric topologist, and
giving us a slightly more topological way to think of what the
Seiberg–Witten invariants are counting.

Inspired by Taubes’ near-symplectic vision discussed in the preceding
section, Auroux, Donaldson and Katzarkov
[e17]
generalized
Donaldson’s Lefschetz fibration result to show that every
near-symplectic 4-manifold, after blowing up, has the structure of
a *broken Lefschetz fibration* (BLF), which is like a Lefschetz
fibration over __\( S^2 \)__ but also allows for a 1-dimensional locus of
singularities of a particular model, called *indefinite folds*.
This 1-dimensional singular set is exactly the same zero locus
where the near symplectic form vanishes. Following our noses, we
naturally wondered next how to construct these BLF’s “explicitly”
from, for example, a handle decomposition of a 4-manifold. In the
end
[3]
we discovered that if we allowed both
orientation preserving and orientation reversing local models for the
Lefschetz type singularities (we called the resulting fibrations
*broken achiral Lefschetz fibrations*, or BALFs), then we could
construct BALFs on *all* closed oriented 4-manifolds.

The construction was similar in spirit to our construction of near-symplectic forms: First we found a good way to split the given 4-manifold into two pieces on each of which we could construct partial fibrations and then we figured out the appropriate boundary conditions to govern these fibrations on the separating 3-manifold. (Experienced low-dimensional topologists will not be surprised that this boundary condition is the structure of an “open book decomposition”.) Then we massaged our constructions carefully so as to arrange that this boundary data agreed and we could glue the fibrations together. As before, it would be disingenuous to describe our construction as explicit because of this last step, which passed from open book decompositions to contact structures by way of the Giroux correspondence [e14] and then appealed again to Eliashberg’s classification [e8] of overtwisted contact structures.

Aside from this caveat about explicitness or the lack thereof, another thread the reader may be picking up is the increasing mixing of categories here: The study of almost complex structures, symplectic structures and Lefschetz fibrations already lies (a little bit uncomfortably) somewhere between complex algebraic geometry and differential topology. Throwing in circles along which new types of degenerations arise, and in the fibration case having both isolated complex type singularities as well as 1-dimensional decidedly nonholomorphic singularities, and then allowing the isolated singularities to flip orientations, really starts to seem like a bit of a stretch and an awkward mix of perspectives. The next phase in our collaboration started to clean this up.

#### 6. Morse 2-functions

The auxiliary structures of relevance here are, in a broad sense,
stable smooth maps from manifolds of some dimension to other manifolds
of lower dimension. A map is stable if small perturbations do not
change its essential qualitative features; more precisely, the
original map and the perturbed are equal after pre- and post-composing
with isotopies of the domain and range. Thus the function __\( y=x^2 \)__ is
stable in the world of smooth maps from __\( \mathbb{R} \)__ to __\( \mathbb{R} \)__
while the function __\( y=x^3 \)__ is not. Stable maps from smooth manifolds
to __\( \mathbb{R} \)__ (and sometimes to other 1-manifolds such as the
circle or a closed interval) are called *Morse functions* and have
been tremendously useful tools for probing the topology of smooth
manifolds in general, especially when used in conjunction with
gradient-like vector fields to give handle decompositions. Stable maps
to dimension 2 are also fairly well understood but have not been
used as extensively as Morse functions as a tool to probe the topology
of manifolds.

Perutz
[e20]
studied the problem of defining, in the
broken Lefschetz fibration setting, something analogous to the counts
of pseudoholomorphic multisections for Lefschetz fibrations studied by
Usher, motivated by the possibility that these would then again
recover Seiberg–Witten invariants, or at least be invariants even if
one did not know that they agreed with preexisting invariants. To
show directly that these counts, called Lagrangian matching
invariants, are in fact 4-manifold invariants, and did not depend
upon the choice of BLF, one would need to understand how to move from
one BLF on a given 4-manifold to another. In other words, one needs
not only existence results for B(A)LFs, but also *uniqueness*
results.

While we were thinking about this (working together at the African Institute of Mathematical Sciences in Muizenberg, South Africa, probably in 2007) we found out about Lekili’s insightful observation [e21] that if one wanted to connect two BLF’s by a 1-parameter family of smooth functions, one should first observe that Lefschetz singularities are actually not stable in the world of smooth maps. A small perturbation in the world of smooth maps will make a Lefschetz singularity into a much more complicated circle of singularities. Thus a generic 1-parameter family of smooth functions connecting two Lefschetz fibrations or BALFs should be expected to pass through functions that are much more general and do not have any Lefschetz singularities at all. The local behavior of stable smooth maps from 4-manifolds to 2-manifolds has in fact been understood for a long time and Lekili’s paper [e21] has an excellent appendix that runs through the general machinery for understanding stable maps between various dimensions [e3] applied in the case of dimensions 4 and 2. At least this is where I learned this material, Rob probably basically had already absorbed most of this by osmosis over the years and just needed a little reminder from Lekili’s article.

Lekili essentially answered the question of how to think about local
moves connecting one BALF to another, but the key problem was that in
the intermediate stages one might wander quite far away from the world
of BALFs and move through maps that are nothing like small
perturbations of BALFs. In particular, one might run into what are
called *definite folds*, discussed in more detail below. Whether
one could avoid definite folds is analogous to the question of
whether, in a 1-parameter family of functions connecting two given
Morse functions with the same number of minima and maxima, one can
avoid introducing extra minima or maxima along the way. In general
this can be done, modulo some obvious counterexamples and
low-dimensional exceptions. Because of the analogy with Morse theory,
Rob and I began calling stable maps to dimension 2 *Morse
2-functions*, thinking of them as vaguely like a 2-category
version of Morse functions, whatever that might mean. (Here we were
very much inspired by conversations with
Peter Teichner
at MSRI in the
spring of 2010.)

The key idea of a Morse 2-function is that locally a Morse
2-function __\( F \)__ on an __\( n \)__-manifold looks like a generic
1-parameter family __\( f_t \)__ of Morse functions on an
__\( (n-1) \)__-manifold, so that there are local coordinates so that __\( F(t,p)
= (t,f_t(p)) \)__ where __\( t \)__ is a single time parameter, __\( p \)__ is an
__\( (n-1) \)__-dimensional “spatial” coordinate, and __\( f_t \)__ is a generic
path of functions connecting two Morse functions. However, globally
there is no well-defined time direction, either in domain or range.
Generic 1-parameter families of Morse functions will be honest
Morse functions for all but finitely many times, and at finitely many
times will experience births or deaths of pairs of critical points, or
the coincidence of two critical points having the same critical value
at an instant. The tracks of the critical points for the honest Morse
functions are called *folds*, and are 1-dimensional in domain and
range, while the birth/death events are called
*cusps*; the
coincidences of having two critical points with the same critical
value are folds whose images in the range cross. The generic behavior
of homotopies between Morse 2-functions is exactly locally modeled
on the generic behavior of homotopies between homotopies between Morse
functions, in other words the moves of “Cerf theory”.

If the preceding paragraph did not mean much to the reader, the basic
idea is well illustrated with a picture of a Morse 2-function on a
surface. Figure 6 shows a Morse 2-function on a
torus. In fact, any time one draws a picture of a surface on a piece
of paper, one is of course presenting a map from that surface to
__\( \mathbb{R}^2 \)__ and, assuming genericity, this will be a Morse
2-function. The folds are literally the places where the surface
folds over, and the cusps are where folds “switch directions” in
some sense; these are all clearly visible in this illustration. The
preimage of a nonsingular point is an even number of points in the
torus, and the number of these points jumps by 2 when crossing a
fold. The only difference in higher dimensions in that different
dimensions and codimensions can fold in opposite directions along a
fold, and the preimages of nonsingular points are higher dimensional
submanifolds, not collections of points. In dimension 4, the
preimages of points are surfaces, and the topology of these surfaces
changes as one crosses a fold.

Note that the index of a fold ranges from 0 to __\( n-1 \)__, but is only
well-defined up to switching __\( k \)__ with __\( n-1-k \)__. A fold of index 0
(equivalently __\( n-1 \)__) is called a
*definite fold* and all other folds
are called
*indefinite*. Thus on 4-manifolds there are only really
two types of folds: definite index 0 (equivalently 3) folds and
indefinite index 1 (equivalently 2) folds. (On a 2-manifold,
all folds are definite.) One reason definite folds in dimension 4
might be undesirable is that they in general lead to disconnected
fibers, since crossing a definite fold in the index 0 direction
creates a new __\( S^2 \)__-component of the fiber. This (and __\( S^2 \)__-fibers
in general perhaps) is bad from a symplectic geometry and invariant
constructing perspective, as explained to us by
Katrin Wehrheim,
another major motivator for us in our early work in this subject.

The upshot is that we showed
[4]
how to
construct indefinite, fiber-connected Morse 2-functions on a given
__\( n \)__-manifold in a given homotopy class of maps, when __\( n > 3 \)__ and a
natural __\( \pi_1 \)__ condition is satisfied, and we also showed that any
two such Morse 2-functions can be connected by a generic homotopy
maintaining the indefinite and fiber-connected properties, again when
__\( n > 3 \)__. Of course, all along we were primarily interested in the case
__\( n=4 \)__ and, having settled this question, we now started to think
seriously about what to do with Morse 2-functions in general on
smooth 4-manifolds, how to put them into particularly nice forms,
and how to use them to appropriately probe the topology of the
manifolds involved.

#### 7. Trisections

I started my position at the University of Georgia (UGA) in August
2011. One thing Rob was always fantastic about was traveling to visit
me wherever I ended up, understanding that my family life with three
kids made it harder for me to up and travel at the drop of a hat. And
Rob had a long time connection to UGA going back to the early days of
the Georgia Topology Conference, so he started visiting me at UGA
pretty regularly (and making sure to combine each such visit with some
social time with old retired friends
Jim Cantrell
and
John Hollingsworth).
Over the course of a few visits in late 2011 and early
2012 we were busy drawing pictures of __\( \mathbb{R}^2 \)__-valued Morse
2-functions on homotopy 4-spheres arising from balanced
presentations of the trivial group not known to be Andrews–Curtis
trivializable. Our grand hope of course was to find an invariant that
could show that these homotopy 4-spheres were not diffeomorphic to
__\( S^4 \)__, and we had an idea that would involve counting certain surfaces
with boundary on the folds. (We called these surfaces
*flow charts*
because they would be swept out by flow lines of gradient-like vector
fields in local charts; this is a loose end idea that never panned out
for us, just in case the story was seeming too tidy with suspiciously
few false starts and dead ends.)

In the process we developed a very systematic and standardized way to
organize the image of a __\( \mathbb{R}^2 \)__-valued Morse 2-function
and, suddenly, one day trisections jumped out at us! In hindsight so
much of it is obvious but at the time it seemed somehow unusual and
dramatic. The key point is that we were busy building Morse
2-functions “from left to right”, starting from handle
decompositions of 4-manifolds, with the 0- and the 1-handles
at the left contributing a very standard part of the Morse
2-function, the 3- and 4-handles at the right contributing a
mirrored standard piece, and the 2-handles in the middle doing
something interesting. Then suddenly trisections jumped out at us when
we realized one day in my office at UGA that we could split things a
little differently by separating the cusps into three groups.

The basic idea is well illustrated by a half-dimensional cartoon. (I
believe that Rob taught all of his students and collaborators the
value of the half-dimensional analogy when working in dimension 4.)
In Figure 7, we show a sequence of Morse
2-functions on the torus, each obtained by performing some
procedure to the preceding one, and ending with one which might seem
unnecessarily complicated. However, if one then trisects the torus
using the last Morse 2-function, as illustrated in
Figures 8 and 9,
we see that in fact the torus is decomposed into three disks (careful
inspection shows that these disks are hexagons), and this is the basic
“trisection” of __\( T^2 \)__. We essentially mimic this procedure with more
care in dimension 4 to produce trisections of arbitrary
4-manifolds.

We released two versions of our paper on the arXiv with quite a gap between them and a significant difference. One of Rob’s life lessons for me was that if you prove an existence theorem then you should prove a uniqueness theorem. Our first version (May 2012) only had the existence result and then it took us more than a year to nail down the uniqueness result (September 2013). Somewhere in between those two dates was a hardworking session at the Max Planck Institute for Mathematics in Bonn, in which Andras Stipsicz had several helpful comments about how to proceed with uniqueness. Once we had the uniqueness result it was stupidly obvious, and the (for us) really startling fact was just how closely it mirrored the uniqueness result for Heegaard splittings of 3-manifolds.

Having “discovered” trisections, it did also become clear to us that
in many ways we were obviously rediscovering things others already
knew quite well. Of particular note is the fact that
Birman
and
Craggs
[e4]
essentially already thought about
trisections of 4-manifolds from the perspective of mapping class groups of surfaces,
using the term *triadic 4-manifolds*. (At least,
this term appears in a section heading in that paper, although it
never appears in the actual textual body of the paper. Perhaps they
had an intuition that there was something more going on with these
structures on 4-manifolds or even, as is so often the case, knew
much more than they wrote in their paper.) More recently, Oszváth
and Szabó were very obviously working with these structures in the
guise of Heegaard triples
[e19]
Our main
existence result is more or less implicit in Oszváth and Szabó’s
work, and one could interpret the thrust of our work as being to
understand the raw *topological* implications of the fact that
every 4-manifold can be described by a Heegaard triple, rather than
simply viewing Heegaard triples as a tool to understanding various
maps in Heegaard Floer theory. However the uniqueness result, and the
study of trisections as actual decompositions of 4-manifolds,
unifies the subject and sets the stage for many interesting directions
of study.

#### 8. Group trisections

Many other mathematicians have made striking contributions to the
theory of trisections since our first paper. Each of this
contributions is worth a lengthy discussion, but since we are focusing
on Rob Kirby’s work here, I’ll briefly describe the most important
extension to the theory that Rob and I were involved in, namely the
development of *group trisections*
[6]
in
collaboration with Aaron Abrams.

One of the first things we thought about with trisections was
understanding how much information about a smooth 4-manifold one
retains when applying algebraic topological functors to a trisection,
(to all the pieces and their intersections and the various inclusion
maps). Since Freedman taught us that in some special cases the
algebraic topology of 4-manifolds seems to basically record
homeomorphism type, we expected somehow that by passing from smooth
trisections to the algebraic topological shadows of trisections we
would pass from smooth information to topological information and then
be able to understand something about how trisections of homeomorphic
but not diffeomorphic manifolds are related. In fact it turned out,
after Aaron Abrams reminded Rob and me of some basic facts from
3-manifold topology, that the most basic algebraic topology functor
of all, __\( \pi_1 \)__, loses *no information at all*! The inclusion maps
between the triple intersections, double intersections, single pieces
and total manifold making up a trisection fit into a cube of spaces
and, when the __\( \pi_1 \)__ functor is applied, one gets a cube of groups
involving one surface group, six free groups, and the fundamental
group of the total 4-manifold and from this cube of groups one can
completely recover the smooth 4-manifold and its trisection.

We thus defined a
*group trisection* of a given group __\( G \)__ as a cube
of groups and
homomorphisms between them such that at one vertex we
have the fundamental group of a closed, oriented surface, at the
opposite vertex we have the group __\( G \)__, and at the six intermediate
vertices we have free groups. The maps all flow from the surface group
in the direction of __\( G \)__, all the maps are surjective, and all the six
faces of the cube are pushouts. The main result is that a trisected
group gives a trisected 4-manifold and vice versa. This is
striking; for example, a trisected group carries with it an integral
quadratic form, namely the intersection form of the associated
4-manifold.

The crucial piece of 3-manifold topology that Aaron reminded
us of
is the following “standard fact” (for a proof
see
[e12]):
Any surjective group
homomorphism from the
fundamental group of a genus __\( g \)__ surface to a free group of rank __\( g \)__
arises as the map induced by the inclusion of the surface as the
boundary of a genus __\( g \)__ handlebody, and the handlebody filling of that
surface is uniquely determined by the group surjection. This, together
with
Laudenbach
and
Poénaru’s
result
[e2],
that every
diffeomorphism of a connected sum of __\( S^1 \times S^2 \)__s extends across
a boundary connected sum __\( S^1 \times B^3 \)__s, is really all you need to
prove that a trisected 4-manifold can be completely recovered from
the associated group trisection.

In fact, everything we did is foreshadowed in, and inspired by, John Stallings’ famous paper [e1] “How not to prove the Poincaré conjecture”, which begins with the memorable line “I have committed the sin of falsely proving Poincaré’s Conjecture. But that was in another country and until now, no one has known about it.” One consequence of the dictionary between trisections of groups and trisections of 4-manifolds is that the smooth 4-dimensional Poincaré conjecture can be reformulated entirely group theoretically, just as Stallings reformulated the 3-dimensional Poincaré conjecture purely group theoretically. I argued unsuccessfully with my coauthors in favor of titling our group trisections paper “How not to prove the smooth 4-dimensional Poincaré conjecture”. I suspect that Rob, in particular, rejected my proposal because, deep down, he is a fundamentally optimistic mathematician and, whether the conjecture is true or false, he’s always willing to consider the possibility that some new idea really might be the right way forward to settling the question once and for all.

*David Gay was born in 1968 in Suacoco County in
upcountry Liberia, grew up in Liberia, England, Lesotho and
Massachusetts, ended up with a PhD in Mathematics from UC Berkeley in
1999, and has been a professor at the University of Georgia since
2011. He has thought about topology and taught mathematics at many
levels around the world, with postdoctoral positions at the University
of Arizona, the Nankai Institute of Mathematics, and the University of
Quebec, a Senior Lectureship at the University of Cape Town, and, most
recently, a visiting position as Hirzebruch Research Chair at the Max
Planck Institute of Mathematics in Bonn in 2019–20.
*