Trisections of 4-manifolds

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Universit`

a degli Studi di Pisa

DIPARTIMENTO DI MATEMATICA

Corso di Laurea in Matematica

Tesi di laurea magistrale

Trisections of 4-manifolds

Candidato

Giovanni Italiano

Relatore

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Introduction

A typical approach in low-dimensional topology is to decompose a manifold into simple pieces, in order to analyse them separately; in this way, it is sometimes possible to break down a hard problem into simpler ones. An example of this kind of techniques are Heegaard splittings: these are decompositions of a closed, oriented 3-manifold into two handlebodies, glued along their boundaries.

In this thesis we present the theory of trisections, an interesting decomposition for 4-manifolds that can be considered to be a 4-dimensional analogue to Heegaard splittings. Trisections have been introduced few years ago by David Gay and Robion Kirby in [2], and consist in dividing a 4 manifold X into three pieces X1, X2 and X3, diffeomorphic

to 4-dimensional handlebodies, with the property that each pairwise intersection Hij =

Xi ∩ Xj is diffeomorphic to a 3 dimensional handlebody and the triple intersection

Σ = X1∩X2∩X3 is a genus g orientable surface. In particular, the pairwise intersections

provide Heegaard splittings ∂Xi = Hij ∪ Hki. A very important fact about trisections

is that these Heegaard splittings suffice to determine the whole trisection; this allows to translate some 4-dimensional properties in the more understood context of tridimensional manifolds.

Gay and Kirby proved in [2] that every closed oriented 4-manifold admits a trisection; and showed how different trisections of the same manifold are related. It is easy to check that trisections are not unique: there is a stabilization operation that changes the trisection, increasing the genus of the central surface. However, any two trisections of the same manifold have a common stabilization.

An interesting topic about trisections is the theory of how embedded surfaces in 4-manifolds behave with respect to trisections. In [9], Jeffrey Meier and Alexander Zupan showed that any surface embedded in S4 _{can be put in “good” position with}

respect to the trivial trisection, and in [10] they improved the result, extending it to every trisection of any 4-manifold. This allows to study surfaces in 4-manifolds from a tridimensional point of view, and led to a topological proof of the Thom Conjecture (by Peter Lambert-Cole, in [7]).

The outline of the thesis is the following:

In Chapter 1 we present the tools that are necessary to understand the theory of trisections, with a special focus on Heegaard splittings, Kirby diagrams for handle de-compositions of 4-manifolds and the theory of Morse functions and Morse 2-functions.

Chapter 2 is devoted to present the definition of trisections, some basic properties and some simple examples.

In Chapter 3 we will prove the Existence Theorem, and the theorem of uniqueness up to stabilizations and isotopies.

Chapter 4 will be dedicated to the presentation of the theory of bridge trisections for knotted surfaces.

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1

Preliminaries

The goal of this chapter is to briefly present the topics and tools that we are going to use further in the thesis. Our focus will be aimed mainly towards the theory of handle decomposition of 3- and 4-manifolds, a more complete and comprehensive discussion can be find in [4]. The final part of the chapter is dedicated to present the Theory of Morse 2-functions (developed by Gay and Kirby in [3]), that will be the fundamental tool for the proof of existence of trisections.

1.1 Handle decomposition

The handle decomposition is a way to study a differentiable manifold by dividing it into smaller pieces: we imagine the manifold to be built by subsequent attachments of simple parts, called handles. This decomposition will allow us to represent manifolds by combinatorial pictures, from which topological properties can often be easily deduced.

Definition 1.1.1. A n-dimensional k-handle is a copy of Dk× Dn−k _{to be attached to}

a (possibly empty) n-manifold M along a smooth embedding f : ∂Dk× Dn−k _{→ ∂M .}

The disks Dk× {0} and {0} × Dn−k _{are called respectively the core and the cocore of}

the handle, the sphere ∂Dk× {0} = Sk−1 _{× {0} is called the attaching sphere (while}

∂Dk× Dn−k _{is called the attaching region), and {0} × ∂D}n−k _{= {0} × S}n−k−1 _{is called}

the belt sphere of the handle.

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Definition 1.1.2. A handle decomposition of a compact manifold M is a sequence of

handle attachments that, starting from the empty manifold, results into M . Sometimes, the handle decomposition is taken by choosing some boundary components ∂−M and

starting from ∂−M × I.

Remark 1.1.3. Handles are indeed attached along (a tubular neighbourhood of) their

attaching sphere:

• 0-handles have an empty attaching sphere, so they can only be attached to an empty manifold; the attachment of a 0-handle consists in the appearance of a new connected component in M .

• 1-handles are the only ones having a disconnected attaching sphere; this implies that they are the only one that can lower the amount of connected components.

• n-handles are the only ones whose attaching region is the whole boundary; this implies that any handle decomposition of a closed manifold has at least one n-handle.

• It is possible to turn upside down a handle decomposition of a closed manifold, by interpreting each k-handle as a n − k-handle and reversing the order of the attachments. In this dual interpretation, the attaching sphere and the belt sphere are swapped, as well as the core and the cocore.

Using Morse theory, it is possible to prove that every compact manifold admits a handle decomposition. This result can be refined:

Theorem 1.1.4. Every compact n-manifold M admits a handle decomposition with a

single 0-handle and at most one n-handle. Moreover, the order of attachment can be modified in a way such that handles are attached in increasing index order, and such that handles with the same index are attached simultaneously.

1.1.1 Handle moves

A manifold can have several different handle decompositions: for example it is always possible to substitute a 0-handle with two 0-handles connected by a 1-handle such as in Figure 1.2.

Figure 1.2: Three different handle decompositions of D3. The first one consist in a 0-handle, the second one in two 0-handles connected by a 1-handle, and the third one has a 1-handle and a 2-handle in cancelling position.

However, any two different handle decompositions of the same manifold are connected by isotopies and moves of the following types:

• Adding/Deleting a pair of cancelling handles (Figure 1.2): A k-handle h_k and a (k + 1)-handle h_k+1 are said to be in cancelling position if the attaching

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1.2. Heegaard splittings

sphere of hk+1 intersects the belt sphere of hk transversely in a single point. The

attachment of a pair of handles in cancelling position is equivalent to perform a boundary connected sum of a D3, so it does not change the manifold.

• Handle slides: Given two handles with the same index 0 < k < n, it is possible to slide one of them over the other as shown in Figure 1.3. The move consists in isotoping the attaching sphere of the first handle, pushing it over the belt sphere of the second one.

Figure 1.3: Example of a handle slide of 2-dimensional 1-handles.

1.2 Heegaard splittings

In this section we will present Heegaard splittings, which are a decomposition of a three manifold into two very simple pieces, called handlebodies. It could seem surprising that every closed 3-manifold can be divided in such simple pieces but, as we will see in the proof of Theorem 1.2.5, Heegaard splittings are just a reformulation of handle decompositions.

From now on, we will indicate with the symbol # the connected sum of two manifolds (i.e. eliminating a ball from both manifolds, and attaching them along the spheric boundary component we have created), and with the symbol #_∂the boundary connected sum of two manifolds (a generalization of the previous operation to manifolds with non-empty boundary). The symbols #k and #k_∂ mean we are performing a connected sum of k copies of the manifold.

Definition 1.2.1. The n-dimensional handlebody of genus k is the compact orientable

manifold #k_∂Dn−1× S1_.

Remark 1.2.2. A genus k handlebody has a handle decomposition with one 0-handle and k 1-handles.

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Remark 1.2.3. The boundary of a genus k handlebody is ∂(#k_∂Dn−1× S1_{) = #}k_(Sn−2_×

S1). In the particular case of 3-dimensional handlebodies, the boundary of a genus k handlebody is a surface of genus k.

Given two compact manifolds M₁, M₂ and a homeomorphism f : ∂M₁ → ∂M₂, we can glue together M1 and M2, obtaining a new closed manifold M = M1∪f M2. When

M1 and M2 are 3-dimensional handlebodies, this decomposition is called a Heegaard

splitting for M :

Definition 1.2.4. Let M be a closed, oriented 3-manifold, and let g ∈ N. A Heegaard

splitting for M is a decomposition of M in two submanifolds M = H₁∪ H₂ satisfying the following properties:

1. Hi is a 3-dimensional handlebody of genus g;

2. Σ := H1∩ H2 = ∂H1 = ∂H2 is a surface of genus g.

Theorem 1.2.5. Any closed, orientable 3-manifold M admits a Heegaard splitting.

Proof. M admits a handle decomposition with only one 0-handle and one 3-handle. The

manifold obtained by attaching all the 1-handles to the 0-handle is a handlebody, and its genus is equal to the number of 1-handles. In the same way, by seeing them upside down, the manifold obtained by attaching all the 2-handles to the 3-handle is a handlebody, and its genus is equal to the number of 2-handles. Then, the manifold M is obtained exactly by glueing those two handlebodies along their boundary.

Definition 1.2.6. The Heegaard genus of a closed orientable 3-manifold is the minumum

g such there exist a genus g Heegaard splitting of M .

Example. The standard equatorial sphere divides S3 in two copies of the 3-ball D3, providing a genus 0 Heegaard splitting of the 3-sphere.

The complement of standard embedded solid torus in S3 (i.e. tubular neighborhood of an unknot) is a solid torus. Together, these tori give a genus 1 Heegaard splitting, which we will call the standard genus 1 splitting of the S3. From the point of view of handle decompositions, this new splitting is obtained by adding a cancelling pair of 1-and 2-h1-andles to the genus 0 one.

Example. Lens spaces can be obtained by glueing two solid tori along their boundaries:

this decomposition is a genus 1 Heegaard splitting.

1.2.1 Heegaard diagrams

It is possible to encode all the information of a Heegaard splitting (up to isotopy) in the combinatorial data of a genus g orientable surface and two g-tuples of non-separating curves (α₁, . . . αg) and (β1, . . . βg).

Definition 1.2.7. Let H be a 3-dimensional handlebody of genus g, D = {D1, ..., Dg} a

set of properly embedded disks in H, and let N (D₁), . . . , N (D_g) be tubular neighbour-hoods of the disks. D is said to be a set of compressing disks if H \ (N (D1) ∪ · · · ∪ N (Dg))

is a 3-ball. The boundaries of these disks can be identified with some disjoint simple closed curves (α₁, . . . , αg) in ∂H, called a set of defining curves for the handlebody H.

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1.2. Heegaard splittings

(a) S3 (b) S2× S1 _{(c) L(1, 1)}

Figure 1.4: Some examples of genus 1 Heegaard diagrams

Let N be a 3-manifold with a boundary component S ' S_g, H a 3-dimensional handlebody of genus g, f : ∂H → S a diffeomorphism, and D = {D1, ..., Dg} a set of

compressing disks for H. Finally, let (α1, . . . , αg) be the images through f of the defining

set of curves corrisponding to D. We can obtain the manifold M = N ∪_fH from N and

the curves (α1, . . . , αg) by attaching g 2-handles along the curves α1, . . . , αg and capping

the manifold with a 3-handle.

By applying the previous construction to glue two handlebodies to the manifold

N = Sg × [−1, 1], we can observe that a Heegaard splitting is determined by two

g-tuples of curves (α1, . . . , αg) in Sg× {−1} and (β1, . . . , βg) in Sg× {1}. We can push

those curves in the central surface S_g× {0} to obtain what is called a Heegaard diagram for the splitting:

Definition 1.2.8. A Heegaard diagram is a triple (Σ, α, β) where Σ is a closed oriented

surface of genus g, α = (α₁, . . . , αg) and β = (β1, . . . , βg) are two g-tuples of disjoint

closed simple curves such that cutting Σ along either α or β leaves the surface connected.

Notation. We will draw the diagram using red and blue to distinguish α and β curves.

It is possible to obtain a Heegaard diagram from a Heegaard splitting by taking defining sets of curves for the two handlebodies on the central surface of the splitting. Conversely, given a Heegaard diagram, it is possible to recover the corresponding Hee-gaard splitting by taking Σ × [−1, 1], pushing the α curves on Σ × {−1}, the β curves on Σ × {1} and attaching 2- and 3-handles as in the construction previously described.

The operation of connected sum can be extended in a natural way to Heegaard splittings: it is sufficient to choose a standard bisected ball in both manifolds and perform the connected sum using that ball. In this way, given Heegaard splittings of M and N with diagrams (Σ, α, β) and (Σ0, α0, β0), there is a natural Heegaard splitting for M #N : its handlebodies are the boundary connected sum of the handlebodies for M and N and a corresponding diagram is (Σ#Σ0, α ∪ α0, β ∪ β0) (connected sum of the diagrams for M and N ).

1.2.2 Stabilization

It is easy to see that Heegaard splittings are not unique (different handle decompositions of a manifold result in different splittings). We now present the stabilization, a key operation in the classification of splittings for a given manifold.

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Definition 1.2.9. A stabilization is a move that turns a genus g Heegaard splitting

M = H1∪ H2 into a genus g + 1 Heegaard splitting of the same manifold by taking a

boundary parallel, unknotted arc in H₂ and adding a (closed) tubular neighbourhood of it to H1 (thus subtracting it from H2). This operation does not depend on the arc

chosen, and is equivalent to adding a pair of cancelling 1- and 2- handles to the handle decomposition of M .

Remark 1.2.10. Stabilization is equivalent to perform a connected sum with a standard

genus 1 Heegaard splitting of S3. In particular, a diagram for the stabilization can be obtained performing a connected sum with the diagram in Figure 1.4a.

We will say that two Heegaard splittings are stably equivalent if they become isotopic after applying some stabilizations to each of them. The following theorem, proved by K. Reidemeister [11] and J. Singer [12] shows that stabilizations and isotopies are sufficient to classify all Heegaard splittings:

Theorem 1.2.11 (Reidemeister-Singer). Two Heegaard splittings of the same manifold

are stably equivalent.

Remark 1.2.12. It could be necessary to apply stabilization to both splittings, since

there exist manifolds which admit non-isotopic splitting of the same genus. This will not concern us, since this phenomenon does not arise with the manifolds we are interested in.

1.2.3 Heegaard splittings for #k_(S2_{× S}1₎

The manifolds of the type #k(S2× S1_{) are the boundary of some 4-dimensional}

handle-body, so they will play an important role in the study of trisections. In this section, we will provide a classification of their Heegaard splittings up to isotopy.

Definition 1.2.13. Let 0 ≤ k ≤ g be two integers. The (g, k)-standard Heegaard

diagram is a diagram obtained by connected sum of k copies of the standard genus 1 diagram for S2× S1 _{and k copies of the standard genus 1 diagram for S}3_{. This diagram}

corresponds to a genus g splitting for #k(S2 × S1_{): it is the one already described,}

stabilized g − k times.

This definition can be slightly weakened to be more versatile:

Definition 1.2.14. Let 0 ≤ k ≤ g be two integers. Two g-tuples of curves α and β in

a Heegaard diagram are said to be in (g, k)-standard position if all the intersections are transverse and (possibly after rearranging the indices) the following conditions hold:

• α_i k β_i for i ≤ k;

• |αi∩ βj| = δij for i > k and 1 ≤ j ≤ g.

A diagram with these properties can be changed to a (g, k)-standard Heegaard diagram using isotopies and handle slides, so it represents the standard genus g splitting for #k(S2× S1_).

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1.3. Kirby calculus

Figure 1.5: The (g, k)-standard Heegaard diagram

A classical result from Waldhausen provides a complete classification of all Heegaard splittings of the 3-sphere:

Theorem 1.2.15 (Waldhausen, [13]). Every Heegaard splitting of S3 is a stabilization of the one of genus 0.

Using the theory developed in [8], this result can be generalized to all the boundaries of 4-dimensional handlebodies:

Theorem 1.2.16. The manifold #k(S2× S1_{) has a unique genus g Heegaard splitting}

for each g ≥ k, described by the (g, k)-standard Heegaard diagram.

1.3 Kirby calculus

The aim of this section is to display a system to present a handle decomposition of a compact orientable 4-manifold, using a bidimensional picture together with some com-binatorial structure.

1.3.1 Handle attachment in dimension four

We have said that in order to define the attachment of a handle it is necessary to have an embedding defined on the attaching region; however, if the dimension does not exceed 3, it is sufficient to specify an embedding defined just on the attaching sphere.

In dimension 4 this is still true for all handles except 2-handles: in this case the embedding of the attaching sphere gives a knot into the 3-manifold ∂M , and in order to extend the embedding to the attaching region, it is necessary to choose a trivialization of its normal bundle. The choice of such a trivialization can be done by selecting a preferred longitude in the tubular neighbourhood of the knot; if ∂M = S3 this choice can be expressed by an integer number (the linking number of the knot and its longitude).

Theorem 1.3.1. Given a compact 4-manifold, there is at most one closed manifold that

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Remark 1.3.2. By turning the handle decomposition upside down, attaching 3- and

4-handles is equivalent to glue some 4-dimensional handlebodies to the manifold along their boundaries. This is possible if and only if each boundary component of the manifold is diffeomorphic to #k(S2× S1_).

The previous theorem is based on the fact that every orientation preserving self-diffeomorphism of #k(S2_{× S}1_{) can be obtained as composition of isotopies and handle}

slides of #k(D3× S1_{) = D}4_{∪ 1-handles. The proof of this fact uses basic 3-manifold}

theory, and can be found in [8]. The theorem implies that to describe a closed 4 manifold via its handle decomposition it is sufficient to show how the 0-, 1- and 2-handles are attached, and then state that some 3- and 4- handle are used to close the manifold.

1.3.2 Kirby diagrams

To represent our handle decomposition, we will assume that there is only one 0-handle, and use its boundary S3 _{= R}3 ∪ {∞} to draw the attaching regions of the 1- and 2-handles.

The attaching region of each 1-handle is D3t D3_{, and it can be represented as a pair}

of (round) balls. To separate different 1-handles, we will arrange them as in Figure 1.6: each pair is symmetric with respect to the plane {x = 0}, and the balls are identified using the symmetry with respect to that plane.

We can now draw the attaching region of the 2-handles as a framed link. The framing will be depicted by assigning an integer number to each component of the link, as anticipated in the beginning of this section.

Unfortunately, the notation of 1 handles as pair of ball causes some problems related to the framings of 2-handles, so we will present also a different one. The main idea is that, since a 1-handle can be cancelled with the attachment of an appropriate 2-handle, the attachment of a 1-handle can be represented as the elimination of a 2-handle: this can be done by taking a disk D in the boundary of the manifold, pushing its interior into the manifold, and removing a tubular neighbourhood of D. In our diagrams we will draw the boundary of the disk to be eliminated as a dotted circle, to distinguish it from 2-handles.

It could be helpful to visualize the 1-handle given by the dotted circle as two balls squeezed together to resemble the shape of an Oreo cookie: with some further flattening, we can substitute them with a disk (whose boundary is represented by the dotted circle), almost forgetting that they used to be two distinct objects. Using this model, it is easy to see that a curve passing through the dotted circle is passing through the corresponding 1-handle.

1.3.3 Kirby moves

We now provide a small summary describing how handle moves change a Kirby diagram. A more detailed description can be find in [4], Chapter 5.

• Handle slides: We will describe the situation just for 2-handles. In a diagram, sliding h1over h2 (with attaching spheres determined by the knots K1 and K2, and

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1.3. Kirby calculus

Figure 1.6: The same Kirby diagram, using the two different notations for 1-handles.

We consider the parallel curve K₂0 determining the framing for K₂: this curve bounds a disk even after the attachment of h2. Sliding h1 by isotoping K1 over

that disk produces the diagram in Figure 1.7. We can perform a connected sum using any band (disjoint from the rest of the attaching link) connecting K₁ and

K₂0 to get a handle slide. With this operation, the framing of K1 changes to

n1+ n2± 2lk(K1, K2) (the ± depends on how many semi-twists the band has).

• 1/2-handle cancellation: A 1-handle h1 and a 2-handle h2 are in cancelling

position if the attaching sphere of h₂ intersects the belt sphere of h₁ transversely in a single point (regardless of the framing of h2). However, in order to be able

to simply erase the two handles from the diagram (without other changes), it is necessary that h₂ is the only 2 handle whose attaching sphere intersects the belt sphere of h1. This situation can always be reached using a sequence of handle

slides over h2 to separate all the other 2-handles from h1.

• 2/3-handle cancellation: Any unknotted, unlinked, 0-framed component of the attaching link for 2-handles can be (and in case we close the manifold, will be) cancelled by a 3-handle. As a consequence of this, any unknotted, unlinked, 0-framed 2-handle in a Kirby diagram for a closed manifold can be eliminated without changing the resulting manifold.

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Figure 1.8: Handle decomposition of S1× S3_{. We have written ∪ 3-handles ∪ 4-handles}

intending that the manifold should be closed by using those handles.

1.3.4 Examples

We will show some examples of Kirby diagrams for simple manifolds, to better under-stand how these pictures work.

• S4_{: The equatorial sphere decompose S}4_{as the union of a 0-handle and a 4-handle.}

The diagram for this decomposition is an empty picture ∪ 3-handles ∪ 4-handles.

• S1_{× S}3_{: Writing S}3 _{as the union of two disks glued along their boundary, we}

have the decomposition of S1 × S3 _{as the double of the handlebody S}1 _{× D}3_.

Writing one of the handlebodies as a 0-handle and a 1-handle and the other one as a 4-handle and a 3-handle, we get a diagram representing a single 1-handle, ∪ 3-handles, ∪ 4-handles, as shown in Figure 1.8.

• T × D2_{: We start with the 2 dimensional handle decomposition for the torus T}

given in Figure 1.9. Following the attaching sphere of the 2-handle (clockwise, starting from the black dot) we cross the blue handle, the red one, then the blue one in the opposite direction, and finally the red one in the opposite direction. Figure 1.10a is a 4-dimensional replica of the previous picture (equivalent to the same handle decomposition, times D2), since the green curve crosses the 1-handles in the same order with framing 0. Finally, using an isotopy, we can arrange the diagram to be as in Figure 1.10b.

• Sg× D2: The argument presented above can be generalized to surfaces of arbitrary

genus, providing the Kirby diagram in Figure 1.11.

1.4 Morse 2-functions

In this section, we will briefly summarize the theory of Morse functions and Morse 2-functions, that will be largely used in Chapter 3. A more comprehensive discussion can be found in [1] and [3].

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1.4. Morse 2-functions

Figure 1.9: Handle decomposition of the torus S1_{× S}1 _{with one 0-handle (white), two}

1-handles (blue and red) and one 2-handle (yellow). The green curve is the attaching sphere of the 2-handle.

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Figure 1.10: Kirby diagrams for T × D2.

Figure 1.11: Kirby diagram for Sg× D2. The dots denote the fact that there are g pairs

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1.4.1 Morse functions

Definition 1.4.1. Let M be a n-manifold, N an oriented 1-manifold, and g : M → N a

smooth function. The function g is said to be locally Morse if, for each critical point p of g, there exist k ≤ n and coordinates in a neighbourhood of p and g(p) with respect to which g(x1, . . . , xn) = x21+ · · · + x2k− x2k+1− · · · − x2n. The number k is called the index

of the critical point p. A Morse function is a map g : M → N which is proper, locally Morse and such that distinct critical points have different critical values.

Remark 1.4.2. We will often consider the case when the manifold N is I = [0, 1] or R. In

the first case, we imply that M is a cobordism from ∂₊M = g−1(0) to ∂−M = g−1(1).

Morse functions are strictly related to handle decompositions. Given a Morse func-tion f , we can interpret the line R as a set of times (i.e. at the time t we have constructed the manifold f−1([−∞, t])): in this point of view, near regular points the manifold is just a product of a n − 1 manifold times I, while index-k points correspond to attachments of a k-handle to the manifold.

1.4.2 Homotopies of Morse functions

We want to discuss homotopies gtbetween Morse function that are not necessarily Morse

functions for some intermediate times. We will consider functions G : I × M → I × N of the form G(t, p) = (t, g_t(p)), with a special emphasis on their fold locus Z_G (i.e. the set of critical points of the functions gt), and we will restrict to a particular class of

functions:

Definition 1.4.3. A homotopy g_t : Mn → N1 _{(t ∈ [0, 1]) is said to be a generic}

homotopy between Morse functions if it satisfies the following properties:

• The functions g0 and g1 are Morse functions;

• The function g_t is Morse, except for finitely many values of t;

• At a value t∗ such that gt∗ is not Morse, exactly one of the following events occurs:

1. Two critical values cross: the function gt∗ is locally Morse but not Morse,

and the fold locus Z_G∩ [t∗− , t∗+ ] × M is composed by some arcs: G acts

as an embedding on every arc except two of them, which are mapped with a transverse intersection in a point. This singularity is called 1-parameter

crossing.

2. A pair of cancelling critical points is born (or dies): for each t ∈ [t∗− , t∗+ ],

the function gt is Morse outside of a ball, and there are cordinates inside the

ball such that g_tis modelled as

gt(x1, · · · , xm) = −x21− · · · − x2k+ x3k+1± (t − t∗)xk+1+ x2k+2+ · · · + x2m.

Note that for t 6= t∗ the function gt is Morse, and crossing t∗ the number

of critical points inside the ball changes from 0 to 2. The fold locus Z_G∩ [t∗− , t∗+ ] × M is composed by some arcs: all arcs but one have endpoints

mapped by G on the two opposite vertical sides, and one is mapped with both endpoints on the same side, in the form of a semicubical cusp. This

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Figure 1.12: An example of Cerf graphic with three cusps and two crossings.

type of singularity is called 1-parameter birth (1-parameter death if the time is reversed).

Generic homotopies between Morse functions can be represented by diagrams (called Cerf graphics) showing the trajectory of the critical values and their corresponding in-dices. An example of such diagram is given in Figure 1.12.

Finally, to simplify the notation, we give this definition:

Definition 1.4.4. An arc of Morse functions is a homotopy g_t that is Morse for all t.

1.4.3 Homotopies of homotopies of Morse functions

We are now interested in discussing homotopies gs,tbetween generic homotopies of Morse

functions, with the relaxed condition that they do not need to be generic for all values of the parameter s. As before, we can consider the functions G_s: I ×M → I ×N defined by

Gs(t, p) = (t, gs,t(p)) and G : I × I × M → I × I × N defined by G(s, t, p) = (s, t, gs,t(p)),

and their singular loci ZGs and ZG. To help visualize the singularities that can arise

in homotopies between homotopies, we will provide some pictures to emphasize what happens in the fold locus around a point gs∗,t∗(p). Each figure will be composed by

three squares representing, from left to right, the fold locus for s < s∗, s = s∗ and

s > s∗: in each square will be drawn a graphic such the one in Figure 1.12, with the

horizontal axis representing the time t, and each vertical slice representing a Morse function.

Definition 1.4.5. Given two manifold Mm and N1, and two generic homotopies g0,t :

M → N (between Morse functions g0,0 and g0,1) and g1,t : M → N (between Morse

functions g1,0 and g1,1), we say that a 2-parameter family {gs,t}s,t∈I is a generic

ho-motopy between generic homotopies between Morse functions if it satisfies the following

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• {gs,0}s∈[0,1] and {gs,1}s∈[0,1] are arcs of Morse functions.

• Except for finitely many values of s, {gs,t}tis a generic homotopy of Morse functions

between gs,0 and gs,1.

• For each value s∗ of s for which {gs∗,t}t∈ [0, 1] is not a generic homotopy, there is

a value t∗ such that {gs∗,t}t∈ [0, t∗) and {gs∗,t}t∈ (t∗, 1] are generic homotopies.

Moreover, around each one of these points (s∗, t∗) ∈ I × I, exactly one of the

following events occurs (possibly with the parameters s and/or t reversed):

1. 2-parameter coincidence: The family {gs∗,t}t fails to be a generic homotopy

just because two of the events described in Definition 1.4.3 happen simulta-neously at the time t∗. The 2-parameter crossing has to be transverse, in the

sense that the event happening before the other one for s > s∗ must happen

after the other one for s < s∗ (see Figure 1.13).

Figure 1.13: An example of 2-parameter coincidence: at s = s∗ the crossing happens at

the same time of a birth of critical points

2. Reidemeister-II fold crossing: The function g_s∗,t∗ is locally Morse, but {g_s∗,t}_t is not a generic homotopy because the fold locus ZGs∗ is mapped with a

single non-transverse quadratic double point. As before, this should happen transversally in s, in the sense that for s < s∗ there are no double points

around the soon-to-be quadratic point, and for s > s+the folds intersect into

two transverse double points. This is shown in Figure 1.13.

Figure 1.14: A crossing of two folds with a quadratic double point. Notice that there is no constraint for the indices of the two folds. In each vertical slice involved, the corresponding Morse function swaps two critical values: this implies that there must be some condition on the attaching sphere of the corresponding handles.

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3. Reidemeister-III fold crossing: The function gs∗,t∗is locally Morse, but {gs∗,t}t

is not a generic homotopy because the fold locus Z_G_s∗ is mapped with a single transverse triple point. As before, this should happen transversally in s, in the same sense of the previous cases. An example of this situation is shown in Figure 1.15.

Figure 1.15: A crossing with a triple point. As before, there is no constraint on the indices of the folds.

4. Cusp-fold crossing: The family {gs∗,t}tfails to be a generic homotopy because

there is a birth (or death) point having the same critical value of another critical point. This means that the fold locus ZGs∗ is mapped with a

non-transverse double point involving a cusp and a non-cusp. As always, we require this phenomenon to happen transversally with respect to s, so that the cusp crosses the fold. An example is shown in Figure 1.16.

Figure 1.16: A left cusp crossing a fold from right to left. The only constraint of the indices is given by the fact that a cusp must have folds of consecutive indices k and k + 1.

5,6,7. Singularities: The function gs∗,t∗ is Morse except in a point p, and for s, t

such that |s − s∗| < δ, |t − t∗| < there are coordinates around p and gs∗,t∗(p)

with respect to which g_s∗,t∗ is modelled in a specific way, depending on the type of the singularity. Moreover, we request that there are no other critical points of gs,t in a inverse image of a neighbourhood of gs∗,t∗(p).

5. Birth/Death of an eye singularity: The local model is gs∗,t∗(x1, . . . , xm) =

−Pk

1x2i + x3k+1+ (t − t∗)2xk+1− (s − s∗)xk+1+Pmk+2x2i. This is shown in

Figure 1.17: at s = s∗ there is a birth of two cusps joint together in the

shape of an eye (this means that, for s > s∗, in the homotopy (gs,t)ta pair of

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Figure 1.17: The birth of an eye. Since they are joined by cusps, the two folds must have consecutive indices.

6. Merge/Unmerge singularity: The local model is given by gs∗,t∗(x1, . . . , xm) =

−Pk

1x2i + x3k+1− (t − t∗)2xk+1− (s − s∗)xk+1+Pmk+2x2i. This is shown in

Figure 1.18, and consists into two cusps (with the same indices) that merge together (or two parallel folds with consecutive indices that separate into two cusps, if s is reversed).

Figure 1.18: Two cusps merge together: for s < s∗ a cancelling pair dies and it is

created again, while for s > s∗ the cancelling pair continues to live (playing the role that

each cancelling pair had in the previous situation). As always, the two folds must have consecutive indices.

7. Birth/Death of a swallowtail singularity: The local model is g_s∗,t∗(x₁, . . . , xm) =

−Pk

1x2i + x4k+1− (s − s∗)x2k+1+ (t − t∗)xk+1+Pmk+2x2i. This is shown in

Figure 1.19: a fold develops two new cusps and a crossing.

Figure 1.19: The birth of a upwards swallowtail: two cusps and a crossing are added to the fold. As always, the folds separated by cusps must have consecutive indices.

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homotopies gs,t such that for all fixed values of the parameter s is a generic homotopy

(in the parameter t).

1.4.4 Morse 2-functions

We are now ready to define Morse 2-functions:

Definition 1.4.7. A Morse 2-function is a smooth proper map G : X → Σ from a

n-manifold X to a 2-n-manifold Σ, such that every point q ∈ Σ has a compact neighbourhood

S such that there exist a (n − 1)-manifold M and two diffeomorphisms φ : S → I × I

and ϕ : G−1(S) → I × M such that the map φ ◦ G ◦ ϕ−1: I × M → I × I is of the form (t, p) 7→ (t, gt(p)) for a generic homotopy of Morse functions gt: M → I.

Critical points for G are called fold points if they admit a model homotopy that is Morse at that point, and are called cusp points if they are modelled as a homotopy with a birth/death at that point.

If the surface has boundary ∂Σ 6= ∅, we can see it as a cobordism between (possibly empty) 1-manifolds N0 and N1. In this case, we request that also X is a cobordism

between M₀ and M₁, and that the Morse 2 function G preserves the cobordism structure (i.e. G−1(Ni) = Mi and G|Mi : Mi → Ni) is a Morse function). In the case Σ is a

cobordism between cobordisms (the main example being Σ = I × I), we request such a structure also in X, and that G preserves it.

Remark 1.4.8. Unlike the case of generic homotopies between Morse functions, the index

of a fold is not well defined: it could happen that there are two different models in which the fold have different index. However, the index is well defined once we have chosen a transverse direction to the fold (meaning that such direction correspond to the upwards direction in the local model).

In a figure picturing the singular locus of a Morse 2-function, we will draw a transverse arrow on each fold and write a number to indicate the index relative to the direction.

Definition 1.4.9. A 1-parameter family Gs: X → Σ is said to be a generic homotopy

between Morse 2-functions if for each point q ∈ Σ and for each s ∈ I there exist > 0, a

compact neighbourhood S of q, a (n−1)-manifold M , a diffeomorphism φ : S → I ×I and a 1-parameter family of diffeomorphisms ϕs: G−1s (S) → I × M such that for |s −s| < 

the family φ ◦ Gs◦ ϕ−1s : I × M → I × I is of the form (t, p) 7→ (t, gs,t(p)) for a generic

homotopy between homotopies between Morse functions g_s,t.

In practice, we will interpret the crossings and the singularities in the previous section as moves that we can execute on a Morse 2-function, to change the shape of the fold locus without changing the manifold X. Sometimes these moves require additional conditions: for example a Reidemeister-II move swaps two critical points in the (n − 1)-dimensional Morse point of view, and this is not always achievable (to have an example of this, it is sufficient to consider a cancelling pair of 1-2-handles, that are forced to be attached in a specific order).

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2

Trisections

2.1 Definition

This chapter is dedicated to the presentation of the definition and basic properties of trisections. The original definition by D. Gay and R. Kirby in [2] involves a decompo-sition of the 4-manifold in three handlebodies of the same genus; however we will view it from a slightly more general point of view, allowing the three handlebodies to have different genera.

Definition 2.1.1. Let X be a closed, connected, oriented 4-manifold, and let k1, k2, k3

and g be integers such that 0 ≤ ki ≤ g. A (g; k1, k2, k3)-trisection of M is a decomposition

of X in three submanifolds X = X₁∪ X₂∪ X₃ satisfying the following properties: 1. X_i is a 4-dimensional handlebody of genus k_i for each i = 1, 2, 3;

2. Hij = Xi∩ Xj is a 3-dimensional handlebody of genus g for each i 6= j;

3. Σ = X₁∩ X₂∩ X₃ is a close orientable surface of genus g.

The number g is called the genus of the trisection, and S = H₁₂ ∪ H₂₃ ∪ H₃₁ is called the spine of the trisection. If the 4-dimensional handlebodies have the same genus

k = k1= k2 = k3, the trisection is said to be balanced.

Remark 2.1.2. The boundary of each 4-dimensional handlebody ∂X_i = #ki_(S2_{× S}1_{) is}

a closed, connected, oriented 3-manifold, and (Σ, Hij, Hil) is a Heegaard splitting for it.

Remark 2.1.3. Given a balanced (g, k)-trisection of a manifold X, it is easy to show

that χ(X) = 2 + g − 3k. By this formula, k is determined by g and the manifold X, so we will often write balanced trisection of genus g instead of balanced (g, k)-trisection. Moreover, it is worth noting that the genus of a balanced trisection of a fixed manifold is determined mod 3.

Theorem 2.1.4. A trisection is determined by its 3 dimensional spine.

Proof. It is sufficient to note that the trisection can be recovered from its spine S by

attaching the three 4 dimensional handlebodies to S. Since there is only one way to glue handlebodies (see Theorem 1.3.1), everything is completely determined.

We will show in chapter 3 that each connected, oriented, closed 4-manifold admits a (balanced) trisection. Moreover, we will prove an analogous of Theorem 1.2.11, present-ing a stabilization move that connects trisections in the same way Heegaard stabilization connects Heegaard splittings.

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Figure 2.1: A diagrammatic view of a trisection, showing how the pieces fit together

2.2 Trisection diagrams

In a similar way to what we have observed for Heegaard splittings, all the information concerning a trisection can be encoded in a combinatorial diagram.

Definition 2.2.1. A trisection diagram is a quadruple (Σ, α, β, γ) where Σ is a closed

oriented surface of genius g and α = (α₁, . . . , αg), β = (β1, . . . , βg), γ = (γ1, . . . , γg) are

three g-tuples of simple closed curves such that each triple (Σ, α, β), (Σ, β, γ), (Σ, γ, α), gives a Heegaard diagram for #k(S2× S1_).

Notation. We will draw the diagram using red, blue and green to distinguish the α, β

and γ curves.

To obtain a trisection diagram from a trisection, it is sufficient to consider the cen-tral surface and draw a defining set of curves for each one of the three 3-dimensional handlebodies. Conversely, starting from a trisection diagram, it is possible to recover to whole trisected manifold with the following procedure. We start by thickening the surface Σ ∼= Sg to obtain the compact 4-manifold Σ × D2, we proceed to push the α

curves in Σ × {1}, the β curves in Σ × {e2πi3 } and the γ curves in Σ × {e 4πi

3 } and attach

the 2-handles along those curves with the framing induced by the surfaces. Now we close the manifold using 3- and 4-handles (note that there is a unique way to do it, see Theorem 1.3.1).

To prove that we ended up with the right manifold, we will show that, after having attached the 2-handles, it is possible to attach some 3-handles to obtain (a thickening of) the spine of the trisection. This, together with Theorem 1.3.1, would complete the proof, since we already know that the spine can be closed using 3- and 4-handles.

Observe that, after having attached two of the three g-tuples of 2-handles, the re-sulting manifold factorizes as N × [0, 1] where N is a 3-manifold. The construction we

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2.3. Stabilization

described, restricted to N , consists in attaching (3-dimensional) 2-handles to the speci-fied curves in Σ × {0} and Σ × {1}. Since the curves in the diagrams represent a defining set of curves for some handlebody, we can cap each of the two sides with a 3-handle to obtain two handlebodies. From the 4-dimensional point of view, the attachment of the 3-dimensional 3-handle to N is equivalent to the attachment of D3× I along S2 _{× I,}

which corresponds to the attachment of a 4-dimensional 3-handle.

Since each triple (Σ, α, β), (Σ, β, γ), (Σ, γ, α) is a Heegaard diagram for #k(S2× S1_),

it is possible to modify the curves via isotopies and handle slides to obtain a standard diagram for a pair of g-tuples; however, it is not guaranteed that the three pairs of

g-tuples can be put in standard position simultaneously. Once the α and the β curves

are drawn as a standard diagram, the only condition on the choice of the γ curves is that they must be slidable in standard position with each g-tuple α and β, separately.

Similarly to Heegaard splittings, given two trisected manifold there is a trisection on their connected sum that arises in a natural way (it is sufficient to perform the connected sum using a standard trisected 4-ball). A diagram for such trisection can be obtained as the connected sum of two diagrams for the starting trisections.

2.3 Stabilization

Definition 2.3.1. • An unbalanced stabilization (on X1) is a move that changes a

(g, k1, k2, k3)-trisection into a (g + 1, k1+ 1, k2, k3)-trisection in this way: we consider a

boundary parallel arc in H₂₃, take N to be one closed tubular neighborhood of that arc, and replace (X₁, X2, X3) with X10 = X1∪ N , X20 = X2\ ˚N and X30 = X3\ ˚N .

• A balanced stabilization is the move that consists in performing unbalanced stabi-lizations simultaneously on the three handlebodies: we take boundary parallel arcs a_ij in each Hij, consider disjoint closed tubular neighborhoods Nij, and replace (X1, X2, X3)

with (X₁0, X₂0, X₃0) defined by:

• X₁0 = (X1∪ N23) \ ( ˚N12∪ ˚N13)

• X0

2= (X2∪ N13) \ ( ˚N23∪ ˚N12)

• X0

3= (X3∪ N12) \ ( ˚N23∪ ˚N13)

A balanced stabilization acts on a (g, k)-trisection increasing the genus by 3 and k by 1.

Remark 2.3.2. In terms of diagrams, a balanced stabilization is equivalent to perform

a connected sum with the diagram in Figure 2.2b, while an unbalanced stabilization corresponds to a connected sum with one of the three tori composing the picture. This can be used to understand completely how a stabilization acts on the 3-dimensional spine of a trisection.

As always, we will say that two trisections are stably equivalent if they become isotopic after having applied some stabilization to each of them. We will prove in Chapter 3 that any pair of trisections of the same manifold are stably equivalent.

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(a) (b)

Figure 2.2: A tridimensional view of how a trisection stabilization works (left), and a balanced stabilization diagram (right).

2.4 Examples

We now provide some explicit examples (all of genus 1), to help understanding how trisections and trisection diagrams work.

• It is possible to explicitly divide S4 _{⊂ C × R}3 _{into three regions X}

1, X2, X3 defined by Xk= (reit, x, y, z) ∈ S4 : 2πi 3 (k − 1) ≤ t ≤ 2πi 3 k

that form a genus-0 trisection for S4. The diagram is given by S2 with no curves. • Let us consider the map Φ : CP2→ R2 _{given by the formula}

[z₁: z₂ : z₃] 7→ _|z 1| |z₁| + |z₂| + |z₃|; |z₂| |z₁| + |z₂| + |z₃|

The image Φ(CP2) is the right triangle with vertices (0, 0), (0, 1) and (1, 0). The counterimage through Φ of the barycentric subdivision divides CP2 into three regions

X1 = {[z1 : z2 : z3] | |z1| ≥ |z3|, |z1| ≥ |z2|}, X2 = {[z1 : z2 : z3] | |z2| ≥ |z3|, |z2| ≥ |z1|}

and X₃ = {[z₁ : z₂ : z₃] | |z₃| ≥ |z₁|, |z₃| ≥ |z₂|} whose pairwise intersections are

Hij = {[z1 : z2 : z3] | |zi| = |zj| ≥ |zk|}, and with triple intersection X1 ∩ X2 ∩ X3 =

Σ = {[z1 : z2 : z3] | |z1| = |z2| = |z3|}. Using the chart {z1 6= 0} it is easy to see that

Σ ∼_{= {(z, w) ∈ C}2| |z| = 1, |w| = 1} ∼= S1 × S1 _{is a torus, X}

1 ∼= {(z, w) ∈ C2| |z| ≤

1, |w| ≤ 1} ∼= D2× D2_{, is a genus 0 handlebody, and H}

1j ∼= {(z, w) ∈ C2| |z| = 1, |w| ≤

1} ∼= S1 × D2 _{(for j = 2 and j = 3) are two solid tori forming a genus 1 Heegaard}

splitting of ∂X₁. Since the previous discussion applies also to the other two regions, the decomposition CP2 = X1∪ X2∪ X3 is a (1, 0)-trisection of CP2. In order to determine

the diagram of this trisection, it is sufficient to draw a defining curve for each of the three handlebodies H_ij on the central torus {[eiθ1 : eiθ2 : 1] | θ₁, θ2 ∈ [0, 2π]}. The three curves

can be taken to be t 7→ [eit : 1 : 1] for H23, t 7→ [1 : eit : 1] for H13 and t 7→ [1 : 1 : eit]

for H12, the latter being equivalent to t 7→ [eit : eit : 1], thus the trisection diagram

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2.4. Examples

(a) (b)

Figure 2.3: On the left, the standard toric picture of CP2: the counterimage of the three regions give rise to a trisection, whose diagram is pictured on the right.

• We now try to investigate what manifold arises from the trisection diagram in Figure 2.4a. To do this, we repeat the construction described in Section 2.2: starting from a thickening of the central torus T × D2, we attach 2-handles along the curves, producing the diagram in Figure 2.4b. We now observe that one of the two 1-handles is in cancelling position with the new 2 handles. To perform the handle elimination, it is necessary to slide off the other 2-handles first: in Figure 2.4c we have slid the attaching sphere of non-green 2-handles outside the 1-handle using the green one, and after having performed the 1/2-cancellation we end up with with Figure 2.4d. When we attach 3-and 4-h3-andles, the 0-framed 2-h3-andles get cancelled by some three h3-andles, leaving us with the handle decomposition of S1× S3 _{that we have seen in Section 1.3.4.}

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(a) The starting trisection diagram (b) The corresponding Kirby diagram

(c) Kirby diagram after the handle slides (d) Kirby diagram after the 1-2 cancellation

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3

Existence and uniqueness

theorems

In this chapter, we will prove the existence theorem for trisections, and show that the stabilization move described in Definition 2.3.1 can connect every pair of stabilizations of the same manifold.

The existence theorem will be proved in two different ways: the first proof relies on the theory of Morse 2-functions presented in Section 1.4, while the second one uses much simpler tools (not much more than handle decompositions). The naive idea inspiring both proofs is the following: a handle decomposition provides us two handlebodies (the one formed by 0- and 1-handles and the one formed by 4- and 3-handles), and we would like to transform the 2-handles to find the third handlebody. What we really do is borrowing some “connective tissue” from the first handlebody, in order to mold the 2-handles in the desired shape.

3.1 Existence theorem

Theorem 3.1.1. Every closed, connected, oriented 4-manifold X has a balanced

trisec-tion.

Proof. The aim of this proof is to build a Morse 2-function G : X → R2 with some symmetry properties, and use it to generate a subdivision from which the trisection will arise. We will represent our Morse 2-functions by drawing their fold locus, calling t the horizontal axis in R2, and z the vertical axis. A blank box in a picture indicates that in that region there is a Cerf graphic. The time axis for the Cerf graphic is horizontal, so it cannot contain vertical tangency points.

The outline of the proof is the following:

0. We will start from a handle decomposition of the manifold X with only one 0-handle and one 4-0-handle.

1. Using this handle decomposition, we will show that there exists a Morse 2-function

G1 : X → R2 such that the projection t ◦ G1 is a Morse function associated to the

starting handle decomposition, and such that the image of its fold locus is as in Figure 3.1. The vertical tancency points correspond to the critical points of the Morse function t ◦ G1, and the corresponding index is given by the index of the

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2. We will show a way to homotope G1 to another Morse 2-function G2, such that

the image of the fold locus is such as in Figure 3.2. In this case, each of the two Cerf graphics in the white boxes has no cusps. The main differences with the previous locus are that the kinks corresponding to the 2-handles in the handle decomposition have been replaced by double-cusps, and that the Cerf graphic has been splitted into two pieces, each one involving only critical points of the same index (and no cusps).

3. Figure 3.2 can be redrawn in a more symmetric way as in Figure 3.3. This new picture comes with a natural subdivision of R2 into three sectors. Note that the central fiber is an oriented surfaces, thus each sector has exactly g component, where g is the genus of the central surface.

4. Figure 3.3 is a particular case of the shape pictured in Figure 3.4 (we allow for arbitrary Cerf graphics without cusps between the sectors, and the number of cusps in each of the three sectors can be different). At this point we could divide R2 _into

three sectors (R2 = R21∪ R22∪ R32, with R2i = {(r, θ) ∈ R2| −π+4iπ6 ≤ θ ≤ 3π+4iπ

6 }),

take Xi = G−12 (R2i), and we would have Xi= #∼ k_∂i(S1×D3) for different ki(equal to

the number of folds without cusps in that sector), forming an unbalanced trisection. Our last step consists in homotoping G₂ to a new Morse 2-function G₃ having the same number of folds without cusps in each sector: this operation will be performed by adding some eye singularities.

We begin from a handle decomposition of X with one 0-handle, i1 1-handles, i2

2-handles, i₃ 3-handles and one 4-handle. The union of 0- and 1-handles X∗ is the

handlebody #i1_∂(S1× D3_{), and we will see it as I × #}i1

∂(S1× D2), mapping it into I × I

by (t, p) 7→ (t, g(p)), where g is a standard Morse function for #i1_∂(S1× D2_{) (with one}

index 0 critical point and i₁ index 1 critical points). We now postcompose this map with a diffeomorphism from I × I to the half-disk, obtaining a Morse 2-funcion G₁ on

X∗ whose fold locus is such as the one in Figure 3.5. The boundary ∂X∗∼= #i1(S1× S2)

is the counterimage of the red edge in Figure 3.5, and the vertical Morse function on

∂X∗ (i.e. t ◦ G1|∂X∗) is the standard Morse function with i1 index 1 and index 2 critical

points (inducing the standard Heegaard splitting, with central surface S).

We now proceed to extend the Morse 2-function already constructed to the 2-handles. To do so, we consider the attaching link L ⊂ ∂X∗ for 2-handles: in order to be able to

include the attachment in the Morse 2-function G1, we need L to be in a fiber of that

function. Since L can be slightly moved (in ∂X∗) to be disjoint from the cores of the

two handlebodies (i.e. the ascending 1–manifolds of the index 2 critical points and the descending 1–manifolds of the index 1 critical points), we can project L onto the central surface S to give an immersed curved L. With some further isotopy, L ⊂ S can be taken to have at worst transverse double points, and we can add kinks to make the framing given by the handle attachment match the framing induced by the surface S.

We would like to eliminate the crossings, in order to have the link L embedded in S, while preserving the property that the desired framing is the one induced by the surface itself. To do so, we perform a stabilization of the Heegaard splitting of ∂X∗ around each

double point, to separate the two components of the link, as shown in Figure 3.6. These stabilizations can be incorporated in our function G₁, extending it to X∗∪ (∂X∗ × I),

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3.1. Existence theorem

Figure 3.1: The image of the fold locus of the Morse 2-function G₁. The projection to the horizontal axis is a Morse function, and its critical values are written below the axis.

Figure 3.2: The image of the fold locus of the Morse 2-function G₂. The box in this figure are Cerf graphics that do not contain cusps.

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Figure 3.3: A more symmetric drawing for the fold locus of G2. The outermost fold has

index 0 pointing inwards, all the other folds have index 1 pointing inwards. The picture can naturally be divided into three pieces, that form an unbalanced trisection.

Figure 3.4: A generalization of the fold locus showed in the previous picture. After having arranged the folds to have the same number of cusps in each sector, we end up with a balanced trisection.

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Figure 3.5: A Morse 2-function G1 defined on the 0- and 1-handles of X. The two

pictures are obtained one from the other by composing G₁ with a diffeomorphism.

Figure 3.6: How to solve a double-point crossing using a stabilization

by defining it on ∂X∗× [0, 1] as a homotopy from the standard Morse function g0 for

#i1(S1× S2_{) to the stabilized Morse function g}

1. The procedure is described in Figure

3.7.

We now want to attach the 2-handles along their attacching link L in the new stabi-lized surface S∗. The attachment of a 4 dimensional 2-handle along a blackboard-framed

knot K ⊂ L ⊂ ∂X∗ can be viewed as performing I times the attachment of a 3

dimen-sional 2-handle along I × K ⊂ I × S∗ ⊂ ∂X∗. Thus, the attachment can be represented

in the Morse 2-function G1 as described in Figure 3.8. We first add the rectangle

repre-senting the 2-handle, then we map it into a half-disk as we have already done in Figure 3.5. Finally we cross the two folds, as in Figure 3.8-(right), in order to have indices that are coherent with the previous folds (this can be done in the same way; by adding a new rectangle with the crossing, and using the usual diffeomorphism into the half disk).

Note that this operation happens locally around the attaching circle K: thus, the remaining components of the attaching link L continue to be embedded with the right framing into the central surface of the boundary (i.e. the counterimage of a central point

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Figure 3.7: The extension of G1 on a collar of ∂X∗. On the left of the first blue dotted

line there is the previous function G₁, as it was in Figure 3.5. On the right, between the two blue lines, we have a Morse 2-function on ∂X∗ that incidentally is a homotopy of

Morse funcions on ∂X∗. By glueing the two functions together, we obtain a new Morse

2-function for the same manifold, with a different Heegaard splitting in the boundary.

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Figure 3.9: On the left hand side, the image of the fold locus of the function G₁ defined on 0-, 1- and 2-handles, on the right hand side, the image of the fold locus of a function defined in the same way on 4- and 3-handles.

in the right edge). This property allows us to encode in G1 the attachment of the other

2-handles using the same trick.

After having attached all the 2-handles, the fold locus of the new manifold X∗∗

is as in the left part of Figure 3.9. The 4-handle and the 3-handles together form a handlebody, which can be represented with a Morse 2-function having fold locus such as in the right part of Figure 3.9 (this is the same construction we have done in Figure 3.5). The boundary of the two manifolds (the one formed by 0-, 1- and 2-handles and the one formed by 4- and 3-handles) must be the same manifold #i3S1× S2_{, so the two vertical}

Morse functions given by z ◦ G₁ (restricted to the boundaries) can be connected by a homotopy (this is a consequence of Theorem 1.2.16). This implies that we can connect the two parts of 3.9 with a Cerf graphic, obtaining a Morse 2-function defined on X (the counterimage of the added part is just a product ∂X∗∗× I).

The previous construction leaves us with a Morse 2-function G1 fulfilling the

condi-tions requested for the first step; we now have to homotope it to an other function G₂ satisfying the conditions given by the second step.

We start by splitting the Cerf graphic in two parts; the whole procedure is shown in Figure 3.10. At first, we move all the left cusps in the graphic to the left, and the right cusps to the right, pulling them out of the Cerf graphics. The resulting Cerf graphic consists in index-1 and index-2 folds that cross each other, but never change index (since there are no cusps nor vertical tangency): therefore, using Reidemeister-II moves, we can slide all the index-1 folds in the bottom half, and the index-2 folds in the

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Figure 3.10: The procedure to split the Cerf graphics: we start with the top left picture, then we pull the cusps out (top right), we split the graph (bottom left), and finally ws pull the cusps before the kinks (bottom right). The new crossings created with this procedure can be incorporated into the Cerf graphics.

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Figure 3.12: The addition of a new fold (eye singularity) to the Morse 2-function, in order to increase the number of folds without cusps in a given sector.

top half, producing two separate Cerf graphics (we are always allowed to pull down folds with lower index). Finally, the cusps on the left can be pulled further left, before the kinks: this is possible thanks to the fact that each cusp corresponds to a 3-dimensional stabilization, and the attachment of 2-handles is independent of the stabilizations of the boundary.

The procedure to homotope a kink in a pair of cusps is described in Figure 3.11; it consists in introducing a swallowtail singularity at the vertical tangency of the kink, and performing a Reidemeister-II move to the crossing folds (this is possible because in every vertical Morse function, the 2-handle attachment can be postponed after the 1-handle attachment).

Following the outline of the proof, we end up with the fold locus in Figure 3.4, and it is necessary to balance the number of folds without cusps in each sector. This can be done by introducing some eye singularity in the middle, as shown in Figure 3.12. Depending on how the eye is oriented, we add a new fold without cusps to one of the three sectors.

Finally, we need to show that the preimage of each of the three sectors of Figure 3.4 (having k folds without cusps and g − k folds with cusps in each sector) is G−1₃ _(R2_i) = #k_∂(S1× D3_{). Each sector, ignoring the Cerf graph in the box (that doesn’t change the}

manifold), is identical to the fold locus in Figure 3.5; we already know that it represents #k_∂(S1× D3_{) with a standard (g, k)-Heegaard splitting of the boundary.}

3.1.1 Handle decompositions and trisections

Using Morse 2-functions, we can analyse the relationship between handle decompositions and trisections. In particular, we will prove that each trisection carries in a natural way a handle decomposition with some additional structure; and that given a handle decom-position together with that additional structure, it is possible to recover a trisection. We will finally present a second proof of Theorem 3.1.1, showing that is possible to find a handle decomposition with such structure.

Proposition 3.1.2. If X = X1∪ X2∪ X3 is a (g, k)-trisection, then X admits a handle

Trisections of 4-manifolds

Universit`

a degli Studi di Pisa

Trisections of 4-manifolds

Contents

Introduction

1

Preliminaries

1.1

Handle decomposition

1.2

Heegaard splittings

1.3

Kirby calculus

1.4

Morse 2-functions

2

Trisections

2.1

Definition

2.2

Trisection diagrams

2.3

Stabilization

2.4

Examples

3

Existence and uniqueness

theorems

3.1

Existence theorem