## Thesis etd-06282016-121015 |

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Thesis type

Tesi di laurea magistrale

Author

BARBENSI, AGNESE

URN

etd-06282016-121015

Thesis title

Trisections of 4-manifolds

Department

MATEMATICA

Course of study

MATEMATICA

Supervisors

**relatore**Lisca, Paolo

Keywords

- 4-manifolds
- low dimensional topology
- trisections

Graduation session start date

15/07/2016

Availability

Full

Summary

A common theme in topology is to reduce hard problems by decomposing

a manifold into simpler and standard pieces. Classical examples

in this direction include handle decompositions, Heegaard splittings of

3-manifolds, surgery theory and so on.

This thesis will present the concept of trisections for smooth, closed

and oriented 4-manifolds. Trisections were recently defined by David

T. Gay and Robion Kirby. A trisection is a decomposition of a

4-manifold X into three pieces X1, X2 and X3, each diffeomorphic to

a 4-dimensional handlebody; the triple intersection X1∩X2∩X3 is a closed

surface Σ, and each pair of double intersections (Xi∩Xj, Xk∩Xj ) provides

a Heegaard splitting for the boundary ∂Xj with Heegaard surface Σ.

Trisections can be thought of as a 4-dimensional analogue of Heegaard

splittings. As in the case of 3-manifolds, we are decomposing

into standard pieces, and we can recover the underlying manifold from

combinatorial data associated to the decomposition. In the original

definition the three 4-dimensional handlebodies are meant to be of the

same genus. However, in this thesis we are going to work in the slightly

general context of unbalanced trisections.

The authors proved that every smooth, closed and oriented

4-manifolds admits a trisection. Trisections are far from being unique;

however, two trisections representing the same 4-manifold X are related

by simple stabilization moves. We will show that uniqueness of a

trisection is guaranteed up to stabilizations.

The proofs of existence and uniqueness rely on results concerning

Morse 2-functions. These arise as a generalization of homotopies between

regular Morse functions. In other words, a Morse 2-function is

a generic map from a smooth n-manifold X to a 2-manifold, satisfying

some additional constraints. All the relevant tools, mostly hinging

upon Cerf theory, were developed by Gay and Kirby.

Between 2015 and 2016, trisections have been a very active topic

of research. Several authors managed to prove partial classification

results, to extend to the non-empty boundary case and

a group-theoretic approach to trisections was obtained in a recent paper. We will

give a complete classification of trisected 4-manifolds up to genus 2

(Theorem 4.4).

We will also present some developments concerning the compact

case, that is currently work in progress by Juanita Pinzon Caicedo and

Nick Castro. Lastly we will talk about the group-theoretic reformulation

due to Gay and Kirby.

a manifold into simpler and standard pieces. Classical examples

in this direction include handle decompositions, Heegaard splittings of

3-manifolds, surgery theory and so on.

This thesis will present the concept of trisections for smooth, closed

and oriented 4-manifolds. Trisections were recently defined by David

T. Gay and Robion Kirby. A trisection is a decomposition of a

4-manifold X into three pieces X1, X2 and X3, each diffeomorphic to

a 4-dimensional handlebody; the triple intersection X1∩X2∩X3 is a closed

surface Σ, and each pair of double intersections (Xi∩Xj, Xk∩Xj ) provides

a Heegaard splitting for the boundary ∂Xj with Heegaard surface Σ.

Trisections can be thought of as a 4-dimensional analogue of Heegaard

splittings. As in the case of 3-manifolds, we are decomposing

into standard pieces, and we can recover the underlying manifold from

combinatorial data associated to the decomposition. In the original

definition the three 4-dimensional handlebodies are meant to be of the

same genus. However, in this thesis we are going to work in the slightly

general context of unbalanced trisections.

The authors proved that every smooth, closed and oriented

4-manifolds admits a trisection. Trisections are far from being unique;

however, two trisections representing the same 4-manifold X are related

by simple stabilization moves. We will show that uniqueness of a

trisection is guaranteed up to stabilizations.

The proofs of existence and uniqueness rely on results concerning

Morse 2-functions. These arise as a generalization of homotopies between

regular Morse functions. In other words, a Morse 2-function is

a generic map from a smooth n-manifold X to a 2-manifold, satisfying

some additional constraints. All the relevant tools, mostly hinging

upon Cerf theory, were developed by Gay and Kirby.

Between 2015 and 2016, trisections have been a very active topic

of research. Several authors managed to prove partial classification

results, to extend to the non-empty boundary case and

a group-theoretic approach to trisections was obtained in a recent paper. We will

give a complete classification of trisected 4-manifolds up to genus 2

(Theorem 4.4).

We will also present some developments concerning the compact

case, that is currently work in progress by Juanita Pinzon Caicedo and

Nick Castro. Lastly we will talk about the group-theoretic reformulation

due to Gay and Kirby.

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