• 4-manifolds
• low dimensional topology
• trisections

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.