• 4-manifolds
• trisection

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, and consist in dividing a 4-manifold X into three pieces X_1 , X_2 and X_3, diffeomorphic to 4-dimensional handlebodies, with the property that each pairwise intersection H_ij=X_i ∩ X_j is diffeomorphic to a 3-dimensional handlebody and the triple intersection Σ=X_1 ∩ X_2 ∩ X_3 is a genus g orientable surface. In particular, the pairwise intersections provide Heegaard splittings ∂X_i = H_ij ∪ H_ki.

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 3-dimensional manifolds.
Gay and Kirby proved 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 2017, Jeffrey Meier and Alexander Zupan showed that any surface embedded in S^4 can be put in “good” position with respect to the trivial trisection, and in 2018 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 2018).