Speaker
Description
A trisection is the analogue of a Heegaard splitting for a smooth 4-manifold, and it consists of splitting a 4-manifold into three 4-dimensional handlebodies. In a Heegaard splitting, the inclusion of the surface into the two 3-dimensional handlebodies induces a commutative square at the level of fundamental groups, where the fundamental group of the 3-manifold is the pushout. Analogously, a trisection of a 4-manifold induces a group trisection of its π1, i.e. a commutative cube of fundamental groups.
Abrams, Gay and Kirby showed that group trisections up to isomorphism and stabilization are in one-to-one correspondence with smooth 4-manifolds. However, group trisections are very complicated group-theoretical objects and it is quite hard to work with them. This observation leads to the following natural question: what happens when we abelianize and look at the homology groups? What information about the 4-manifold is lost in the process?
Therefore, we define abelian trisections: these are objects that contain all the information about the abelianized group trisection. It is known that from an abelian trisection we can deduce the homology and the intersection form of the 4-manifold.
We will see that, when the homology is torsion free, there is a unique abelian trisection (up to isomorphism and stabilization) with prescribed homology and intersection form. We will then explain the difficulties of the torsion case, and give a complete classification of abelian trisections up to isomorphism and stabilization.
This is joint work with Paolo Aceto.