Trisections of 4-manifolds with Boundary

TL;DR

This paper provides a method to construct trisection diagrams of 4-manifolds with boundary from handle decompositions and open book decompositions, extending to multiple boundary components and their gluing.

Contribution

It makes the proof of relative trisections more explicit using handles and extends the theory to 4-manifolds with multiple boundary components.

Findings

Constructed trisection diagrams from handle decompositions and open books

Extended existence results to multiple boundary components

Described how to glue trisected 4-manifolds with multiple boundaries

Abstract

Given a handle decomposition of a 4-manifold with boundary, and an open book decomposition of the boundary, we show how to produce a trisection diagram of a trisection of the 4-manifold inducing the given open book. We do this by making the original proof of the existence of relative trisections more explicit, in terms of handles. Furthermore, we extend this existence result to the case of 4-manifolds with multiple boundary components, and show how trisected 4-manifolds with multiple boundary components glue together.