The algebraic topology of 4-manifolds multisections

TL;DR

This paper develops methods to compute algebraic invariants of 4-manifolds using multisection diagrams, generalizing trisections, and relates these to boundary open book decompositions, simplifying calculations significantly.

Contribution

It introduces a framework to derive homology, torsion, and intersection forms of 4-manifolds directly from multisection diagrams, extending previous trisection techniques.

Findings

Homology and torsion computed via a complex from the diagram

Intersection form expressed through the central surface

Multisections induce open book decompositions with monodromy action described

Abstract

A multisection of a 4-manifold is a decomposition into 1-handlebodies intersecting pairwise along 3-dimensional handlebodies or along a central closed surface; this generalizes the Gay-Kirby trisections. We show how to compute the twisted absolute and relative homology, the torsion and the twisted intersection form of a 4-manifold from a multisection diagram. The homology and torsion are given by a complex of free modules defined by the diagram and the intersection form is expressed in terms of the intersection form on the central surface. We give efficient proofs, with very few computations, thanks to a retraction of the (possibly punctured) 4-manifold onto a CW-complex determined by the multisection diagram. Further, a multisection induces an open book decomposition on the boundary of the 4-manifold; we describe the action of the monodromy on the homology of the page from the multisection diagram.