The algebraic topology of $4$-manifold multisections

Delphine Moussard and Trenton Schirmer

Vol. 327 (2023), No. 1, 139–166

DOI: 10.2140/pjm.2023.327.139

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 equivariant 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.

Keywords

trisection, trisection diagram, homology of $4$-manifolds, intersection form

Mathematical Subject Classification

Primary: 57K41

Secondary: 57K40

Milestones

Received: 29 December 2021

Revised: 11 January 2024

Accepted: 12 January 2024

Published: 18 February 2024