Multisections of piecewise linear manifolds

TL;DR

This paper extends the concept of trisections from 4-manifolds to all dimensions, showing that every closed piecewise linear manifold can be decomposed into handlebodies with controlled intersections, broadening topological decomposition techniques.

Contribution

It generalizes the notion of multisections from 4-manifolds to all dimensions using triangulations, providing a new framework for decomposing piecewise linear manifolds.

Findings

Every closed PL n-manifold admits a multisection.

Multisections decompose manifolds into handlebodies with low-dimensional intersections.

The approach applies to all dimensions, generalizing previous 4-manifold results.

Abstract

Recently Gay and Kirby described a new decomposition of smooth closed $4$-manifolds called a trisection. This paper generalises Heegaard splittings of $3$-manifolds and trisections of $4$-manifolds to all dimensions, using triangulations as a key tool. In particular, we prove that every closed piecewise linear $n$-manifold has a multisection, i.e. can be divided into $k+1$ $n$-dimensional $1$-handlebodies, where $n=2k+1$ or $n=2k$, such that intersections of the handlebodies have spines of small dimensions. Several applications, constructions and generalisations of our approach are given.