Generalised trisections in all dimensions

J Hyam Rubinstein, Stephan Tillmann

Abstract

This paper describes a generalisation of Heegaard splittings of 3-manifolds and trisections of 4-manifolds to all dimensions, using triangulations as a key tool. In particular, every closed piecewise linear \(n\)-manifold 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.

Keywords: manifold, multisection, triangulation, colouring, CAT(0) cubing.

AMS Subject Classification: Primary 52B70; secondary 53C21, 57N15.