Generalized trisections in all dimensions

Abstract

This paper describes a generalization 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 generalizations of our approach are given.