Computing trisections of 4-manifolds.

We describe an algorithm to compute trisections of orientable four-manifolds using arbitrary triangulations as input. This results in explicit complexity bounds for the trisection genus of a 4-manifold in terms of the number of pentachora (4-simplices) in a triangulation.

Generalized trisections in all dimensions.

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 [Formula: see text] n-dimensional 1-handlebodies, where [Formula: see text] or [Formula: see text], such that intersections of the handlebodies have spines of small dimensions. Several applications, constructions, and generalizations of our approach are given.