## 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.

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

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 Monday, February 19, 2018