#### Volume 20, issue 6 (2016)

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Trisecting $4$–manifolds

### David T Gay and Robion Kirby

Geometry & Topology 20 (2016) 3097–3132
##### Abstract

We show that any smooth, closed, oriented, connected $4$–manifold can be trisected into three copies of ${♮}^{k}\left({S}^{1}×{B}^{3}\right)$, intersecting pairwise in $3$–dimensional handlebodies, with triple intersection a closed $2$–dimensional surface. Such a trisection is unique up to a natural stabilization operation. This is analogous to the existence, and uniqueness up to stabilization, of Heegaard splittings of $3$–manifolds. A trisection of a $4$–manifold $X$ arises from a Morse $2$–function $G:X\to {B}^{2}$ and the obvious trisection of ${B}^{2}$, in much the same way that a Heegaard splitting of a $3$–manifold $Y$ arises from a Morse function $g:Y\to {B}^{1}$ and the obvious bisection of ${B}^{1}$.

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##### Keywords
4-manifold, trisection, Heegaard splitting, Heegaard triple, Morse 2-function
##### Mathematical Subject Classification 2010
Primary: 57M50, 57M99
Secondary: 57R45, 57R65