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 ${B}^{1}$.

Keywords
4-manifold, trisection, Heegaard splitting, Heegaard triple, Morse 2-function
Mathematical Subject Classification 2010
Primary: 57M50, 57M99
Secondary: 57R45, 57R65
Publication
Received: 4 December 2013
Revised: 21 January 2016
Accepted: 18 February 2016
Published: 21 December 2016
Proposed: Yasha Eliashberg
Seconded: Benson Farb, Dmitri Burago
Authors
 David T Gay Euclid Lab 160 Milledge Terrace Athens, GA 30606 United States Department of Mathematics University of Georgia Athens, GA 30602 United States Robion Kirby Department of Mathematics University of California, Berkeley Berkeley, CA 94720-3840 United States