Volume 20, issue 6 (2016)

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Trisecting $4$–manifolds

David T Gay and Robion Kirby

Geometry & Topology 20 (2016) 3097–3132

We show that any smooth, closed, oriented, connected 4–manifold can be trisected into three copies of k(S1 × B3), 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 B2 and the obvious trisection of B2, in much the same way that a Heegaard splitting of a 3–manifold Y arises from a Morse function g: Y B1 and the obvious bisection of B1.

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4-manifold, trisection, Heegaard splitting, Heegaard triple, Morse 2-function
Mathematical Subject Classification 2010
Primary: 57M50, 57M99
Secondary: 57R45, 57R65
Received: 4 December 2013
Revised: 21 January 2016
Accepted: 18 February 2016
Published: 21 December 2016
Proposed: Yasha Eliashberg
Seconded: Benson Farb, Dmitri Burago
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