Volume 22, issue 3 (2018)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 22
Issue 6, 3145–3760
Issue 5, 2511–3144
Issue 4, 1893–2510
Issue 3, 1267–1891
Issue 2, 645–1266
Issue 1, 1–644

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Subscriptions
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Ethics Statement
Author Index
To Appear
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Group trisections and smooth $4$–manifolds

Aaron Abrams, David T Gay and Robion Kirby

Geometry & Topology 22 (2018) 1537–1545
Abstract

A trisection of a smooth, closed, oriented 4–manifold is a decomposition into three 4–dimensional 1–handlebodies meeting pairwise in 3–dimensional 1–handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3–dimensional handlebodies, the 4–dimensional handlebodies and the closed 4–manifold, with homomorphisms between them induced by inclusion, form a commutative diagram of epimorphisms, which we call a trisection of the 4–manifold group. A trisected 4–manifold thus gives a trisected group; here we show that every trisected group uniquely determines a trisected 4–manifold. Together with Gay and Kirby’s existence and uniqueness theorem for 4–manifold trisections, this gives a bijection from group trisections modulo isomorphism and a certain stabilization operation to smooth, closed, connected, oriented 4–manifolds modulo diffeomorphism. As a consequence, smooth 4–manifold topology is, in principle, entirely group-theoretic. For example, the smooth 4–dimensional Poincaré conjecture can be reformulated as a purely group-theoretic statement.

Keywords
trisection, group theory, finitely presented groups, $4$–manifolds, Morse $2$–functions, Heegaard splitting
Mathematical Subject Classification 2010
Primary: 57M05
Secondary: 20F05
References
Publication
Received: 1 June 2016
Accepted: 19 August 2017
Published: 16 March 2018
Proposed: Peter Teichner
Seconded: Walter Neumann, András I Stipsicz
Authors
Aaron Abrams
Mathematics Department
Washington and Lee University
Lexington, VA
United States
David T Gay
Euclid Lab and Department of Mathematics
University of Georgia
Athens, GA
United States
Robion Kirby
Department of Mathematics
University of California
Berkeley, CA
United States