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Group trisections and smooth $4$–manifolds

Aaron Abrams, David T Gay and Robion Kirby

Geometry & Topology 22 (2018) 1537–1545

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.

trisection, group theory, finitely presented groups, $4$–manifolds, Morse $2$–functions, Heegaard splitting
Mathematical Subject Classification 2010
Primary: 57M05
Secondary: 20F05
Received: 1 June 2016
Accepted: 19 August 2017
Published: 16 March 2018
Proposed: Peter Teichner
Seconded: Walter Neumann, András I Stipsicz
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