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2018 Group trisections and smooth $4$–manifolds
Aaron Abrams, David T Gay, Robion Kirby
Geom. Topol. 22(3): 1537-1545 (2018). DOI: 10.2140/gt.2018.22.1537

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.

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Aaron Abrams. David T Gay. Robion Kirby. "Group trisections and smooth $4$–manifolds." Geom. Topol. 22 (3) 1537 - 1545, 2018. https://doi.org/10.2140/gt.2018.22.1537

Information

Received: 1 June 2016; Accepted: 19 August 2017; Published: 2018
First available in Project Euclid: 31 March 2018

zbMATH: 06864262
MathSciNet: MR3780440
Digital Object Identifier: 10.2140/gt.2018.22.1537

Subjects:
Primary: 57M05
Secondary: 20F05

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.22 • No. 3 • 2018
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