Geometry & Topology

Group trisections and smooth $4$–manifolds

Aaron Abrams, David T Gay, and Robion Kirby

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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.

Article information

Geom. Topol., Volume 22, Number 3 (2018), 1537-1545.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M05: Fundamental group, presentations, free differential calculus
Secondary: 20F05: Generators, relations, and presentations

trisection group theory finitely presented groups $4$–manifolds Morse $2$–functions Heegaard splitting


Abrams, Aaron; Gay, David T; Kirby, Robion. Group trisections and smooth $4$–manifolds. Geom. Topol. 22 (2018), no. 3, 1537--1545. doi:10.2140/gt.2018.22.1537.

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