## Geometry & Topology

### Group trisections and smooth $4$–manifolds

#### 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.

#### Article information

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

Dates
Accepted: 19 August 2017
First available in Project Euclid: 31 March 2018

https://projecteuclid.org/euclid.gt/1522461622

Digital Object Identifier
doi:10.2140/gt.2018.22.1537

Mathematical Reviews number (MathSciNet)
MR3780440

Zentralblatt MATH identifier
06864262

#### Citation

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. https://projecteuclid.org/euclid.gt/1522461622

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