Geometry & Topology
- Geom. Topol.
- Volume 22, Number 3 (2018), 1537-1545.
Group trisections and smooth $4$–manifolds
A trisection of a smooth, closed, oriented –manifold is a decomposition into three –dimensional –handlebodies meeting pairwise in –dimensional –handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the –dimensional handlebodies, the –dimensional handlebodies and the closed –manifold, with homomorphisms between them induced by inclusion, form a commutative diagram of epimorphisms, which we call a trisection of the –manifold group. A trisected –manifold thus gives a trisected group; here we show that every trisected group uniquely determines a trisected –manifold. Together with Gay and Kirby’s existence and uniqueness theorem for –manifold trisections, this gives a bijection from group trisections modulo isomorphism and a certain stabilization operation to smooth, closed, connected, oriented –manifolds modulo diffeomorphism. As a consequence, smooth –manifold topology is, in principle, entirely group-theoretic. For example, the smooth –dimensional Poincaré conjecture can be reformulated as a purely group-theoretic statement.
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
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 57M05: Fundamental group, presentations, free differential calculus
Secondary: 20F05: Generators, relations, and presentations
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