#### Volume 22, issue 3 (2018)

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Group trisections and smooth $4$–manifolds

### Aaron Abrams, David T Gay and Robion Kirby

Geometry & Topology 22 (2018) 1537–1545
##### 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.

##### Keywords
trisection, group theory, finitely presented groups, $4$–manifolds, Morse $2$–functions, Heegaard splitting
Primary: 57M05
Secondary: 20F05