|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
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 –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. |
Keywords
trisection, group theory, finitely presented groups,
$4$–manifolds, Morse $2$–functions, Heegaard splitting
|
Mathematical Subject Classification 2010
Primary: 57M05
Secondary: 20F05
|
References |
Publication
Received: 1 June 2016
Accepted: 19 August 2017
Published: 16 March 2018
Proposed: Peter Teichner
Seconded: Walter Neumann, András I Stipsicz
|
Authors |
| Aaron Abrams | |
| Mathematics Department Washington and Lee University Lexington, VA United States |
|
| David T Gay | |
| Euclid Lab and Department of
Mathematics University of Georgia Athens, GA United States |
|
| Robion Kirby | |
| Department of Mathematics University of California Berkeley, CA United States |
|
