Group trisections and smooth 4-manifolds

TL;DR

This paper establishes a correspondence between trisections of smooth 4-manifolds and certain group structures, suggesting that 4-manifold topology can be understood entirely through group theory.

Contribution

It introduces the concept of group trisections and proves a bijection between these and smooth 4-manifolds, linking topology with algebraic group structures.

Findings

Every trisected group uniquely determines a 4-manifold.

The classification of smooth 4-manifolds can be approached via group theory.

The smooth 4-dimensional Poincaré conjecture can be reformulated group-theoretically.

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 Poincar\'e conjecture can be reformulated as a purely group theoretic statement.