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.