School of Mathematics & Statistics

Symmetries of Manifolds

We study the group of symmetries of manifolds, i.e. the homeomorphisms of a given manifold to itself. There is specific focus in Glasgow on symmetries of low-dimensional manifolds, namely those of dimensions 2, 3, and 4. The connected components of the homeomorphism group is the mapping class group of the manifold. We seek to compute mapping class groups and study their properties, for example comparing with the smooth mapping class group of diffeomorphisms, or the structure preserving groups of contactomorphisms or symplectomorphisms. We also aim to understand the homotopy type of homeomorphism groups, more generally, which can lead to classifications of fibre bundles. The study of symmetries of manifolds uses Morse theory, surgery theory, homotopy theory, and geometric structures on manifolds, and has connections to dynamics and geometric group theory.