Submetries are generalizations of Riemannian submersions and quotients by proper isometric group actions.

They naturally arise in a number of areas such as collapsing with various curvature bounds, singular Riemannian foliations and various rigidity problems in geometry. We study geometric and topological structure of submetries of manifolds. I will also discuss possible applications to collapsing with lower curvature bounds. This is joint work with Alexander Lytchak.