Spaces of Immersions and Their Homotopy Groups

If you have ever seen a model of a Klein bottle, this is an immersion of a closed non-orientable surface into 3-dimensional Euclidean space. The set of all immersions from one manifold into another can be equipped with a topology and analyzing the connected components of this space allows us to distinguish immersions up to regular homotopy. In this talk, we will identify the connected components, as well as the higher homotopy groups, of the space of immersions from a closed orientable surface into an arbitrary parallelizable manifold. An application of this work is that every immersion of a 2-torus into a hyperbolic manifold is regularly homotopic to a self-cover of a tubular neighborhood of some closed geodesic.