Immersions of surfaces

Abstract

To determine the existence of a regular homotopy between two immersions, f, g : M --> N, is equivalent to showing that they lie in the same path component of the space Imm(M, N). We identify the connected components, pi 0 Imm(W g, M), of the space of immersions from a closed, orientable, genus-g surface W g into a parallelizable manifold M. We also identify the higher homotopy groups of Imm(W g, M) in terms of the homotopy groups of M and the Stiefel space V 2 (n). We then use this work to characterize immersions from tori into hyperbolic manifolds as self covers of a tubular neighborhood of a closed geodesic up to regular homotopy. Finally, we identify the homotopy-type of the space of framed immersions from the torus to itself.