Applying Manifold Calculus To Totally Nonparallel Immersions

Abstract

Manifold calculus provides a way to approximate functors from the open sets of a manifold to spaces by a tower of polynomial functors. In this thesis, we explain how this works in detail for the functor of embeddings, a case which is already well-understood. Following that, we make steps towards adapting the approach used there to the functor of totally nonparallel immersions. These are immersions from a manifold into Euclidean space such that the images of the tangent spaces at any two points have no parallel lines. We find that the first stage of the tower in this case is given by formal semifree immersions, and we provide conjectures for the second stage in analogy to the embeddings case.