Abstract

Imagine that you have a nested manifold: a small manifold inside a big manifold. You know that the big manifold is a boundary and that the small manifold is a boundary. With this information, when can you assert that your nested manifold is a nested boundary? We delve in this problem via a nested Pontryagin-Thom construction: the study of nested manifolds up to cobordism amounts to the study of homotopy groups of... bananas! I will discuss this result and some of its surprising consequences.