A cobordism W between compact manifolds M and M’ is an h-cobordism if the inclusions of M and M’ into W are both homotopy equivalences. These sort of cobordisms play an important role in the classification of high-dimensional manifolds, as h-cobordant manifolds are often diffeomorphic.

With this in mind, given two h-cobordant manifolds M and M’, how different can their diffeomorphism groups Diff(M) and Diff(M’) be? The homotopy groups of these two spaces are the same “up to extensions” in a range of strictly positive degrees. Contrasting this, I will present examples of h-cobordant manifolds in high-dimensions with different mapping class groups. In doing so, I will review the classical theory of h-cobordisms and introduce several moduli spaces of manifolds that shed light on this question.