Complexity of h-cobordisms

In-person only.

Two n-manifolds X_0,X_1 are h-cobordant if exists an n+1 dimensional manifold whose boundary is the union of X_0 (with opposite orientation) and X_1, such that the inclusion maps are homotopy equivalences. Diffeomorphic manifolds are clearly h-cobordant and Smale’s celebrated h-cobordism theorem states that the converse holds when n>= 5 and X_i is simply connected. On the other hand, Donaldson showed that this is not true for n=4. Still, it is possible to define a notion of complexity of an h-cobordism to measure the failure of it from being trivial. This notion has relations to embedded surfaces in the X_i and to their automorphism group. I will talk about some old and recent result about complexity of h-cobordisms of dimension (4+1).