Algebraic K-theory of smooth compact manifolds provides a homotopical lift of the classical h-cobordism theorem and serves as a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. I will give an overview of this story and recent progress on an equivariant homotopical lift of the h-cobordism theorem developed in joint work with Goodwillie, Igusa, and Malkiewich.