Abstract

A fundamental link between Waldhausen's algebraic K-theory of spaces (A-theory) and manifold topology is given by an identification of A-theory in terms of stable homotopy and the stable smooth h-cobordism space. This important result has had many applications in the study of diffeomorphisms of manifolds. In more recent years, the theory of cobordism categories has provided a different approach to the study of diffeomorphism groups with spectacular applications. In collaboration with W. Steimle , we revisit the classical Waldhausen K-theory in light of these developments and investigate new connections and applications. In this talk, I will first discuss a cobordism-type model for A-theory, and then I will focus on the h-cobordism category, the cobordism category of h-cobordisms between smooth manifolds with boundary, and its relationship to the classical h-cobordism space of a compact smooth manifold. This is joint work with W. Steimle.