Guest lectures and seminars - Page 112

The classical s-cobordism theorem classifies completely h-cobordisms from a fixed manifold, but it does not tell us much about the relationship between the two ends. In the talk I will present some old and new results about this. I will also discuss how this relates to a seemingly different problem: what can we say abobut two compact manifolds M and N if we know that MxR and NxR are diffeomorphic? This is joint work with Slawomir Kwasik, Tulane, and Jean-Claude Hausmann, Geneva.

I will survey the connection between the space H(M) of h-cobordisms on a given manifold M, several categories of spaces containing M, Waldhausens algebraic K-theory A(M), and the algebraic K-theory of the suspension ring spectrum S[?M] of the loop space of M. The results extend the h-cobordism theorem of Smale and the s-cobordism theorem of Barden, Mazur and Stallings to a parametrized h-cobordism theorem, valid in a stable range established by Igusa, first discussed by Hatcher and finally proved and published by Waldhausen, Jahren and myself.