# John Rognes : The stable parametrized h-cobordism theorem

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.