Spaces of PL manifolds and categories of simple maps / Friedhelm Waldhausen, Bjorn Jahren, and John Rognes.

Includes bibliographical references (pages 175-178) and index.

Introduction:
1. The stable parametrized h-cobordism theorem --
2. On simple maps --
3. The non-manifold part --
4. The manifold part...

"Since its introduction by Friedhelm Waldhausen in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing Waldhausen's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a "desingularization," improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections."--Publisher's website.