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.