TY - BOOK AU - Waldhausen,Friedhelm AU - Jahren,Bjorn AU - Rognes,John TI - Spaces of PL manifolds and categories of simple maps T2 - Annals of mathematics studies SN - 9780691157764 (pbk. : acidfree paper) U1 - 514.22 23 PY - 2013/// CY - Princeton, Oxford PB - Princeton University Press KW - Piecewise linear topology KW - Mappings (Mathematics) N1 - 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 N2 - "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 ER -