Spaces of PL Manifolds and Categories of Simple Maps (AM-186) / / Bjørn Jahren, John Rognes, Friedhelm Waldhausen.

• Superior document: Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020
• VerfasserIn: Waldhausen, Friedhelm,; Jahren, Bjørn,; Rognes, John,
• Place / Publishing House: Princeton, NJ : : Princeton University Press, , [2013]; ©2013
• Year of Publication: 2013
• Edition: Course Book
• Language: English
• Series: Annals of Mathematics Studies ; 186
• Physical Description: 1 online resource (192 p.) :; 5 line illus.
• Other title: Frontmatter --; Contents --; Introduction --; 1. The stable parametrized h-cobordism theorem --; 2. On simple maps --; 3. The non-manifold part --; 4. The manifold part --; Bibliography --; Symbols --; Index
• Format: Mode of access: Internet via World Wide Web.
• ISBN: 9781400846528; 9783110494914; 9783110442502
• DOI: 10.1515/9781400846528?locatt=mode:legacy
• Access: restricted access
• Hierarchical level: Monograph
• Statement of Responsibility: Bjørn Jahren, John Rognes, Friedhelm Waldhausen.

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.