Spaces of PL Manifolds and Categories of Simple Maps (Am-186) Contributor(s): Waldhausen, Friedhelm (Author), Jahren, Bjørn (Author), Rognes, John (Author) |
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ISBN: 0691157766 ISBN-13: 9780691157764 Publisher: Princeton University Press OUR PRICE: $88.35 Product Type: Paperback - Other Formats Published: April 2013 |
Additional Information |
BISAC Categories: - Mathematics | Transformations - Mathematics | Topology - General |
Dewey: 514.22 |
LCCN: 2012038155 |
Series: Annals of Mathematics Studies (Paperback) |
Physical Information: 0.6" H x 6.1" W x 9.1" (0.60 lbs) 192 pages |
Descriptions, Reviews, Etc. |
Publisher Description: 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. |