eTextbook: Spaces of PL Manifolds and Categories of Simple Maps (AM-186)
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Author(s): Waldhausen, Friedhelm; Jahren, BjÃ¸rn; Rognes, John
ISBN 10: 0691157758. See all ISBN's
ISBN 13: 9780691157757
Publisher: Princeton University Press
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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.
All ISBN's for this title:
ISBN 10: 0691157758, ISBN 13: 9780691157757, eISBN 10: 1400846528, eISBN 13: 97814008465281. List price can refer to either the digital or print list price, whichever is available to Kno.
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