Spaces of PL Manifolds and Categories of Simple Maps : AM-186  - Waldhausen, Friedhelm

Spaces of PL Manifolds and Categories of Simple Maps : AM-186

Friedhelm Waldhausen

Yayınevi: Princeton University Press

Yayın tarihi: 04/2013

ISBN: 9780691157764

Yazar : Bjørn Jahren John Rognes

İngilizce | 192 Sayfa | 15,49x22,86x1,52 cm.

Tür: Matematik-İstatistik

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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.


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