ENGLISH

Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

By Björn Jahren , Friedhelm Waldhausen , John Rognes

Book information

Publisher
Princeton University Press
Year
2013
ISBN
0691157758, 9780691157757
Language
english
Format
PDF
Filesize
1 MB (1422901 bytes)
Series
Annals of Mathematics Studies
Pages
192\192
Time added
2016-06-09 14:19:44
Language & Grammar;WritingReference;Almanacs & Yearbooks;Atlases & Maps;CareResearch & Publishing Guides;Topology;Geometry & T

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.

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