Spaces of PL Manifolds and Categories of Simple Maps (AM-186) Paperback / softback
Part of the Annals of Mathematics Studies series
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.
- Format: Paperback / softback
- Pages: 192 pages, 5 line illus.
- Publisher: Princeton University Press
- Publication Date: 28/04/2013
- Category: Functional analysis & transforms
- ISBN: 9780691157764
- Hardback from £128.05
- PDF from £63.00