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

$32.00

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. Additional ISBNs 9780691157757|9780691157764|9781400847587, 0691157758|0691157766|1400847583, 9780691157757|9780691157764, 0691157758|0691157766, 9780691157757, 0691157758Spaces of PL Manifolds and Categories of Simple Maps (AM-186) by Friedhelm Waldhausen; Bj�rn Jahren; John Rognes and Publisher Princeton University Press. ISBN: 9781400846528, 1400846528. The print version of this textbook is ISBN: 9780691157764, 0691157766.

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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. Additional ISBNs 9780691157757|9780691157764|9781400847587, 0691157758|0691157766|1400847583, 9780691157757|9780691157764, 0691157758|0691157766, 9780691157757, 0691157758Spaces of PL Manifolds and Categories of Simple Maps (AM-186) by Friedhelm Waldhausen; Bj�rn Jahren; John Rognes and Publisher Princeton University Press. ISBN: 9781400846528, 1400846528. The print version of this textbook is ISBN: 9780691157764, 0691157766.