Spaces of PL manifolds and categories of simple maps
 Author/Creator
 Waldhausen, Friedhelm, 1938
 Language
 English.
 Publication
 Princeton : Princeton University Press, 2013.
 Physical description
 184 pages ; 24 cm.
 Series
 Annals of mathematics studies ; no. 186.
Access
Available online

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QA1 .A665 NO.186

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QA1 .A665 NO.186
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Contributors
 Contributor
 Jahren, Bjørn, 1945
 Rognes, John.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 175178) and index.
 Contents

 Introduction 1 1. The stable parametrized hcobordism theorem 7 1.1. The manifold part 7 1.2. The nonmanifold part 13 1.3. Algebraic Ktheory of spaces 15 1.4. Relation to other literature 20 2. On simple maps 29 2.1. Simple maps of simplicial sets 29 2.2. Normal subdivision of simplicial sets 34 2.3. Geometric realization and subdivision 42 2.4. The reduced mapping cylinder 56 2.5. Making simplicial sets nonsingular 68 2.6. The approximate lifting property 74 2.7. Subdivision of simplicial sets over DELTAq 83 3. The nonmanifold part 99 3.1. Categories of simple maps 99 3.2. Filling horns 108 3.3. Some homotopy fiber sequences 119 3.4. Polyhedral realization 126 3.5. Turning Serre fibrations into bundles 131 3.6. Quillen's Theorems A and B 134 4. The manifold part 139 4.1. Spaces of PL manifolds 139 4.2. Spaces of thickenings 150 4.3. Straightening the thickenings 155 Bibliography 175 Symbols 179 Index 181.
 (source: Nielsen Book Data)
 Publisher's Summary
 Since its introduction by Friedhelm Waldhausen in the 1970s, the algebraic Ktheory 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 hcobordism theorem, derived from a homotopy equivalence between a space of PL hcobordisms 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.
(source: Nielsen Book Data)
Bibliographic information
 Publication date
 2013
 Responsibility
 Friedhelm Waldhausen, Bjørn Jahren, and John Rognes.
 Series
 Annals of mathematics studies ; no. 186
 ISBN
 9780691157757 (hardcover : acidfree paper)
 0691157758 (hardcover : acidfree paper)
 9780691157764 (pbk. : acidfree paper)
 0691157766 (pbk. : acidfree paper)