Spaces of PL Manifolds and Categories of Simple Maps (Annals of Mathematics Studies)

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.

Friedhelm Waldhausen is professor emeritus of mathematics at Bielefeld University. Bjørn Jahren is professor of mathematics at the University of Oslo. John Rognes is professor of mathematics at the University of Oslo.

Introduction 1

1. The stable parametrized h-cobordism theorem 7

1.1. The manifold part 7

1.2. The non-manifold part 13

1.3. Algebraic K-theory 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 non-singular 68

2.6. The approximate lifting property 74

2.7. Subdivision of simplicial sets over Δ^q 83

3. The non-manifold 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