eBook

Spaces of PL manifolds and categories of simple maps

Waldhausen, Friedhelm, Rognes, John, Jahren, Bjørn

2013,2015

Overview

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.

• Publisher: Princeton University Press
• Subject: Adjoint functors / Algebraic K-theory / Atlas (topology) / Automorphism / Bundle map / Canonical map / Cartesian product / Cobordism / Codimension / Cofibration / Commutative property / Connected space / Continuous function / Contractible space / CW complex / Diagram (category theory) / Differentiable manifold / Embedding / Equivalence class / Euclidean space / Factorization / Fiber bundle / Frame bundle / Functor / Fundamental group / General position / Geometric topology / H-cobordism / Handle decomposition / Homotopy / Homotopy colimit / Homotopy fiber / Ideal (ring theory) / Inclusion map / Initial and terminal objects / Kan extension / Lefschetz duality / Limit (category theory) / Loop space / Mapping cone / Mapping cylinder / Mappings (Mathematics) / Mathematical induction / MATHEMATICS / MATHEMATICS / General / MATHEMATICS / Geometry / General / Metric space / Moduli space / Monotonic function / Morphism / Natural transformation / Open set / Partially ordered set / Piecewise linear topology / Polyhedron / Pullback (category theory) / Sheaf (mathematics) / Simplex / Simplicial category / Simplicial complex / Simplicial set / Smoothing / Special case / Spectral sequence / Submanifold / Subset / Tangent bundle / Theorem / Topological manifold / Topological space / Topology / Transversality (mathematics) / Universal coefficient theorem / Whitehead torsion / Yoneda lemma
• ISBN: 0691157766, 9780691157764, 0691157758, 9780691157757, 9781400846528, 1400846528