Abstract | Keywords | Export | Availability | Bookmark

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Since its introduction by Friedhelm Waldhausen in the 1970's, 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.

Piecewise linear topology. --- Mappings (Mathematics) --- Maps (Mathematics) --- Functions --- Functions, Continuous --- Topology --- Transformations (Mathematics) --- PL topology --- Manifolds (Mathematics) --- CAT Whitehead space. --- CAT manifold. --- Euclidean neighborhood retract. --- Euclidean polyhedra. --- PL Serre fibration. --- PL Whitehead space. --- PL homeomorphism. --- PL manifold. --- Quillen's Theorem A. --- Quillen's Theorem B. --- absolute neighborhood retract. --- algebraic K-theory of spaces. --- algebraic K-theory. --- approximate lifting property. --- cofibration. --- degeneracy operator. --- desingularization. --- finite simplicial set. --- h-cobordism theorem. --- homotopy equivalence. --- homotopy fiber sequence. --- horn. --- manifold space. --- manifold. --- n-manifolds. --- normal subdivision. --- polyhedral realization. --- q-simplex. --- reduced mapping cylinder. --- simple map. --- simplicial set. --- space. --- stabilization map. --- stable parametrized h-cobordism theorem. --- tangent microbundle. --- thickening. --- triangulation. --- weak homotopy equivalence. --- Čech homotopy type.