Ebook: Spaces of PL Manifolds and Categories of Simple Maps (AM-186)
- Genre: Mathematics
- Tags: Reference, Almanacs & Yearbooks, Atlases & Maps, Careers, Catalogs & Directories, Consumer Guides, Dictionaries & Thesauruses, Encyclopedias & Subject Guides, English as a Second Language, Etiquette, Foreign Language Study & Reference, Genealogy, Quotations, Survival & Emergency Preparedness, Test Preparation, Words Language & Grammar, Writing Research & Publishing Guides, Topology, Geometry & Topology, Mathematics, Science & Math, Transformations, Mathematics, Science & Math, Mathematics, Algebra & Trigonometry, Calcu
- Series: Annals of Mathematics Studies
- Year: 2013
- Publisher: Princeton University Press
- Language: English
- pdf
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.