New Titles  |  FAQ  |  Keep Informed  |  Review Cart  |  Contact Us Quick Search (Advanced Search ) Browse by Subject General Interest Logic & Foundations Number Theory Algebra & Algebraic Geometry Discrete Math & Combinatorics Analysis Differential Equations Geometry & Topology Probability & Statistics Applications Mathematical Physics Math Education

Parametrized Homotopy Theory
J. P. May, The University of Chicago, IL, and J. Sigurdsson, University of Sheffield, England
 SEARCH THIS BOOK:
Mathematical Surveys and Monographs
2006; 441 pp; hardcover
Volume: 132
ISBN-10: 0-8218-3922-5
ISBN-13: 978-0-8218-3922-5
List Price: US$102 Member Price: US$81.60
Order Code: SURV/132

Connective Real $$K$$-Theory of Finite Groups - Robert R Bruner and J P C Greenlees

This book develops rigorous foundations for parametrized homotopy theory, which is the algebraic topology of spaces and spectra that are continuously parametrized by the points of a base space. It also begins the systematic study of parametrized homology and cohomology theories.

The parametrized world provides the natural home for many classical notions and results, such as orientation theory, the Thom isomorphism, Atiyah and Poincaré duality, transfer maps, the Adams and Wirthmüller isomorphisms, and the Serre and Eilenberg-Moore spectral sequences. But in addition to providing a clearer conceptual outlook on these classical notions, it also provides powerful methods to study new phenomena, such as twisted $$K$$-theory, and to make new constructions, such as iterated Thom spectra.

Duality theory in the parametrized setting is particularly illuminating and comes in two flavors. One allows the construction and analysis of transfer maps, and a quite different one relates parametrized homology to parametrized cohomology. The latter is based formally on a new theory of duality in symmetric bicategories that is of considerable independent interest.

The text brings together many recent developments in homotopy theory. It provides a highly structured theory of parametrized spectra, and it extends parametrized homotopy theory to the equivariant setting. The theory of topological model categories is given a more thorough treatment than is available in the literature. This is used, together with an interesting blend of classical methods, to resolve basic foundational problems that have no nonparametrized counterparts.