Mathematical Surveys and Monographs 1997; 249 pp; softcover Volume: 47 ISBN10: 0821843036 ISBN13: 9780821843031 List Price: US$71 Member Price: US$56.80 Order Code: SURV/47.S
 This book introduces a new pointset level approach to stable homotopy theory that has already had many applications and promises to have a lasting impact on the subject. Given the sphere spectrum \(S\), the authors construct an associative, commutative, and unital smash product in a complete and cocomplete category of "\(S\)modules" whose derived category is equivalent to the classical stable homotopy category. This construction allows for a simple and algebraically manageable definition of "\(S\)algebras" and "commutative \(S\)algebras" in terms of associative, or associative and commutative, products \(R\wedge _SR \longrightarrow R\). These notions are essentially equivalent to the earlier notions of \(A_{\infty }\) and \(E_{\infty }\) ring spectra, and the older notions feed naturally into the new framework to provide plentiful examples. There is an equally simple definition of \(R\)modules in terms of maps \(R\wedge _SM\longrightarrow M\). When \(R\) is commutative, the category of \(R\)modules also has an associative, commutative, and unital smash product, and its derived category has properties just like the stable homotopy category. These constructions allow the importation into stable homotopy theory of a great deal of pointset level algebra. Readership Graduate students and research mathematicians interested in algebraic topology. Reviews "Very well organized ... The exposition is quite clear, with just the right amount of motivational comments. All algebraic topologists should obtain some familiarity with the contents of this book."  Mathematical Reviews Table of Contents  Introduction
 Prologue: the category of \({\mathbb L}\)spectra
 Structured ring and module spectra
 The homotopy theory of \(R\)modules
 The algebraic theory of \(R\)modules
 \(R\)ring spectra and the specialization to \(MU\)
 Algebraic \(K\)theory of \(S\)algebras
 \(R\)algebras and topological model categories
 Bousfield localizations of \(R\)modules and algebras
 Topological Hochschild homology and cohomology
 Some basic constructions on spectra
 Spaces of linear isometries and technical theorems
 The monadic bar construction
 Epilogue: The category of \({\mathbb L}\)spectra under \(S\)
 Appendix A. Twisted halfsmash products and function spectra
 Bibliography
 Index
