Axiomatic stable homotopy theory
About this Title
Mark Hovey, John H. Palmieri and Neil P. Strickland
Publication: Memoirs of the American Mathematical Society
Publication Year 1997: Volume 128, Number 610
ISBNs: 978-0-8218-0624-1 (print); 978-1-4704-0195-5 (online)
MathSciNet review: 1388895
MSC (1991): Primary 55U35; Secondary 18E30, 18G35, 55N20, 55P42, 55P60, 55U15
This book gives an axiomatic presentation of stable homotopy theory. It starts with axioms defining a “stable homotopy category”; using these axioms, one can make various constructions—cellular towers, Bousfield localization, and Brown representability, to name a few. Much of the book is devoted to these constructions and to the study of the global structure of stable homotopy categories.
Next, a number of examples of such categories are presented. Some of these arise in topology (the ordinary stable homotopy category of spectra, categories of equivariant spectra, and Bousfield localizations of these), and others in algebra (coming from the representation theory of groups or of Lie algebras, as well as the derived category of a commutative ring). Hence one can apply many of the tools of stable homotopy theory to these algebraic situations.
Provides a reference for standard results and constructions in stable homotopy theory.
Discusses applications of those results to algebraic settings, such as group theory and commutative algebra.
Provides a unified treatment of several different situations in stable homotopy, including equivariant stable homotopy and localizations of the stable homotopy category.
Provides a context for nilpotence and thick subcategory theorems, such as the nilpotence theorem of Devinatz-Hopkins-Smith and the thick subcategory theorem of Hopkins-Smith in stable homotopy theory, and the thick subcategory theorem of Benson-Carlson-Rickard in representation theory.
This book presents stable homotopy theory as a branch of mathematics in its own right with applications in other fields of mathematics. It is a first step toward making stable homotopy theory a tool useful in many disciplines of mathematics.
Graduate students and research mathematicians interested in algebraic topology, representation theory, and algebraic geometry.
Table of Contents
- 1. Introduction and definitions
- 2. Smallness, limits and constructibility
- 3. Bousfield localization
- 4. Brown representability
- 5. Nilpotence and thick subcategories
- 6. Noetherian stable homotopy categories
- 7. Connective stable homotopy theory
- 8. Semisimple stable homotopy theory
- 9. Examples of stable homotopy categories
- 10. Future directions