Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem
Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem
Michael J. Hopkins, Harvard University, Massachusetts
Douglas C. Ravenel, University of Rochester, New York
NZD$261.95
inc GSTThe long-standing Kervaire invariant problem in homotopy theory arose from geometric and differential topology in the 1960s and was quickly recognised as one of the most important problems in the field. In 2009 the authors of this book announced a solution to the problem, which was published to wide acclaim in a landmark Annals of Mathematics paper. The proof is long and involved, using many sophisticated tools of modern (equivariant) stable homotopy theory that are unfamiliar to non-experts. This book presents the proof together with a full development of all the background material to make it accessible to a graduate student with an elementary algebraic topology knowledge. There are explicit examples of constructions used in solving the problem.
Also featuring a motivating history of the problem and numerous conceptual and expository improvements on the proof, this is the definitive account of the resolution of the Kervaire invariant problem.
- A complete account of the authors' milestone solution to the Kervaire invariant problem in homotopy theory
- Includes background results in stable homotopy theory that have not previously appeared in the literature
- Accessible to graduate students with an elementary knowledge of algebraic topology
Reviews & endorsements
'… the book succeeds in simultaneously being readable as well as presenting a complex result, in providing tools without being lost in details, and in showing an exciting journey from classical to (at the time of this review) modern stable homotopy theory. Thus, we can expect that it will find a home on many topologists' bookshelves.' Constanze Roitzheim, zbMATH
'The purpose of the book under review is to give an expanded and systematic development of the part of equivariant stable homotopy theory required by readers wishing to understand the proof of the Kervaire Invariant Theorem. The book fully achieves this design aim. The book ends with a 130-page summary of the proof of the theorem, and having this as a target shapes the entire narrative.' J. P. C. Greenlees, MathSciNet
Product details
July 2021Hardback
9781108831444
888 pages
250 × 175 × 47 mm
0.164kg
Available
Table of Contents
- 1. Introduction
- Part I. The Categorical Tool Box:
- 2. Some Categorical Tools
- 3. Enriched Category Theory
- 4. Quillen's Theory of Model Categories
- 5. Model Category Theory Since Quillen
- 6. Bousfield Localization
- Part II. Setting Up Equivariant Stable Homotopy Theory:
- 7. Spectra and Stable Homotopy Theory
- 8. Equivariant Homotopy Theory
- 9. Orthogonal G-spectra
- 10. Multiplicative Properties of G-spectra
- Part III. Proving the Kervaire Invariant Theorem:
- 11. The Slice Filtration and Slice Spectral Sequence
- 12. The Construction and Properties of $MU_{\R}$
- 13. The Proofs of the Gap, Periodicity and Detection Theorems
- References
- Table of Notation
- Index.
