Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem

A complete and definitive account of the authors' resolution of the Kervaire invariant problem in stable homotopy theory.

Michael A. Hill (Author), Michael J. Hopkins (Author), Douglas C. Ravenel (Author)

9781108831444, Cambridge University Press

Hardback, published 29 July 2021

888 pages
25 x 17.5 x 4.7 cm, 0.164 kg

'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

The 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.

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.

Subject Areas: Algebraic topology [PBPD], Mathematics [PB]