Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem (New Mathematical Monographs, Series Number 40)

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. Front matter Copyright Dedications Contents 1 Introduction 1.1 The Kervaire Invariant Theorem and the Ingredients of Its Proof 1.2 Background and History 1.3 The Foundational Material in This Rook 1.4 Highlights of Later Chapters 1.5 Acknowledgments Part I: The Categorical Tool Box 2 Some Categorical Tools 2.1 Basic Definitions and Notational Conventions 2.2 Natural Transformations. Adjoint Functors and Monads 2.3 Limits and Colimits as Adjoint Functors 2.4 Ends and Coends 2.5 Kan Extensions 2.6 Monoidal and Symmetric Monoidal Categories 2.7 2-Categories and Beyond 2.8 Grothendieck Fibrations and Opfibrations 2.9 Indexed Monoidal Products 3 Enriched Category Theory 3.1 Basic Definitions 3.2 Limits, Colimits. Ends and Coends in Enriched Categories 3.3 The Day Convolution 3.4 Simplicial Sets and Simplicial Spaces 3.5 The Homotopy Extension Property, h-Cofibrations and Nondegenerate Base Points 4 Quillen’s Theory of Model Categories 4.1 Basic Definitions 4.2 Three Classical Examples of Model Categories 4.3 Homotopy in a Model Category 4.4 Nonhomotopical and Derived Functors 4.5 Quillen Functors and Quillen Equivalences 4.6 The Suspension and Loop Functors 4.7 Fiber and Cofiber Sequences 4.8 The Small Object Argument 5 Model Category Theory since Quillen 5.1 Homotopical Categories 5.2 Cofibrantly and Compactly Generated Model Categories 5.3 Proper Model Categories 5.4 The Category of Functors from a Small Category to a Cofibrantly Generated Model Category 5.5 Monoidal Model Categories 5.6 Enriched Model Categories 5.7 Stable and Exactly Stable Model Categories 5.8 Homotopy Limits and Colimits 6 Bousfield Localization 6.1 It’s All about Fibrant Replacement 6.2 Bousfield Localization in More General Model Categories 6.3 When Is Left Bousfield Localization Possible? Part II: Setting Up Equivariant Stable Homotopy Theory 7 Spectra and Stable Homotopy Theory 7.1 Hovey’s Generalization of Spectra 7.2 The Functorial Approach to Spectra 7.3 Stabilization and Model Structures for Hovey Spectra 7.4 Stabilization and Model Structures for Smashable Spectra 8 Equivariant Homotopy Theory 8.1 Finite G-Sets and the Bumside Ring of a Finite Group 8.2 Mackey Functors 8.3 Some Formal Properties of G-Spaces 8.4 G-CW Complexes 8.5 The Homology of a G-CW Complex 8.6 Model Structures 8.7 Some Universal Spaces 8.8 Elmendorf's Theorem 8.9 Orthogonal Representations of G and Related Structures 9 Orthogonal G-Spectra 9.1 Categorical Properties of Orthogonal G-Spectra 9.2 Model Structures for Orthogonal G-Spectra 9.3 Naive and Genuine G-Spectra 9.4 Homotopical Properties of G-Spectra 9.5 A Homotopical Approximation to the Category of G-Spectra 9.6 Homotopical Properties of Indexed Wedges and Indexed Smash Products 9.7 The Norm Functor 9.8 Change of Group and Smash Product 9.9 The RO(G)-Graded Homotopy of HZ 9.10 Fixed Point Spectra 9.11 Geometric Fixed Points 10 Multiplicative Properties of G-Spectra 10.1 Equivariant '/'-Diagrams 10.2 Indexed Smash Products and Cofibrations 10.3 The Arrow Category and Indexed Corner Maps 10.4 Indexed Smash Products and Trivial Cofibrations 10.5 Indexed Symmetric Powers 10.6 Iterated Indexed Symmetric Powers 10.7 Commutative Algebras in the Category of G-Spectra 10.8 K-Modules in the Category of Spectra 10.9 Indexed Smash Products of Commutative Rings 10.10 Twisted Monoid Rings Part III: Proving the Kervaire in Variant Theorem 11 The Slice Filtration and Slice Spectral Sequence 11.1 The Filtration behind the Spectral Sequence 11.2 The Slice Spectral Sequence 11.3 Spherical Slices 11.4 The Slice Tower. Symmetric Powers and the Norm 12 The Construction and Properties of MUr 12.1 Real and Complex Spectra 12.2 The Real Bordism Spectrum 12.3 Algebra Generators for πu*MU((G)) 12.4 The Slice Structure of MU((G)) 13 The Proofs of the Gap, Periodicity and Detection Theorems 13.1 A Warm-Up: The Slice Spectral Sequence for MUr 13.2 The Gap Theorem 13.3 The Periodicity Theorem 13.4 The Detection Theorem References Table of Notations Index "This unique book on modern topology looks well beyond traditional treatises, and explores spaces that may, but need not, be Hausdorff. This is essential for domain theory, the cornerstone of semantics of computer languages, where the Scott topology is almost never Hausdorff. For the first time in a single volume, this book covers basic material on metric and topological spaces, advanced material on complete partial orders, Stone duality, stable compactness, quasi-metric spaces, and much more. An early chapter on metric spaces serves as an invitation to the topic (continuity, limits, compactness, completeness) and forms a complete introductory course by itself"-- Provided by publisher