Equivariant stable homotopy theory and the Kervaire invariant problem

Description

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 Book 1.4 Highlights of Later Chapters 1.5 Acknowledgments 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? 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 Burnside 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 T-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 R-Modules in the Category of Spectra 10.9 Indexed Smash Products of Commutative Rings 10.10 Twisted Monoid Rings III Proving the Kervaire Invariant 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 MU_R 12.1 Real and Complex Spectra 12.2 The Real Bordism Spectrum 12.3 Algebra Generators for (π_*)^u(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 MU_R 13.2 The Gap Theorem 13.3 The Periodicity Theorem 13.4 The Detection Theorem References Table of Notations Index