Equivariant stable homotopy theory and the Kervaire invariant problem:
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
Cambridge, UK ; New York, NY
Cambridge University Press
2021
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| Series: | New mathematical monographs
40 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | ix, 888 Seiten |
| ISBN: | 9781108831444 9781108932943 |
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MARC
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| 245 | 1 | 0 | |a Equivariant stable homotopy theory and the Kervaire invariant problem |c Michael A. Hill, University of California, Los Angeles ; Michael J. Hopkins, Harvard University ; Douglas C. Ravenel, University of Rochester |
| 264 | 1 | |a Cambridge, UK ; New York, NY |b Cambridge University Press |c 2021 | |
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| 700 | 1 | |a Hopkins, Michael J. |d 1958- |0 (DE-588)1237728711 |4 aut | |
| 700 | 1 | |a Ravenel, Douglas C. |d 1947- |0 (DE-588)1237728789 |4 aut | |
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Record in the Search Index
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| adam_text | 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 PART ONE THE CATEGORICAL TOOL BOX page 1 1 6 17 24 48 49 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 51 53 72 91 119 129 134 156 161 164 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, fi-Cofibrations and Nondegenerate Base Points 190 191 209 221 228 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 244 246 256 263 4 234 vii
Contents viii 4.4 4.5 4.6 4.7 4.8 5 6 Nonhomotopical and Derived Functors Quillen Functors and Quillen Equivalences The Suspension and Loop Functors Fiber and Cofiber Sequences The Small Object Argument 272 27g 284 288 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 321 5.5 Monoidal Model Categories 5.6 Enriched Model Categories 5.7 Stable and Exactly Stable Model Categories 5.8 Homotopy Limits and Colimits 294 299 308 320 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? 391 393 395 403 PART TWO SETTING UP EQUIVARIANT STABLE HOMOTOPY THEORY 335 350 366 371 409 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 411 422 432 453 474 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 495 504 510 519 528 531 538 547 549 551 9 Orthogonal G-Spectra 9.1 Categorical Properties of Orthogonal G-Spectra 9.2 Model Structures
for Orthogonal G-Spectra 566 568 584
Contents 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 10 Naive and Genuine G-Spectra Homotopical Properties of G-Spectra A Homotopical Approximation to the Category of G-Spectra Homotopical Properties of Indexed Wedges and Indexed Smash Products 615 The Norm Functor Change of Group and Smash Product The RO (G)-Graded Homotopy of HZ Fixed Point Spectra Geometric Fixed Points 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 PART THREE PROVING THE KERVAIRE INVARIANT THEOREM ix 590 599 608 617 622 624 639 643 663 665 666 670 671 674 684 687 690 694 703 711 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 713 714 731 739 744 12 The Construction and Properties of MU^ 12.1 Real and Complex Spectra 12.2 The Real Bordism Spectrum 12.3 Algebra Generators for 12.4 The Slice Structure of MU^G^ 750 752 761 772 ТП 13 The Proofs of the Gap, Periodicity and Detection Theorems 13.1 A Warm-Up: The Slice Spectral Sequence for Af Ur 13.2 The Gap Theorem 13.3 The Periodicity Theorem 13.4 The Detection Theorem 793 794 799 801 817 References Table
of Notations Index 830 842 856
|
| adam_txt |
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 PART ONE THE CATEGORICAL TOOL BOX page 1 1 6 17 24 48 49 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 51 53 72 91 119 129 134 156 161 164 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, fi-Cofibrations and Nondegenerate Base Points 190 191 209 221 228 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 244 246 256 263 4 234 vii
Contents viii 4.4 4.5 4.6 4.7 4.8 5 6 Nonhomotopical and Derived Functors Quillen Functors and Quillen Equivalences The Suspension and Loop Functors Fiber and Cofiber Sequences The Small Object Argument 272 27g 284 288 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 321 5.5 Monoidal Model Categories 5.6 Enriched Model Categories 5.7 Stable and Exactly Stable Model Categories 5.8 Homotopy Limits and Colimits 294 299 308 320 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? 391 393 395 403 PART TWO SETTING UP EQUIVARIANT STABLE HOMOTOPY THEORY 335 350 366 371 409 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 411 422 432 453 474 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 495 504 510 519 528 531 538 547 549 551 9 Orthogonal G-Spectra 9.1 Categorical Properties of Orthogonal G-Spectra 9.2 Model Structures
for Orthogonal G-Spectra 566 568 584
Contents 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 10 Naive and Genuine G-Spectra Homotopical Properties of G-Spectra A Homotopical Approximation to the Category of G-Spectra Homotopical Properties of Indexed Wedges and Indexed Smash Products 615 The Norm Functor Change of Group and Smash Product The RO (G)-Graded Homotopy of HZ Fixed Point Spectra Geometric Fixed Points 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 PART THREE PROVING THE KERVAIRE INVARIANT THEOREM ix 590 599 608 617 622 624 639 643 663 665 666 670 671 674 684 687 690 694 703 711 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 713 714 731 739 744 12 The Construction and Properties of MU^ 12.1 Real and Complex Spectra 12.2 The Real Bordism Spectrum 12.3 Algebra Generators for 12.4 The Slice Structure of MU^G^ 750 752 761 772 ТП 13 The Proofs of the Gap, Periodicity and Detection Theorems 13.1 A Warm-Up: The Slice Spectral Sequence for Af Ur 13.2 The Gap Theorem 13.3 The Periodicity Theorem 13.4 The Detection Theorem 793 794 799 801 817 References Table
of Notations Index 830 842 856 |
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| index_date | 2024-07-03T18:06:34Z |
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| spelling | Hill, Michael A. 1980- (DE-588)1237728525 aut Equivariant stable homotopy theory and the Kervaire invariant problem Michael A. Hill, University of California, Los Angeles ; Michael J. Hopkins, Harvard University ; Douglas C. Ravenel, University of Rochester Cambridge, UK ; New York, NY Cambridge University Press 2021 © 2021 ix, 888 Seiten txt rdacontent n rdamedia nc rdacarrier New mathematical monographs 40 Homotopy theory Hopkins, Michael J. 1958- (DE-588)1237728711 aut Ravenel, Douglas C. 1947- (DE-588)1237728789 aut Erscheint auch als Online-Ausgabe 978-1-108-91727-8 New mathematical monographs 40 (DE-604)BV035420183 40 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032863068&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hill, Michael A. 1980- Hopkins, Michael J. 1958- Ravenel, Douglas C. 1947- Equivariant stable homotopy theory and the Kervaire invariant problem New mathematical monographs Homotopy theory |
| title | Equivariant stable homotopy theory and the Kervaire invariant problem |
| title_auth | Equivariant stable homotopy theory and the Kervaire invariant problem |
| title_exact_search | Equivariant stable homotopy theory and the Kervaire invariant problem |
| title_exact_search_txtP | Equivariant stable homotopy theory and the Kervaire invariant problem |
| title_full | Equivariant stable homotopy theory and the Kervaire invariant problem Michael A. Hill, University of California, Los Angeles ; Michael J. Hopkins, Harvard University ; Douglas C. Ravenel, University of Rochester |
| title_fullStr | Equivariant stable homotopy theory and the Kervaire invariant problem Michael A. Hill, University of California, Los Angeles ; Michael J. Hopkins, Harvard University ; Douglas C. Ravenel, University of Rochester |
| title_full_unstemmed | Equivariant stable homotopy theory and the Kervaire invariant problem Michael A. Hill, University of California, Los Angeles ; Michael J. Hopkins, Harvard University ; Douglas C. Ravenel, University of Rochester |
| title_short | Equivariant stable homotopy theory and the Kervaire invariant problem |
| title_sort | equivariant stable homotopy theory and the kervaire invariant problem |
| topic | Homotopy theory |
| topic_facet | Homotopy theory |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032863068&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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