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E-grāmata: Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem

(Harvard University, Massachusetts), (University of Rochester, New York), (University of California, Los Angeles)
  • Formāts: PDF+DRM
  • Sērija : New Mathematical Monographs
  • Izdošanas datums: 29-Jul-2021
  • Izdevniecība: Cambridge University Press
  • Valoda: eng
  • ISBN-13: 9781108912907
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Equivariant Stable Homotopy Theory and the Kervaire Invariant ProblemPalielināt
  • Formāts: PDF+DRM
  • Sērija : New Mathematical Monographs
  • Izdošanas datums: 29-Jul-2021
  • Izdevniecība: Cambridge University Press
  • Valoda: eng
  • ISBN-13: 9781108912907
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"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"--

This is the definitive account of the resolution of the Kervaire invariant problem, a major milestone in algebraic topology. It develops all the machinery that is needed for the proof, and details many explicit constructions and computations performed along the way, making it suitable for graduate students as well as experts in homotopy theory.

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.

Recenzijas

' 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

Papildus informācija

A complete and definitive account of the authors' resolution of the Kervaire invariant problem in stable homotopy theory.
1 Introduction
1(48)
1.1 The Kervaire Invariant Theorem and the Ingredients of Its Proof
1(5)
1.2 Background and History
6(11)
1.3 The Foundational Material in This Book
17(7)
1.4 Highlights of Later
Chapters
24(24)
1.5 Acknowledgments
48(1)
PART ONE THE CATEGORICAL TOOL BOX
49(360)
2 Some Categorical Tools
51(139)
2.1 Basic Definitions and Notational Conventions
53(19)
2.2 Natural Transformations, Adjoint Functors and Monads
72(19)
2.3 Limits and Colimits as Adjoint Functors
91(28)
2.4 Ends and Coends
119(10)
2.5 Kan Extensions
129(5)
2.6 Monoidal and Symmetric Monoidal Categories
134(22)
2.7 2-Categories and Beyond
156(5)
2.8 Grothendieck Fibrations and Opfibrations
161(3)
2.9 Indexed Monoidal Products
164(26)
3 Enriched Category Theory
190(54)
3.1 Basic Definitions
191(18)
3.2 Limits, Colimits, Ends and Coends in Enriched Categories
209(12)
3.3 The Day Convolution
221(7)
3.4 Simplicial Sets and Simplicial Spaces
228(6)
3.5 The Homotopy Extension Property, h-Cofibrations and Nondegenerate Base Points
234(10)
4 Quillen's Theory of Model Categories
244(50)
4.1 Basic Definitions
246(10)
4.2 Three Classical Examples of Model Categories
256(7)
4.3 Homotopy in a Model Category
263(6)
4.4 Nonhomotopical and Derived Functors
269(3)
4.5 Quillen Functors and Quillen Equivalences
272(6)
4.6 The Suspension and Loop Functors
278(6)
4.7 Fiber and Conner Sequences
284(4)
4.8 The Small Object Argument
288(6)
5 Model Category Theory since Quillen
294(97)
5.1 Homotopical Categories
299(9)
5.2 Cofibrantly and Compactly Generated Model Categories
308(12)
5.3 Proper Model Categories
320(1)
5.4 The Category of Functors from a Small Category to a Cofibrantly Generated Model Category
321(14)
5.5 Monoidal Model Categories
335(15)
5.6 Enriched Model Categories
350(16)
5.7 Stable and Exactly Stable Model Categories
366(5)
5.8 Homotopy Limits and Colimits
371(20)
6 Bousfield Localization
391(18)
6.1 It's All about Fibrant Replacement
393(2)
6.2 Bousfield Localization in More General Model Categories
395(8)
6.3 When is Left Bousfield Localization Possible?
403(6)
PART TWO SETTING UP EQUIVARIANT STABLE HOMOTOPY THEORY
409(302)
7 Spectra and Stable Homotopy Theory
411(84)
7.1 Hovey's Generalization of Spectra
422(10)
7.2 The Functorial Approach to Spectra
432(21)
7.3 Stabilization and Model Structures for Hovey Spectra
453(21)
7.4 Stabilization and Model Structures for Smashable Spectra
474(21)
8 Equivariant Homotopy Theory
495(71)
8.1 Finite G-Sets and the Burnside Ring of a Finite Group
504(6)
8.2 Mackey Functors
510(9)
8.3 Some Formal Properties of G-Spaces
519(9)
8.4 G-CW Complexes
528(3)
8.5 The Homology of a G-CW Complex
531(7)
8.6 Model Structures
538(9)
8.7 Some Universal Spaces
547(2)
8.8 Elmendorf's Theorem
549(2)
8.9 Orthogonal Representations of G and Related Structures
551(15)
9 Orthogonal G-Spectra
566(97)
9.1 Categorical Properties of Orthogonal G-Spectra
568(16)
9.2 Model Structures for Orthogonal G-Spectra
584(6)
9.3 Naive and Genuine G-Spectra
590(9)
9.4 Homotopical Properties of G-Spectra
599(9)
9.5 A Homotopical Approximation to the Category of G-Spectra
608(7)
9.6 Homotopical Properties of Indexed Wedges and Indexed Smash Products
615(2)
9.7 The Norm Functor
617(5)
9.8 Change of Group and Smash Product
622(2)
9.9 The /?0(G)-Graded Homotopy of HZ
624(15)
9.10 Fixed Point Spectra
639(4)
9.11 Geometric Fixed Points
643(20)
10 Multiplicative Properties of G-Spectra
663(48)
10.1 Equivariant 7-Diagrams
665(1)
10.2 Indexed Smash Products and Cofibrations
666(4)
10.3 The Arrow Category and Indexed Corner Maps
670(1)
10.4 Indexed Smash Products and Trivial Cofibrations
671(3)
10.5 Indexed Symmetric Powers
674(10)
10.6 Iterated Indexed Symmetric Powers
684(3)
10.7 Commutative Algebras in the Category of G-Spectra
687(3)
10.8 fl-Modules in the Category of Spectra
690(4)
10.9 Indexed Smash Products of Commutative Rings
694(9)
10.10 Twisted Monoid Rings
703(8)
PART THREE PROVING THE KERVAIRE INVARIANT THEOREM
711(131)
11 The Slice Filtration and Slice Spectral Sequence
713(37)
11.1 The Filtration behind the Spectral Sequence
714(17)
11.2 The Slice Spectral Sequence
731(8)
11.3 Spherical Slices
739(5)
11.4 The Slice Tower, Symmetric Powers and the Norm
744(6)
12 The Construction and Properties of M Ur
750(43)
12.1 Real and Complex Spectra
752(9)
12.2 The Real Bordism Spectrum
761(11)
12.3 Algebra Generators for πuMU((G))
772(5)
12.4 The Slice Structure of MU((G))
777(16)
13 The Proofs of the Gap, Periodicity and Detection Theorems
793(1)
13.1 A Warm-Up: the Slice Spectral Sequence for MUr
794(5)
13.2 The Gap Theorem
799(2)
13.3 The Periodicity Theorem
801(16)
13.4 The Detection Theorem
817(13)
References
830(12)
Table of Notations 842(14)
Index 856
Michael A. Hill is Professor at the University of California, Los Angeles. He is the author of several papers on algebraic topology and is an editor for journals including Mathematische Zeitschrift and Transactions of the American Mathematical Society. Michael J. Hopkins is Professor at Harvard University. His research concentrates on algebraic topology. In 2001, he was awarded the Oswald Veblen Prize in Geometry from the AMS for his work in homotopy theory, followed by the NAS Award in Mathematics in 2012 and the Nemmers Prize in Mathematics in 2014. Douglas C. Ravenel is the Fayerweather Professor of Mathematics at the University of Rochester. He is the author of two influential previous books in homotopy theory and roughly 75 journal articles on stable homotopy theory and algebraic topology.
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