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Equivariant Homotopy And Cohomology Theory
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Book Synopsis Equivariant Homotopy and Cohomology Theory by : J. Peter May
Download or read book Equivariant Homotopy and Cohomology Theory written by J. Peter May and published by American Mathematical Soc.. This book was released on 1996 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume introduces equivariant homotopy, homology, and cohomology theory, along with various related topics in modern algebraic topology. It explains the main ideas behind some of the most striking recent advances in the subject. The works begins with a development of the equivariant algebraic topology of spaces culminating in a discussion of the Sullivan conjecture that emphasizes its relationship with classical Smith theory. The book then introduces equivariant stable homotopy theory, the equivariant stable homotopy category, and the most important examples of equivariant cohomology theories. The basic machinery that is needed to make serious use of equivariant stable homotopy theory is presented next, along with discussions of the Segal conjecture and generalized Tate cohomology. Finally, the book gives an introduction to "brave new algebra", the study of point-set level algebraic structures on spectra and its equivariant applications. Emphasis is placed on equivariant complex cobordism, and related results on that topic are presented in detail.
Book Synopsis Equivariant Homotopy and Cohomology Theory by : J. Peter May
Download or read book Equivariant Homotopy and Cohomology Theory written by J. Peter May and published by . This book was released on 1996 with total page 366 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume introduces equivariant homotopy, homology, and cohomology theory, along with various related topics in modern algebraic topology. It explains the main ideas behind some of the most striking recent advances in the subject. The book begins with a development of the equivariant algebraic topology of spaces culminating in a discussion of the Sullivan conjecture that emphasizes its relationship with classical Smith theory. It then introduces equivariant stable homotopy theory, the equivariant stable homotopy category, and the most important examples of equivariant cohomology theories. T.
Book Synopsis Equivariant Homotopy and Cohomology Theory by : M. Cole
Download or read book Equivariant Homotopy and Cohomology Theory written by M. Cole and published by American Mathematical Soc.. This book was released on 1996-01-01 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume introduces equivariant homotopy, homology, and cohomology theory, along with various related topics in modern algebraic topology. It explains the main ideas behind some of the most striking recent advances in the subject. The works begins with a development of the equivariant algebraic topology of spaces culminating in a discussion of the Sullivan conjecture that emphasizes its relationship with classical Smith theory. The book then introduces equivariant stable homotopy theory, the equivariant stable homotopy category, and the most important examples of equivariant cohomology theories. The basic machinery that is needed to make serious use of equivariant stable homotopy theory is presented next, along with discussions of the Segal conjecture and generalized Tate cohomology. Finally, the book gives an introduction to "brave new algebra", the study of point-set level algebraic structures on spectra and its equivariant applications. Emphasis is placed on equivariant complex cobordism, and related results on that topic are presented in detail.
Book Synopsis Equivariant Stable Homotopy Theory by : L. Gaunce Jr. Lewis
Download or read book Equivariant Stable Homotopy Theory written by L. Gaunce Jr. Lewis and published by Springer. This book was released on 2006-11-14 with total page 548 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a foundational piece of work in stable homotopy theory and in the theory of transformation groups. It may be roughly divided into two parts. The first part deals with foundations of (equivariant) stable homotopy theory. A workable category of CW-spectra is developed. The foundations are such that an action of a compact Lie group is considered throughout, and spectra allow desuspension by arbitrary representations. But even if the reader forgets about group actions, he will find many details of the theory worked out for the first time. More subtle constructions like smash products, function spectra, change of group isomorphisms, fixed point and orbit spectra are treated. While it is impossible to survey properly the material which is covered in the book, it does boast these general features: (i) a thorough and reliable presentation of the foundations of the theory; (ii) a large number of basic results, principal applications, and fundamental techniques presented for the first time in a coherent theory, unifying numerous treatments of special cases in the literature.
Book Synopsis Equivariant Cohomology Theories by : Glen E. Bredon
Download or read book Equivariant Cohomology Theories written by Glen E. Bredon and published by Springer. This book was released on 2006-11-14 with total page 72 pages. Available in PDF, EPUB and Kindle. Book excerpt: a
Book Synopsis Introductory Lectures on Equivariant Cohomology by : Loring W. Tu
Download or read book Introductory Lectures on Equivariant Cohomology written by Loring W. Tu and published by Princeton University Press. This book was released on 2020-03-03 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives a clear introductory account of equivariant cohomology, a central topic in algebraic topology. Equivariant cohomology is concerned with the algebraic topology of spaces with a group action, or in other words, with symmetries of spaces. First defined in the 1950s, it has been introduced into K-theory and algebraic geometry, but it is in algebraic topology that the concepts are the most transparent and the proofs are the simplest. One of the most useful applications of equivariant cohomology is the equivariant localization theorem of Atiyah-Bott and Berline-Vergne, which converts the integral of an equivariant differential form into a finite sum over the fixed point set of the group action, providing a powerful tool for computing integrals over a manifold. Because integrals and symmetries are ubiquitous, equivariant cohomology has found applications in diverse areas of mathematics and physics. Assuming readers have taken one semester of manifold theory and a year of algebraic topology, Loring Tu begins with the topological construction of equivariant cohomology, then develops the theory for smooth manifolds with the aid of differential forms. To keep the exposition simple, the equivariant localization theorem is proven only for a circle action. An appendix gives a proof of the equivariant de Rham theorem, demonstrating that equivariant cohomology can be computed using equivariant differential forms. Examples and calculations illustrate new concepts. Exercises include hints or solutions, making this book suitable for self-study.
Book Synopsis Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem by : Michael A. Hill
Download or read book Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem written by Michael A. Hill and published by Cambridge University Press. This book was released on 2021-07-29 with total page 881 pages. Available in PDF, EPUB and Kindle. Book excerpt: A complete and definitive account of the authors' resolution of the Kervaire invariant problem in stable homotopy theory.
Book Synopsis Global Homotopy Theory by : Stefan Schwede
Download or read book Global Homotopy Theory written by Stefan Schwede and published by Cambridge University Press. This book was released on 2018-09-06 with total page 847 pages. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive, self-contained approach to global equivariant homotopy theory, with many detailed examples and sample calculations.
Book Synopsis Equivariant Ordinary Homology and Cohomology by : Steven R. Costenoble
Download or read book Equivariant Ordinary Homology and Cohomology written by Steven R. Costenoble and published by Springer. This book was released on 2017-01-02 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: Filling a gap in the literature, this book takes the reader to the frontiers of equivariant topology, the study of objects with specified symmetries. The discussion is motivated by reference to a list of instructive “toy” examples and calculations in what is a relatively unexplored field. The authors also provide a reading path for the first-time reader less interested in working through sophisticated machinery but still desiring a rigorous understanding of the main concepts. The subject’s classical counterparts, ordinary homology and cohomology, dating back to the work of Henri Poincaré in topology, are calculational and theoretical tools which are important in many parts of mathematics and theoretical physics, particularly in the study of manifolds. Similarly powerful tools have been lacking, however, in the context of equivariant topology. Aimed at advanced graduate students and researchers in algebraic topology and related fields, the book assumes knowledge of basic algebraic topology and group actions.
Book Synopsis Nilpotence and Periodicity in Stable Homotopy Theory by : Douglas C. Ravenel
Download or read book Nilpotence and Periodicity in Stable Homotopy Theory written by Douglas C. Ravenel and published by Princeton University Press. This book was released on 1992-11-08 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nilpotence and Periodicity in Stable Homotopy Theory describes some major advances made in algebraic topology in recent years, centering on the nilpotence and periodicity theorems, which were conjectured by the author in 1977 and proved by Devinatz, Hopkins, and Smith in 1985. During the last ten years a number of significant advances have been made in homotopy theory, and this book fills a real need for an up-to-date text on that topic. Ravenel's first few chapters are written with a general mathematical audience in mind. They survey both the ideas that lead up to the theorems and their applications to homotopy theory. The book begins with some elementary concepts of homotopy theory that are needed to state the problem. This includes such notions as homotopy, homotopy equivalence, CW-complex, and suspension. Next the machinery of complex cobordism, Morava K-theory, and formal group laws in characteristic p are introduced. The latter portion of the book provides specialists with a coherent and rigorous account of the proofs. It includes hitherto unpublished material on the smash product and chromatic convergence theorems and on modular representations of the symmetric group.
Book Synopsis Rational $S^1$-Equivariant Stable Homotopy Theory by : John Patrick Campbell Greenlees
Download or read book Rational $S^1$-Equivariant Stable Homotopy Theory written by John Patrick Campbell Greenlees and published by American Mathematical Soc.. This book was released on 1999 with total page 306 pages. Available in PDF, EPUB and Kindle. Book excerpt: The memoir presents a systematic study of rational S1-equivariant cohomology theories, and a complete algebraic model for them. It provides a classification of such cohomology theories in simple algebraic terms and a practical means of calculation. The power of the model is illustrated by analysis of the Segal conjecture, the behaviour of the Atiyah-Hirzebruch spectral sequence, the structure of S1-equivariant K-theory, and the rational behaviour of cyclotomic spectra and the topological cyclic homology construction.
Book Synopsis Generalized Cohomology by : Akira Kōno
Download or read book Generalized Cohomology written by Akira Kōno and published by American Mathematical Soc.. This book was released on 2006 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: Aims to give an exposition of generalized (co)homology theories that can be read by a group of mathematicians who are not experts in algebraic topology. This title starts with basic notions of homotopy theory, and introduces the axioms of generalized (co)homology theory. It also discusses various types of generalized cohomology theories.
Book Synopsis Proper Equivariant Stable Homotopy Theory by : Dieter Degrijse
Download or read book Proper Equivariant Stable Homotopy Theory written by Dieter Degrijse and published by American Mathematical Society. This book was released on 2023-09-15 with total page 154 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.
Book Synopsis Diagram Cohomology and Isovariant Homotopy Theory by : Giora Dula
Download or read book Diagram Cohomology and Isovariant Homotopy Theory written by Giora Dula and published by American Mathematical Soc.. This book was released on 1994 with total page 97 pages. Available in PDF, EPUB and Kindle. Book excerpt: Obstruction theoretic methods are introduced into isovariant homotopy theory for a class of spaces with group actions; the latter includes all smooth actions of cyclic groups of prime power order. The central technical result is an equivalence between isovariant homotopy and specific equivariant homotopy theories for diagrams under suitable conditions. This leads to isovariant Whitehead theorems, an obstruction-theoretic approach to isovariant homotopy theory with obstructions in cohomology groups of ordinary and equivalent diagrams, and qualitative computations for rational homotopy groups of certain spaces of isovariant self maps of linear spheres. The computations show that these homotopy groups are often far more complicated than the rational homotopy groups for the corresponding spaces of equivariant self maps. Subsequent work will use these computations to construct new families of smooth actions on spheres that are topologically linear but differentiably nonlinear.
Book Synopsis Generalized Tate Cohomology by : John Patrick Campbell Greenlees
Download or read book Generalized Tate Cohomology written by John Patrick Campbell Greenlees and published by American Mathematical Soc.. This book was released on 1995 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: Let [italic capital]G be a compact Lie group, [italic capitals]EG a contractible free [italic capital]G-space and let [italic capitals]E~G be the unreduced suspension of [italic capitals]EG with one of the cone points as basepoint. Let [italic]k*[over][subscript italic capital]G be a [italic capital]G-spectrum. Let [italic capital]X+ denote the disjoint union of [italic capital]X and a [italic capital]G-fixed basepoint. Define the [italic capital]G-spectra [italic]f([italic]k*[over][subscript italic capital]G) = [italic]k*[over][subscript italic capital]G [up arrowhead symbol] [italic capitals]EG+, [italic]c([italic]k*[over][subscript italic capital]G) = [italic capital]F([italic capitals]EG+,[italic]k*[over][subscript italic capital]G), and [italic]t([italic]k[subscript italic capital]G)* = [italic capital]F([italic capitals]EG+,[italic]k*[over][subscript italic capital]G) [up arrowhead symbol] [italic capitals]E~G. The last of these is the [italic capital]G-spectrum representing the generalized Tate homology and cohomology theories associated to [italic]k[subscript italic capital]G. Here [italic capital]F([italic capitals]EG+,[italic]k*[over][subscript italic capital]G) is the function space spectrum. The authors develop the properties of these theories, illustrating the manner in which they generalize the classical Tate-Swan theories.
Book Synopsis Complex Cobordism and Stable Homotopy Groups of Spheres by : Douglas C. Ravenel
Download or read book Complex Cobordism and Stable Homotopy Groups of Spheres written by Douglas C. Ravenel and published by American Mathematical Soc.. This book was released on 2003-11-25 with total page 418 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since the publication of its first edition, this book has served as one of the few available on the classical Adams spectral sequence, and is the best account on the Adams-Novikov spectral sequence. This new edition has been updated in many places, especially the final chapter, which has been completely rewritten with an eye toward future research in the field. It remains the definitive reference on the stable homotopy groups of spheres. The first three chapters introduce the homotopy groups of spheres and take the reader from the classical results in the field though the computational aspects of the classical Adams spectral sequence and its modifications, which are the main tools topologists have to investigate the homotopy groups of spheres. Nowadays, the most efficient tools are the Brown-Peterson theory, the Adams-Novikov spectral sequence, and the chromatic spectral sequence, a device for analyzing the global structure of the stable homotopy groups of spheres and relating them to the cohomology of the Morava stabilizer groups. These topics are described in detail in Chapters 4 to 6. The revamped Chapter 7 is the computational payoff of the book, yielding a lot of information about the stable homotopy group of spheres. Appendices follow, giving self-contained accounts of the theory of formal group laws and the homological algebra associated with Hopf algebras and Hopf algebroids. The book is intended for anyone wishing to study computational stable homotopy theory. It is accessible to graduate students with a knowledge of algebraic topology and recommended to anyone wishing to venture into the frontiers of the subject.
Book Synopsis Algebraic Topology by : Mark E. Mahowald
Download or read book Algebraic Topology written by Mark E. Mahowald and published by American Mathematical Soc.. This book was released on 1989 with total page 366 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book will provide readers with an overview of some of the major developments in current research in algebraic topology. Representing some of the leading researchers in the field, the book contains the proceedings of the International Conference on Algebraic Topology, held at Northwestern University in March, 1988. Several of the lectures at the conference were expository and will therefore appeal to topologists in a broad range of areas. The primary emphasis of the book is on homotopy theory and its applications. The topics covered include elliptic cohomology, stable and unstable homotopy theory, classifying spaces, and equivariant homotopy and cohomology. Geometric topics--such as knot theory, divisors and configurations on surfaces, foliations, and Siegel spaces--are also discussed. Researchers wishing to follow current trends in algebraic topology will find this book a valuable resource.
