Softcover ISBN:  9780821803196 
Product Code:  CBMS/91 
List Price:  $60.00 
Individual Price:  $48.00 
Electronic ISBN:  9781470424510 
Product Code:  CBMS/91.E 
List Price:  $60.00 
Individual Price:  $48.00 

Book DetailsCBMS Regional Conference Series in MathematicsVolume: 91; 1996; 366 ppMSC: Primary 19; 55;
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. 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 pointset 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.
Features: Introduces many of the fundamental ideas and concepts of modern algebraic topology.
 Presents comprehensive material not found in any other book on the subject.
 Provides a coherent overview of many areas of current interest in algebraic topology.
 Surveys a great deal of material, explaining main ideas without getting bogged down in details.
ReadershipGraduate students and research mathematicians interested in algebraic topology.

Table of Contents

Chapters

1. Introduction

Chapter I. Equivariant cellular and homology theory

Chapter II. Postnikov systems, localization, and completion

Chapter III. Equivariant rational homotopy theory (by Georgia Triantafillou)

Chapter IV. Smith theory

Chapter V. Categorical constructions; equivariant applications

Chapter VI. The homotopy theory of diagrams (by Robert J. Piacenza)

Chapter VII. Equivariant bundle theory and classifying spaces

Chapter VIII. The Sullivan conjecture

Chapter IX. An introduction to equivariant stable homotopy

Chapter X. $G{\rm CW}(V)$ complexes and $RO(G)$graded cohomology (by Stefan Waner)

Chapter XI. The equivariant Hurewicz and suspension theorems (by L. Guance Lewis, Jr.)

Chapter XII. The equivariant stable homotopy category

Chapter XIII. $RO(G)$graded homology and cohomology theories

Chapter XIV. An introduction to equivariant $K$theory (by J. P. C. Greenlees)

Chapter XV. An introduction to equivariant cobordism (by S. R. Costenoble)

Chapter XVI. Spectra and $G$spectra; change of groups; duality

Chapter XVII. The Burnside ring

Chapter XVIII. Transfer maps in equivariant bundle theory

Chapter XIX. Stable homotopy and Mackey functors

Chapter XX. The Segal conjecture

Chapter XXI. Generalized Tate cohomology (by J. P. C. Greenlees and J. P. May)

Chapter XXII. Twisted halfsmash products and function spectra (by Michael Cole)

Chapter XXIII. Brave new algebra

Chapter XXIV, Brave new equivariant foundations (by A. D. Elmendorf, L. G. Lewis, Jr., and J. P. May)

Chapter XXV. Brave new equivariant algebra (by J. P. C. Greenlees and J. P. May)

Chapter XXVI. Localization and completion in complex bordism (by J. P. C. Greenlees and J. P. May)

Chapter XXVII. A completion theorem in complex cobordism (by G. Comezana and J. P. May)

Chapter XXVIII. Calculations in complex equivariant bordism (by G. Comezaña)


Reviews

Absolutely necessary to have this guidebook on the desk … applies to the advanced student as well as to the educated scientist. The presentation is clear, reliable, informative and motivating … there is no comparable recent book in algebraic topology … almost certainly guides further research.
Bulletin of the London Mathematical Society 
The exposition and choice of topics by May and his collaborators are well crafted to bring the uninitiated up to speed in a subject that has a long technical past.
Bulletin of the AMS


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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. 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 pointset 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.
Features:
 Introduces many of the fundamental ideas and concepts of modern algebraic topology.
 Presents comprehensive material not found in any other book on the subject.
 Provides a coherent overview of many areas of current interest in algebraic topology.
 Surveys a great deal of material, explaining main ideas without getting bogged down in details.
Graduate students and research mathematicians interested in algebraic topology.

Chapters

1. Introduction

Chapter I. Equivariant cellular and homology theory

Chapter II. Postnikov systems, localization, and completion

Chapter III. Equivariant rational homotopy theory (by Georgia Triantafillou)

Chapter IV. Smith theory

Chapter V. Categorical constructions; equivariant applications

Chapter VI. The homotopy theory of diagrams (by Robert J. Piacenza)

Chapter VII. Equivariant bundle theory and classifying spaces

Chapter VIII. The Sullivan conjecture

Chapter IX. An introduction to equivariant stable homotopy

Chapter X. $G{\rm CW}(V)$ complexes and $RO(G)$graded cohomology (by Stefan Waner)

Chapter XI. The equivariant Hurewicz and suspension theorems (by L. Guance Lewis, Jr.)

Chapter XII. The equivariant stable homotopy category

Chapter XIII. $RO(G)$graded homology and cohomology theories

Chapter XIV. An introduction to equivariant $K$theory (by J. P. C. Greenlees)

Chapter XV. An introduction to equivariant cobordism (by S. R. Costenoble)

Chapter XVI. Spectra and $G$spectra; change of groups; duality

Chapter XVII. The Burnside ring

Chapter XVIII. Transfer maps in equivariant bundle theory

Chapter XIX. Stable homotopy and Mackey functors

Chapter XX. The Segal conjecture

Chapter XXI. Generalized Tate cohomology (by J. P. C. Greenlees and J. P. May)

Chapter XXII. Twisted halfsmash products and function spectra (by Michael Cole)

Chapter XXIII. Brave new algebra

Chapter XXIV, Brave new equivariant foundations (by A. D. Elmendorf, L. G. Lewis, Jr., and J. P. May)

Chapter XXV. Brave new equivariant algebra (by J. P. C. Greenlees and J. P. May)

Chapter XXVI. Localization and completion in complex bordism (by J. P. C. Greenlees and J. P. May)

Chapter XXVII. A completion theorem in complex cobordism (by G. Comezana and J. P. May)

Chapter XXVIII. Calculations in complex equivariant bordism (by G. Comezaña)

Absolutely necessary to have this guidebook on the desk … applies to the advanced student as well as to the educated scientist. The presentation is clear, reliable, informative and motivating … there is no comparable recent book in algebraic topology … almost certainly guides further research.
Bulletin of the London Mathematical Society 
The exposition and choice of topics by May and his collaborators are well crafted to bring the uninitiated up to speed in a subject that has a long technical past.
Bulletin of the AMS