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Equivariant Homotopy and Cohomology Theory

J.P. May University of Chicago, Chicago, IL
A co-publication of the AMS and CBMS
Available Formats:
Softcover ISBN: 978-0-8218-0319-6
Product Code: CBMS/91
List Price: $60.00 Individual Price:$48.00
Electronic ISBN: 978-1-4704-2451-0
Product Code: CBMS/91.E
List Price: $60.00 Individual Price:$48.00
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List Price: $90.00 Click above image for expanded view Equivariant Homotopy and Cohomology Theory J.P. May University of Chicago, Chicago, IL A co-publication of the AMS and CBMS Available Formats:  Softcover ISBN: 978-0-8218-0319-6 Product Code: CBMS/91  List Price:$60.00 Individual Price: $48.00  Electronic ISBN: 978-1-4704-2451-0 Product Code: CBMS/91.E  List Price:$60.00 Individual Price: $48.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$90.00
• Book Details

CBMS Regional Conference Series in Mathematics
Volume: 911996; 366 pp
MSC: 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 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.

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 half-smash 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 guide-book 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
Volume: 911996; 366 pp
MSC: 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 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.

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 half-smash 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 guide-book 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
Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.