**CBMS Regional Conference Series in Mathematics**

Volume: 91;
1996;
366 pp;
Softcover

MSC: Primary 19; 55; 57;
Secondary 18

Print 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

# Equivariant Homotopy and Cohomology Theory

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*J.P. May; M. Cole; G. Comezana; S. Costenoble; A. D. Elmendorf; J. P. C. Greenlees; L. G. Lewis, Jr.; R. J. Piacenza; G. Triantafillou; S. Waner*

A co-publication of the AMS and CBMS

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.

#### Table of Contents

# Table of Contents

## Equivariant Homotopy and Cohomology Theory

Table of Contents pages: 1 2

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents ix10 free
- Introduction 116 free
- Chapter II. Postnikov systems, localization, and completion 2136
- Chapter III. Equivariant rational homotopy theory 2742
- Chapter IV. Smith theory 3348
- Chapter V. Categorical constructions; equivariant applications 3954
- Chapter VI. The homotopy theory of diagrams 4762
- Chapter VII. Equivariant bundle theory and classifying spaces 5974
- Chapter VIII. The Sullivan conjecture 6782
- Chapter IX. An introduction t o equivariant stable homotopy 7994
- Chapter X. G-CW(V) complexes and RO(G)-graded cohomology 89104
- Chapter XI. The equivariant Hurewicz and suspension theorems 97112
- Chapter XII. The equivariant stable homotopy category 111126
- Chapter XIII. RO(G)-graded homology and cohomology theories 129144
- Chapter XIV. An introduction t o equivariant K-theory 143158
- Chapter XV. An introduction to equivariant cobordism 153168
- Chapter XVI. Spectra and G-spectra; change of groups; duality 163178
- 1. Fixed point spectra and orbit spectra 163178
- 2. Split G-spectra and free G-spectra 165180
- 3. Geometric fixed point spectra 166181
- 4. Change of groups and the Wirthmuller isomorphism 167182
- 5. Quotient groups and the Adams isomorphism 169184
- 6. The construction of G/N-spectra from G-spectra 171186
- 7. Spanier-Whitehead duality 173188
- 8. V-duality of G-spaces and Atiyah duality 175190
- 9. Poincaré duality 176191

- Chapter XVII. The Burnside ring 179194
- 1. Generalized Euler characteristics and transfer maps 179194
- 2. The Burnside ring A(G) and the zero stem π[sup(G)][sub(0)](S) 182197
- 3. Prime ideals of the Burnside ring 183198
- 4. Idempotent elements of the Burnside ring 185200
- 5. Localizations of the Burnside ring 186201
- 6. Localization of equivariant homology and cohomology 188203

- Chapter XVIII. Transfer maps in equivariant bundle theory 191206
- Chapter XIX. Stable homotopy and Mackey functors 203218
- Chapter XX. The Segal conjecture 215230
- 1. The statement in terms of completions of G-spectra 215230
- 2. A calculational reformulation 217232
- 3. A generalization and the reduction to finite p-groups 219234
- 4. The proof of the Segal conjecture for finite p-groups 221236
- 5. Approximations of singular subspaces of G-spaces 223238
- 6. An inverse limit of Adams spectral sequences 225240
- 7. Further generalizations; maps between classifying spaces 227242

- Chapter XXI. Generalized Tate cohomology 231246
- Chapter XXII. Twisted half-smash products and function spectra 247262
- Chapter XXIII. Brave new algebra 261276
- 1. The category of S-modules 261276
- 2. Categories of R-modules 263278
- 3. The algebraic theory of R-modules 265280
- 4. The homotopical theory of R-ring spectra 267282
- 5. Categories of R-algebra 271286
- 6. Bousfield localizations of R-modules and algebras 274289
- 7. Topological Hochschild homology and cohomology 277292
- 8. The construction of THH via the standard complex 279294

- Chapter XXIV, Brave new equivariant foundations 283298
- Chapter XXV. Brave new equivariant algebra 299314
- 1. Introduction 299314
- 2. Local and Cech cohomology in algebra 300315
- 3. Brave new versions of local and Cech cohomology 301316
- 4. Localization theorems in equivariant homology 302317
- 5. Completions, completion theorems, and local homology 305320
- 6. A proof and generalization of the localization theorem 307322
- 7. The application to K-theory 310325
- 8. Local Tate cohomology 311326

- Chapter XXVI. Localization and completion in complex bordism 315330
- 1. The localization theorem for stable complex bordism 315330
- 2. An outline of the proof 316331
- 3. The norm map and its properties 318333
- 4. The idea behind the construciton of norm maps 320335
- 5. Global J[sub(∗)]-functors with smash product 322337
- 6. The definition of the norm map 325340
- 7. The splitting of MU[sub(G)] as an algebra 326341

- Chapter XXVII. A completion theorem in complex cobordism 327342
- Chapter XXVIII. Calculations in complex equivariant bordism 333348
- 1. Notations and terminology 333348
- 2. Stably almost complex structures and bordism 334349
- 3. Tangential structures 336351
- 4. Calculational tools 339354
- 5. Statements of the main results 342357
- 6. Preliminary lemmas and families in G x S[sup(1)] 343358
- 7. On the families F[sub(i)] in G x S[sup(1)] 344359
- 8. Passing from G to G x S[sup(1)] and G x Z[sub(k)] 349364

- Bibliography 353368
- Index 361376

Table of Contents pages: 1 2

#### Readership

Graduate students and research mathematicians interested in algebraic topology.

#### 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