Equivariant homotopy and cohomology theory : [Expository lectures from the NSF-CBMS Regional Conference held at the University of Alaska, Fairbanks, AK, August 2-6, 1993] : dedicated to the memory of Robert J. Piacenza

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. Ch. I. Equivariant Cellular And Homology Theory -- Ch. Ii. Postnikov Systems, Localization, And Completion -- Ch. Iii. Equivariant Rational Homotopy Theory / Georgia Triantafillou -- Ch. Iv. Smith Theory -- Ch. V. Categorical Constructions; Equivariant Applications -- Ch. Vi. The Homotopy Theory Of Diagrams / Robert J. Piacenza -- Ch. Vii. Equivariant Bundle Theory And Classifying Spaces -- Ch. Viii. The Sullivan Conjecture -- Ch. Ix. An Introduction To Equivariant Stable Homotopy -- Ch. X. G-cw(v) Complexes And Ro(g)-graded Cohomology / Stefan Waner -- Ch. Xi. The Equivariant Hurewicz And Suspension Theorems / L. Guance Lewis, Jr. -- Ch. Xii. The Equivariant Stable Homotopy Category -- Ch. Xiii. Ro(g)-graded Homology And Cohomology Theories -- Ch. Xiv. An Introduction To Equivariant K-theory / J. P. C. Greenlees -- Ch. Xv. An Introduction To Equivariant Cobordism / S. R. Costenoble Ch. Xvi. Spectra And G-spectra; Change Of Groups; Duality -- Ch. Xvii. The Burnside Ring -- Ch. Xviii. Transfer Maps In Equivariant Bundle Theory -- Ch. Xix. Stable Homotopy And Mackey Functors -- Ch. Xx. The Segal Conjecture -- Ch. Xxi. Generalized Tate Cohomology / J. P. C. Greenlees And J. P. May -- Ch. Xxii. Twisted Half-smash Products And Function Spectra / Michael Cole -- Ch. Xxiii. Brave New Algebra -- Ch. Xxiv. Brave New Equivariant Foundations / A. D. Elmendorf, L. G. Lewis, Jr. And J. P. May -- Ch. Xxv. Brave New Equivariant Algebra / J. P. C. Greenlees And J. P. May -- Ch. Xxvi. Localization And Completion In Complex Bordism / J. P. C. Greenlees And J. P. May -- Ch. Xxvii. A Completion Theorem In Complex Cobordism / G. Comezana And J. P. May -- Ch. Xxviii. Calculations In Complex Equivariant Bordism / G. Comezana. J.p. May ; With Contributions By M. Cole ... [et Al.]. Includes Bibliographical References (p.[353]-360) And Index. Contents......Page 3 Introduction......Page 10 I. Equivariant cellular and homology theory......Page 22 II. Postnikov systems, localization and completion......Page 34 III. Equivariant rational homotopy theory......Page 42 IV. Smith theory......Page 50 V. Categorical constructions. Equivariant applications......Page 56 VI. The homotopy theory of diagrams......Page 66 VII. Equivariant bundle theory and classifying spaces......Page 80 VIII. The Sullivan conjecture......Page 88 IX. An introduction to equivariant stable homotopy......Page 102 X. G-CW(V) complexes and RO(G)-graded cohomology......Page 114 XI. The equivariant Hurewicz and suspension theorems......Page 124 XII. The equivariant stable homotopy category......Page 140 XIII. RO(G)-graded homology and cohomology theories......Page 162 XIV. An introduction to equivariant K-theory......Page 180 XV. An introduction to equivariant cobordism......Page 192 XVI. Spectra and G-spectra. Change of groups. Duality......Page 204 XVII. The Burnside ring......Page 224 XVIII. Transfer maps in equivariant bundle theory......Page 240 XIX. Stable homotopy and Mackey functors......Page 254 XX. The Segal conjecture......Page 270 XXI. Generalized Tate cohomology......Page 288 XXII. Brave new algebra......Page 308 XXIII. Brave new equivariant foundations......Page 336 XXIV. Brave new equivariant algebra......Page 356 XXV. Localization and completion in complex bordism......Page 376 XXVI. Some calculations in complex equivariant bordism......Page 396 Bibliography......Page 420 Index......Page 429