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.
Author(s): J. P. May
Series: Cbms Regional Conference Series in Mathematics 91
Edition: typed, bookmarks
Publisher: American Mathematical Society
Year: 1996
Language: English
Pages: 435
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
