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. It introduces many of the fundamental ideas and concepts of modern algebraic topology. It presents comprehensive material not found in any other book on the subject. It provides a coherent overview of many areas of current interest in algebraic topology. It surveys a great deal of material, explaining main ideas without getting bogged down in details.
Introduction Equivariant cellular and homology theory Postnikov systems, localization, and completion Equivariant rational homotopy theory Smith theory Categorical constructions; equivariant applications The homotopy theory of diagrams Equivariant bundle theory and classifying spaces The Sullivan conjecture An introduction to equivariant stable homotopy $G$-CW$(V)$ complexes and $RO(G)$-graded cohomology The equivariant Hurewicz and suspension theorems The equivariant stable homotopy category $RO(G)$-graded homology and cohomology theories An introduction to equivariant $K$-theory An introduction to equivariant cobordism Spectra and $G$-spectra; change of groups; duality The Burnside ring Transfer maps in equivariant bundle theory Stable homotopy and Mackey functors The Segal conjecture Generalized Tate cohomology Twisted half-smash products and function spectra Brave new algebra Brave new equivariant foundations Brave new equivariant algebra Localization and completion in complex bordism A completion theorem in complex cobordism Calculations in complex equivariant bordism Bibliography Index.