This book presents a systematic study of a new equivariant cohomology theory $t(k_G)^*$ constructed from any given equivariant cohomology theory $k^*_G$, where $G$ is a compact Lie group. Special cases include Tate-Swan cohomology when $G$ is finite and a version of cyclic cohomology when $G = S^1$. The groups $t(k_G)^*(X)$ are obtained by suitably splicing the $k$-homology with the $k$-cohomology of the Borel construction $EG\times _G X$, where $k^*$ is the nonequivariant cohomology theory that underlies $k^*_G$. The new theories play a central role in relating equivariant algebraic topology with current areas of interest in nonequivariant algebraic topology. Their study is essential to a full understanding of such "completion theorems" as the Atiyah-Segal completion theorem in $K$-theory and the Segal conjecture in cohomotopy. When $G$ is finite, the Tate theory associated to equivariant $K$-theory is calculated completely, and the Tate theory associated to equivariant cohomotopy is shown to encode a mysterious web of connections between the Tate cohomology of finite groups and the stable homotopy groups of spheres.

Readership

Research mathematicians.

Table of Contents

• Part I: General theory
• Part II: Eilenberg-Maclane $G$-spectra and the spectral sequences
• Part III: Specializations and calculations
• Part IV: The generalization to families
• Appendix A: Splittings of rational $G$-spectra for a finite group $G$
• Appendix B: Generalized Atiyah-Hirzebruch spectral sequences
• Bibliography
• Index