Memoirs of the American Mathematical Society 1995; 178 pp; softcover Volume: 113 ISBN10: 0821826034 ISBN13: 9780821826034 List Price: US$50 Individual Members: US$30 Institutional Members: US$40 Order Code: MEMO/113/543
 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 TateSwan 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 AtiyahSegal 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: EilenbergMaclane \(G\)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 AtiyahHirzebruch spectral sequences
 Bibliography
 Index
