Electronic ISBN:  9781470401221 
Product Code:  MEMO/113/543.E 
List Price:  $50.00 
MAA Member Price:  $45.00 
AMS Member Price:  $30.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 113; 1995; 178 ppMSC: Primary 19; 20; 55;
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.
ReadershipResearch mathematicians.

Table of Contents

Chapters

I. General theory

II. EilenbergMaclane $G$spectra and the spectral sequences

III. Specializations and calculations

IV. The generalization to families


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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.
Research mathematicians.

Chapters

I. General theory

II. EilenbergMaclane $G$spectra and the spectral sequences

III. Specializations and calculations

IV. The generalization to families