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Generalized Tate Cohomology

Available Formats:
Electronic ISBN: 978-1-4704-0122-1
Product Code: MEMO/113/543.E
List Price: $50.00 MAA Member Price:$45.00
AMS Member Price: $30.00 Click above image for expanded view Generalized Tate Cohomology Available Formats:  Electronic ISBN: 978-1-4704-0122-1 Product Code: MEMO/113/543.E  List Price:$50.00 MAA Member Price: $45.00 AMS Member Price:$30.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 1131995; 178 pp
MSC: 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 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.

Research mathematicians.

• Chapters
• I. General theory
• II. Eilenberg-Maclane $G$-spectra and the spectral sequences
• III. Specializations and calculations
• IV. The generalization to families
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Volume: 1131995; 178 pp
MSC: 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 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.

• II. Eilenberg-Maclane $G$-spectra and the spectral sequences