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Results tagged with equivariant-cohomology
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user 10366
5
votes
Accepted
$RO(G)$-Graded Cohomology Theories
There is no relationship in general. If $E$ is a module over the $G$-equivariant $MU$ spectrum then $E_G^{n+V}(\text{pt})$ is naturally isomorphic to $E^{n+\text{dim}_{\mathbb{R}}(V)}(\text{pt})$ for …
- 56.9k
3
votes
Equivariant cohomology and complex non-degenerate bilinear forms
I'll assume that you mean the Borel cohomology $H^*(EG\times_GM)$. If so, the answer is
$$ \mathbb{Z}[\![c_1,\dotsc,c_n]\!]\otimes\Lambda^*(a_1,\dotsc,a_n), $$
where $|c_k|=2k$ and $|a_k|=2k-1$. To …
- 56.9k
7
votes
Accepted
Equivariant cohomology ring is an integer domain
It is very rare for these rings to be integral domains. To see this, put
$$ f_V(t)=\sum_kc_k(V)t^{\dim(V)-k} \in H^*(BG)[t]. $$
(All cohomology here has rational coefficients.)
It is then standard …
- 56.9k
11
votes
Reference request: Equivariant Topology
\bib{MR1413302}{book}{
author={May, J. P.},
title={Equivariant homotopy and cohomology theory},
series={CBMS Regional Conference Series in Mathematics},
volume={91},
note={With contribu …
1
vote
Do Mackey (co)homology functors factor through derived categories? References with details?
The book "Equivariant Homotopy and Cohomology Theory" by May et al probably counts as the standard reference. I don't think that it answers your last question, however. That is probably best address …
- 56.9k
7
votes
$E^G_\ast(E)$ tensored with the rationals
For any finite abelian $G$ and $H\leq G$ we have a geometric fixed-point functor $\phi^H\colon\text{Sp}_G\to\text{Sp}$ which preserves smash products and sends the equivariant sphere $S^0_G$ to $S^0$. …
- 56.9k
5
votes
Equivariant complex $K$-theory of a real representation sphere
Here is one possible approach, which is specific to the adjoint representation. Let $X$ denote $U(n)$, regarded as a $U(n)$-space by the rule $g.x=gxg^{-1}$. Put $X_k=\{x\in X:\text{rank}(x-1)\leq k …
- 56.9k