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### Vector bundles and equivariant vector spaces

It seems commonly accepted that most of the results of equivariant geometry for vector spaces yield analog result for vector bundles. In so far as I understand it, the reason for that is the ...
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### Geometric construcion of Proj as a quotient by a $\mathbb{G}_m$ action

I'm trying to translate the Proj construction as a kind of quotient by a $\mathbb{G}_m$ action. Here's what I have so far: Let $X=Spec A$ be an affine scheme (after this case is setteled I imagine it ...
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### Localization in equivariant cohomology theory for groups other than ($p$-)tori

Recall the following localization theorem, as stated in Hsiang's Cohomology Theory of Compact Transformation Groups: Theorem. Let $G=(S^1)^k$ be a torus, $X$ a paracompact $G$-space with finite ...
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### Künneth formula for Bredon cohomology theory

Let $G$ be a finite group. Let $X$ and $Y$ be two $G$-CW complexes with known integer graded $G$-equivariant Bredon cohomology with constant coefficient systems. Is there any Künneth formula for this ...
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### How is an $S^1$-equivariant elliptic cohomology theory affected as we continuously vary the underlying elliptic curve?

Grojnowski constructs a $S^1$-equivariant cohomology theory over a complex elliptic curve $E$, designed to trivially satisfy: $$E^*_{S^1}(pt) = E$$ The functor $E^*_{S^1}(-)$ takes in a space $X$ ...
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### cohomology of the orbit space of a group action

Let $M$ be a manifold. Let a finite group $G$ act on $M$ discretely. Let $F$ be a field. Suppose the induced action of $G$ on the cohomology algebra $H^*(M,F)$ is known. We want to obtain $H^*(M/G;F)$...
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### Advantage in Using Cyclic Homology to a compute Equivariant (Co)Homology of Loop Spaces

I am trying to compute equivariant (co)homology of the free loop space of a manifold $M$ that is not a Lie group, $H^{S^1}_*(LM)$ with the natural rotation action of $S^1$ on the loops of the free ...
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Let $X$ be a smooth complex projective variety acted upon by algebraically by a complex torus $T$. Let $F_1,\ldots,F_n$ be the connected components of $X^T$ and assume that the restriction map $$H_T^*(... 0answers 96 views ### Computing the Product Structure in Equivariant Cohomology via a Stratification and Thom-Gysin Let X be a smooth complex projective variety acted upon algebraically by a complex torus T. Suppose that \{X_{\beta}\}_{\beta\in B} is a finite T-invariant stratification of X into smooth ... 0answers 110 views ### Equivariant Poincare Series of Based Loop Group of SU(2) Let \Omega SU(2) denote the based loop group of SU(2), and consider the action of S^1 on \Omega SU(2) as a maximal torus of SU(2). (This is not the "loop rotation" action.) Is there an ... 2answers 447 views ### Equivariant Stratifications of a Variety Let X be a complex variety acted upon algebraically by a complex torus T. Suppose that \{X_{\beta}\}_{\beta\in S} is a finite T-equivariant stratification of X, so that the X_{\beta} are ... 0answers 238 views ### extension of cohomology theories In Rudyak's book it is proved (3.22 page 160) that every additive cohomology theory (in the sense of Eilenberg-Steenrod) defined on the category of finite dimensional pointed CW complexes can be ... 1answer 245 views ### Is there a Whitehead-type theorem in T-equivariant cohomology? Let T be a real torus, and let X and Y be T-spaces. Under what conditions (if any) will the existence of graded H^*_T-algebra isomorphism between the T-equivariant cohomologies of X and ... 1answer 207 views ### GL_k-equivariant cohomology of k\times n matrices I'm having a surprisingly hard time finding references for some facts about GL_k-equivariant cohomology of the space of k\times n matrices. Specifically, I believe the following things to be true: ... 1answer 223 views ### Formality of classifying spaces (for not necessarily connected groups) As should be evident from the title this question has a similar flavor to: Formality of classifying spaces However, unlike Geordie's question, I will be working with torsion free coefficients (say ... 1answer 202 views ### Topology of a Compact Space with Fixed-Point-Free Torus Action Let X be a compact connected smooth manifold and T a compact torus acting smoothly on X without fixed points. What, in general, can be said about the topology of X (ex. rational (co-)homology)?... 2answers 243 views ### Equivariant cohomology and complex non-degenerate bilinear forms Let M = GL(n,\mathbb{C}) be the set of non-degenerate bilinear forms on \mathbb{C}^n (not necessarily symmetric). The general linear group G = GL(n,\mathbb{C}) acts on M in the usual way$$G\...
Let $T$ be a complex torus and $X$ a smooth quasi-projective $T$-variety with finitely many fixed points. Denote by $\varphi:H_{T}(X)\rightarrow H_{T}(X^T)$ the map on equivariant cohomology induced ...