15 votes

Cohomology of quotient by free action

This is true even if the group does not act freely. See Proposition 1.1 of my notes here. I deal with simplicial complexes and work over the rationals, but the statement you give can be proved the ...
  • 40.6k
14 votes
Accepted

"Rotated" version of the Atiyah-Hirzebruch spectral sequence

Good question. I think the answer is yes. The unnamed spectral sequence is usually referred to as the isotropy spectral sequence. For a group $G$ acting on $X$ and an abelian group $A$ of ...
  • 33.6k
12 votes
Accepted

Passing from T-equivariant to G-equivariant cohomology

This isn't a complete answer, but for example, if you know that $H_T^*(X;\mathbb{Z})$ injects into the cohomology of the fixed point set $X^T$, then for $G=GL_n(\mathbb{C})$, the canonical map $H_G^*(...
  • 524
12 votes
Accepted

Borel equivariant homology of a suspension

I assume that by Borel equivariant homology of $X$ you mean the ordinary homology of the "Borel construction" $X\times_G EG$. There is a homotopy cofibration sequence $$ X\times_G EG \to *\...
11 votes

Equivariant cohomology vs. invariant cohomology vs. cohomology of quotient space

In what follows I will assume that $G$ is discrete and that $X$ is a simplicial complex with regular $G$-action (see Bredon's "Introduction to compact transformation groups", Chapter III.1). The ...
  • 33.6k
11 votes
Accepted

Equivariant bundles invisible in K-theory and Borel cohomology

The simplest example of what you are looking for occurs when $G = S^1$ and $X=S^1/C_6$, where $C_6$ is the group of 6th roots of unity. Then the map $$ K_G(X) \rightarrow K(EG\times_G X)$$ identifies ...
10 votes

Allowing $G$-CW complexes to have more general cells

Let me start with the fact that, in one sense, it's true that Type 1 complexes are all that are "needed." That's true in the sense that complexes built from Type 2 and 3 cells have the $G$-homotopy ...
10 votes

What is the circle-equivariant cohomology of the real projective plane

I think one gets $$H^*_{S^1}(\mathbb{RP}^2; \mathbb{F}_2) = \mathbb{F}_2[x, y]/(xy) $$ where $|x|=1$ and $|y|=2$. The module structure over $H^*_{S^1}(pt; \mathbb{F}_2) = \mathbb{F}_2[t]$ is given by $...
9 votes
Accepted

Analogue of Borel--Bott--Weil for General Equivariant Vector Bundles

The article Lie Algebra Cohomology and the Generalized Borel-Weil theorem by Kostant contains generalization of the BBW theorem to equivariant vector bundles over $G/P$ associated to a $G$-...
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9 votes
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Citation: earliest incidence of the Borel localization theorem

Here is the reference trail, according to this source: Borel made the key observation [1] that the cohomology of the fixed point set was closely related to a torsion-free quotient. In the 1960’...
9 votes

T-equivariant cohomology of flag variety

Points 1-3 are available for equivariant cohomology. As an economical link, this paper on the arXiv discusses three well-known available presentations for $T$-equivariant cohomology of the flag ...
9 votes

Reading list for Equivariant Cohomology

I recommend Equivariant de Rham cohomology: theory and applications which in my opinion is a well-made introduction in equivariant cohomology. Moreover the article of Tymoczko "An introduction to ...
8 votes
Accepted

Geometric construcion of Proj as a quotient by a $\mathbb{G}_m$ action

First of all, the ideal $I$ corresponding to the fixed point set is generated by all $A_d$ with $d\ne0$. Thus $I=\bigoplus_d I_d$ with $I_d=A_d$ for $d\ne0$ and $I_0=\sum_{d\ne0}A_dA_{-d}\subseteq A_0$...
8 votes

$RO(G)$-graded homotopy groups vs. Mackey functors

I can answer your first question in some special cases. Let $p$ be a prime and $G=C_p$ the cyclic group of order $p$. If $p=2$, the answer to your question is yes and if $p$ is odd, then it is no. ...
  • 1,688
8 votes
Accepted

cohomology of the orbit space of a group action

If $F$ is a field of characteristic $0$, then $H^k(M/G;F)$ equals the invariants of the action of $G$ on $H^k(M;F)$. For two different proofs of this, see Proposition III.2.4 of Bredon's "...
  • 40.6k
8 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 $S^0$. ...
7 votes

Is there any "deep" relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory

This is very late and you've no doubt learned this in the last five years, but for completeness, the relation is indeed that they are linked by completion and the Chern character, as suggested in one ...
  • 2,947
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 ...
7 votes
Accepted

Is there a kind of Poincare duality for Borel equivariant cohomology?

This kind of thing shows up quite naturally in parameterised stable homotopy theory. Let me translate an idea I know from there into the language in this question. Cap product gives a map $$C^{p}(M ;...
7 votes
Accepted

Homotopy group action and equivariant cohomology theories

From modern perspective this is much more straightforward than the "genuine" version you described above the question. Naive $G$-spaces are just functors $BG\to \cal{S}$ among infinity ...
  • 3,864
7 votes

Frobenius pushforward of an equivariant tautological bundle on the flag variety

EDIT. Corrected the statement ($\sigma$ should be $p-1$ times what I wrote) and answered the question in the comment. In general, the push-forward of a line bundle on the flag variety $G/B$ will not ...
6 votes
Accepted

Change of groups for naive G-spectra

The following will assume that you are using the underlying weak equivalence structure on $G$-spectra and $H$-spectra, so that an equivariant map $X \to Y$ is an equivalence if and only if it is so ...
  • 48.4k
6 votes
Accepted

Computing the equivariant cohomology of a specific $(\mathbb{Z}/2\mathbb{Z})^2$-space

The standard way to compute equivariant cohomology of a $G$-space $X$ is to use the spectral sequence of the fibration $$X\to EG\times_G X\to BG,$$ where the projection is induced by $X\to \ast$. With ...
  • 33.6k
6 votes

Intuition for the construction of the space $M_G=EG\times _G M$

In homotopy theory and in homological algebra, there a general idea that for any given operation that you might want to perform, there's a class of "nice" objects for that operation. It is a good idea ...
6 votes

A question on relative equivariant cohomology

The answer is yes. Because $A$ is equivariantly contractible, the composite of the maps $A\to X \to\ast$ is an equivariant homotopy equivalence, thus applying any cohomology theory gives $H(\ast)\to ...
6 votes

Reading list for Equivariant Cohomology

I would like to point out that the term "equivariant cohomology'' is ambiguous. To those unfamiliar with modern algebraic topology, it means Borel cohomology, the cohomology theory that is the ...
  • 29.3k
6 votes

Deequivariantisation of indecomposable sheaves

Take $X=pt$ and $G=G_m$ and $k$ to have characteristic zero. The equivariant derived category in this case is equivalent to modules for the homology of the circle, ie exterior algebra on a generator ...
5 votes
Accepted

Calculations of cup products in Bredon cohomology

Frankly, there aren't many calculations out there. Most of the work I know of is on the calculation of the $RO(G)$-graded cohomology of a point, of a projective space, or of $B_GO(n)$. Here are some ...
5 votes
Accepted

Differentials in Weil model for equivariant cohomology

Suppose $E \to B $ is a principal $G$-bundle with connection $\omega \in \Omega^1(E,\mathfrak{g})$, and corresponding curvature $\Omega \in \Omega^2(E,\mathfrak{g})$. Then $\Omega^*(E)$ is a ...
5 votes

Reading list for Equivariant Cohomology

For the desired symplectic emphasis, I’d warmly recommend these (of which Panagiotis’ ref. cites #2–5): Atiyah, Michael F.; Bott, Raoul, The moment map and equivariant cohomology, Topology 23, 1-28 (...

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