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4
votes
0answers
113 views

The cohomology of an $S_{3}$ cover of an elliptic curve ramified in one point

Let $E/\mathbb{C}$ be an elliptic curve. Let $C \to E$ be a Galois cover with group $G = S_{3}$ (symmetric group on $3$ elements), ramified in one point. (To clarify: there is a unique point in $E$ ...
0
votes
0answers
80 views

Closed Invariant Forms on Complex Projective $k$-Space

Considering complex projective $k$-space as the homogeneous space $SU_k/U_{k-1}$, is it true that every $SU_k$-invariant form is closed?
1
vote
0answers
89 views

Identifying a cohomology class arising from a Postnikov decomposition of BU(2)

For various reasons I'm currently interested in Principal $U(n)$-bundles $U(n)\hookrightarrow E\xrightarrow{p} S^m\times X_g$ over the Cartesian product of an $m$-sphere and a closed oriented surface ...
0
votes
0answers
37 views

cohomology of a variation of wreath product

Let $C$ the space of points that looks like $(z_1,z_2,\ldots,z_n,z_{\sigma(1)},z_{\sigma(2)},\ldots,z_{\sigma(n)})$ with $z_i\in \mathbb{C}$ and $\sigma$ runs over all the permutations of $S_n$. Is ...
1
vote
1answer
235 views

A Comparison between $\pi_{1}$ of cohomology and cohomology of $\pi_{1}$

Let we have a complex of abelian topological or lie groups $$\ldots \to G_{n}\to G_{n+1}\to \ldots$$ such that the image of $G_{n}$ is a closed subgroup of $G_{n+1}$. Then we have a complex of ...
-2
votes
1answer
84 views

Definition of Milnor exact sequence and complex-oriented generalized cohomology of $\mathbb{C}P^{\infty}$

Consider a complex-oriented multiplicative generalized cohomology theory $h^{*}(X)$. It is complex-oriented, if by the definition the following two conditions hold: 1) There exists an element $t\in ...
5
votes
0answers
193 views

When did “Betti cohomology” come to be used the way it is today? (and how is it used)

This is sort of a mixture of a math and history question. First the math part: thinking about it, I do not actually know how to properly use the term "Betti cohomology". I know I should, but I ...
3
votes
2answers
243 views

The relation between group cohomology and the cohomology of the classifying space

We know that the Borel group cohomology (group cohomology of measurable functions) of a group $G$, ${\cal H}_B^d(G,Z)$, is given by the cohomology of the classifying space: ${\cal ...
1
vote
1answer
110 views

Bockstein homomorphism from $H^d(BG,Z_2)$ to $H^{d+1}(BG,Z)$, and Steenrod Square $Sq^1$

The Theorem 1.5 and 1.6 of Brown, Edgar H., Jr. The cohomology of BSOn and BOn with integer coefficients. Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288. give a general answer for $H^d(BSO_n,Z)$ ...
1
vote
1answer
218 views

Relations between characteristic classes of a group and the Stiefel-Whitney/Pontryagin classes

Let $X$ be a closed manifold and $BG$ be the classifying space of a group $G$ A map from $X$ to $BG$ induce a map from $H^*(BG,Z)$ to $H^*(X,Z)$ by pull back. Let $GH^*(X,Z)$ be the subgroup of ...
15
votes
0answers
226 views

Elliptic $\infty$-line bundles over $B G$

Theorem 5.2 in Jacob Lurie's "Survey of Elliptic Cohomology" (pdf) states the equivalence of two maps $$ B G \longrightarrow B \mathrm{GL}_1(A) $$ for $A$ an $E_\infty$-ring carrying an oriented ...
2
votes
0answers
107 views

generalized Atiyah-Hirzebruch spectral sequence from Postnikov truncation

The Atiyah-Hirzebruch spectral sequence \begin{equation*}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}(E),\end{equation*} computes the generalized homology $h$ of a total space $E$ of a Serre fibration ...
23
votes
1answer
639 views

Making $\mathbb{Q}$-cohomology integral

Let $X$ be an algebraic variety (say, smooth and projective) over $\mathbb{C}$, and fix $$\alpha\in H^i(X^{\text{an}}, \mathbb{Q})$$ with $i>0$. Does there always exist a variety $Y$ and a ...
6
votes
1answer
166 views

cohomology of classifying space of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $r: \Sigma_k\to GL(k)$ be the regular representation by permuting the order of the standard basis of $\mathbb{R}^n$. Let $\rho: ...
14
votes
5answers
547 views

Take contraction wrt a vector field twice and define kernel mod image. Does that give anything interesting?

First, we make the following observation: let $X: M \rightarrow TM $ be a vector field on a smooth manifold. Taking the contraction with respect to $X$ twice gives zero, i.e. $$ i_X \circ i_{X} ...
4
votes
1answer
256 views

Fundamental class in K-theory and orientability

In ordinary homology, the classical results give the following situation: for a compact, connected, topological manifold $M$ of dimension $n$ we have, for each ring $R$, that $H_n(M,M \setminus ...
1
vote
2answers
184 views

The cohomology groups of $\Omega U(n)$

Let $\Omega U(n)$ be the loop space of $U(n)$. Is it true that the cohomology groups $H^*(\Omega U(n); \mathbb{Z})$ are torsion-free? How can one calculate these groups?
0
votes
0answers
41 views

Corestriction map for the relative homology (cohomology) group

Let $G$ be a group and $N$ be its normal subgroup. Is there any concept of corestriction map for the relative homology (cohomology) group $H_n(G,N,-)$ ($H^n(G,N,-)$) such that when $N=G$ it is the ...
2
votes
0answers
152 views

l-adic cohomology and perverse sheaves

Let consider the map $tr:\mathbb{G}_{m}^{n}\rightarrow\mathbb{A}^{1}_{\mathbb{F}_{q}}$ given by the sum of the coordinates and let $\psi:\mathbb{F}_{q}\rightarrow\mathbb{Q}_{l}^{*}$ a non trivial ...
1
vote
0answers
51 views

Which groups may be obtained as $K$-homology groups?

Recently I asked the following question, about the separability of the underlying $C^*$-algebra in the definition of $K$-homology: mathoverflow.net/questions/181361 As far as I understood, ...
1
vote
0answers
168 views

Finite Cohomology and free groups

Let $F$ be a finitely generated nonabelian free profinite group, $p$ a prime number, $L \lhd_o F$ with $[F : L]$ coprime to $p$, $N \lhd_c^\infty F$ contained in $L$ with $L/N$ pro-$p$, and $N \leq H ...
5
votes
0answers
108 views

The action of Steenrod squares on the indeterminate in Bullett-Macdonald identities

The Bullet-Macdonald identity (c.f. On the Adem relations)is the following: $$P(s^2+st)P(t^2)=P(t^2+st)P(s^2).$$ where we have $P(s)=\Sigma _s s^iSq^i$. This is known to be equivalent to the Adem ...
38
votes
2answers
1k views

$H^4(BG,\mathbb Z)$ torsion free for $G$ a connected Lie group

Recently, prompted by considerations in conformal field theory, I was let to guess that for every compact connected Lie group $G$, the fourth cohomology group of it classifying space is torsion free. ...
0
votes
0answers
137 views

natural map from $H^3(BG,\mathbb{Z})$ to $H^3(\underline E G,\mathbb{Z})$?

is there any natural map from natural map from $H^3(BG,\mathbb{Z})$ to $H^3(\underline E G,\mathbb{Z})$? where $G$ is a discrete group, $BG$ is the classifying space, and $\underline E G$ is the ...
1
vote
0answers
196 views

Kaplansky Idempotent conjecture and Extension theory

We consider the Idempotent Kaplansky conjecture with $\mathbb{C}$- coefficients, that is the problem of nontrivial idempotents for group algebra $\mathbb{C}\Gamma$ where $\Gamma$ is a torsion free ...
4
votes
2answers
257 views

Cohomology of the toric variety $X_\Sigma=\mathbb C^2\sqcup \mathbb C^2\big/\left((x,y)_1\sim(x^{-1},y^{-1})_2\right)_{x,y\neq 0}$

I'm writing a thesis on Chow rings of toric varieties and am looking for a reference on the singular cohomology ring of the blowup of $\mathbb C^2$ at xy=0, i.e. at the coordinate axes. The ...
1
vote
0answers
90 views

Differences and relationships between Motivic cohomology and Universal cohomology theory? [duplicate]

Differences and relationships between Motivic cohomology (Beilinson, Lichtenbaum and Voevodsky) and Universal cohomology theory (Grothendieck)?
4
votes
1answer
240 views

Non-vanishing of elements in cohomology of full Flag varieties

Consider the full flag variety $F_n$ consisting of full flags in $\mathbb C^n$. There is a collection of tautological bundles on $F_n$: $0=U_0\subset U_1\subset ...\subset U_{n-1}\subset U_n=\mathbb ...
3
votes
0answers
116 views

Are there any useful Grothendieck topologies for which the H1 of $GL_n$ is not the set of rank $n$ vector bundles

Let n be a positive integer and X a scheme. Then for all the Grothendieck topologies I know (Zariski, etale, fppf) the set $H^1(X,GL_n)$ is the set of (isomorphism classes of) rank $n$ vector bundles. ...
0
votes
1answer
96 views

Cohomology of a fibered surface

Let $R$ be a complete Henselian discrete valuation ring, $\pi:X \to \mathrm{Spec} (R)$ be a smooth, proper, integral, flat $\mathrm{Spec} (R)$-scheme of dimension $2$. Assume that the genus of the ...
4
votes
0answers
106 views

Complexes of arithmetic $\mathcal{D}$-modules with Frobenius structure

This is a question about the category $F\text{-}D^b_\mathrm{coh}(\mathscr{D}^\dagger_{\mathscr{X},\mathbb{Q}})$ of complexes of arithmetic $\mathscr{D}$-modules with Frobenius structure on a smooth ...
4
votes
1answer
200 views

What properties of line bundles can be detected cohomologically?

Let $X$ be a proper, finite type scheme over a field $k$. What useful properties of line bundles (e.g. amplitude, nefness) can be detected cohomologially? For example, in our setting we have the ...
1
vote
0answers
125 views

extension problem for the Atiyah-Hirzebruch spectral sequence

For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow ...
4
votes
2answers
267 views

stability results for the Atiyah-Hirzebruch spectral sequence

For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow ...
7
votes
2answers
520 views

What morphisms induce injective/surjective maps on (Weil) cohomology?

Let $k$ be a field, let $f \colon X \to Y$ be a morphism of $k$-varieties, and assume $X$ and $Y$ are smooth and projective. Let $H(\_)$ be a classical Weil cohomology theory (i.e. one of $\ell$-adic ...
0
votes
1answer
159 views

help with cohomology of $\mathbb{P}^n$ relative to a NCD

Let $H_0, \ldots, H_n$ be $n$ hyperplanes in $\mathbb{P}^n(\mathbb{C})$ with normal crossings and denote by $H$ the union of them. I am trying to understand why (1) $H^n(\mathbb{P}^n(\mathbb{C}), ...
7
votes
2answers
275 views

A cohomology group which depends on the connection

Warning: I am not a differential geometer, so some of the following might not make sense. Background: Let $w: (T\Omega)^k \to \mathbb{R}$ be a $k$-tensor on $\Omega$, an open subset of ...
4
votes
1answer
142 views

Interpretations of differentials in hypercohomology spectral sequences as Yoneda products

I would like to know whether the differentials in a particular hypercohomology spectral sequence can each be interpreted, in some natural way, as Yoneda products between extension groups. More ...
7
votes
0answers
103 views

When is the projection of an induced fibration trivial on cohomology?

Let $p: E\to B$ be a fibration, and let $f: A\to B$ be a continuous map. In my applications, $E$ and $B$ are finite complexes, but $A$ need not be. Form the pullback $$ \begin{array}{ccc} W & \to ...
4
votes
0answers
182 views

Which ring spectra have some kind of exponential map turning addition into multiplication?

This accepted answer to the question about $BU_\otimes$ made me recall that I want to read about this general phenomenon for a long time. What will follow is sort of vernacular but whether it can be ...
9
votes
1answer
301 views

Is there a yoga of effectivity for motives and their realizations?

Inside the 'usual' categories of motives (Chow, Voevodsky ones) there are the categories of effective motives. Similarly, there are effective Hodge structures; they seem to be closely related with ...
3
votes
1answer
129 views

Can André–Quillen homology detect the property of being Gorenstein?

Let $(A,m,k)$ be commutative noetherian local ring. Can one detect if $A$ is a Gorenstein ring from the André–Quillen homologies $H_n(A,k,-)$?
5
votes
0answers
87 views

A cdga for compactly supported cohomology (à la Sullivan's algebra of polynomial forms)

Let $M$ be a smooth manifold, and let $\Omega^\bullet(M)$ be the commutative dg-algebra of differential forms on $M$. It is quasi-isomorphic to the dg-algebra of singular cochains on $M$. If $M$ is no ...
4
votes
2answers
255 views

Recommended textbooks for Hamiltonian group actions?

I am doing a project on Hamiltonian group actions on symplectic manifolds, and my supervisor was able to list several good books on Riemannian geometry to start me off, but he didn't know of any ...
10
votes
0answers
165 views

K-theoretic version of Artin-Mazur formal groups?

An Artin-Mazur formal group is, when it exists, the deformation theory of ordinary cohomology of some degree, on some algebraic variety. My question here is: Has the generalization of the theory of ...
18
votes
0answers
535 views

Calabi-Yau cohomology?

My question here is going to be this -- but I'll give a bit of background to explain myself in a moment: What has been done/what results are available on Calabi-Yau cohomology in degree $n \geq 3$ ...
1
vote
0answers
228 views

Functors with Mayer-Vietoris Sequences

Let $F$ be a contravariant functor from some category of spaces (e.g. smooth manifolds or (compact?) topological Hausdorff spaces), to Abelian groups. Assume that for any open sets $U, V \subseteq X$ ...
1
vote
0answers
82 views

$D^{\infty}$ modules on analytic spaces

In Mebkhout's paper on Local Cohomology of Analytic Spaces, the following theorem is stated: Let $X$ be a complex smooth manifold and $Y$ is an analytic subspace of $X$. Then ...
1
vote
0answers
108 views

(Co)homology of classifying space of spin group $BSpin(n)$

In the answer for question: Homology of classifying space of spin group BSpin(n), it was shown that $H_i(BSpin(\infty),Z)$ is $0,0,0,Z$, for $i=1,2,3,4$. What is $H_i(BSpin(\infty),Z)$ or ...
14
votes
0answers
251 views

Computability of Brauer groups

A friend of mine and I were talking about computable algebra, and this question came up. The answer may already be known, but I couldn't find it with Google: Suppose I have a countable field, $k$. ...