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3
votes
1answer
53 views

symmetric measurable 2-cocycles on compact abelian groups vanish?

Is the following result true? If it is, could you plese give me a reference for it? Thanks in advance! Let $(G, \mu)$ be any compact abelian group with Haar measure $\mu$ (The case I am interested ...
1
vote
0answers
67 views

Profinite groups, directed sets and $H^1$

Usually whenever one reads the definition of profinite group, one starts with an ordered set $I$ which is directed, meaning that for every $i,j\in I$ there is some $k\in I$ such that $i\leq k$ and ...
3
votes
0answers
53 views

Crossed homomorphisms between power series groups

Consider the group $\mathbb{C}[[z]]_1$ of the power series of the form $a_1 z + a_2 z^2 + \cdots$, with $a_1\neq 0$, under the operation of composition, and $\mathbb{C}[[z]]$ as a ...
3
votes
1answer
144 views

The term $H^1(N,A)^{G/N}$ in the inflation-restriction exact sequence

[a repost from SE due to the lack of response] Given a group $G$, let $A$ be a $G$-module and let $N\trianglelefteq G$. If I understand it correctly, the superscript "G/N" in the third term of the ...
1
vote
0answers
147 views

divisible 2nd cohomolgy group $H^2(G,\mathbb{Z}G)$

Recall that a (nontrivial) abelian group $A$ is called divisible if the multiplication by $n$ is surjection $A\to A$ for all $n\in\mathbb{Z}_{>1}$. My question is the following. Is there a ...
6
votes
1answer
278 views

Connected CW complex, isomorphism?

Let $\pi$ be a group and let $K(\pi, 1)$ be a connected CW complex such that $\pi_1(K(\pi, 1)) = \pi$ and $\pi_q(K(\pi, 1)) = 0$ for $q \neq 1$. My question is, are $H_*(K(\pi, 1);A)$ and ...
3
votes
1answer
242 views

What is the (co-)homology of $K(\mathbb{R}_\delta,n)$?

To elaborate on the question from the title, $\mathbb{R}_\delta$ is the additive group of real numbers (without any topology) and $K(\mathbb{R}_\delta,n)$ is an Eilenberg-MacLane space. I would like ...
6
votes
3answers
472 views

Homological vs. cohomological dimension of a group/space

I have several related questions regarding homological vs. cohomological dimension of a space/group (this is not a duplicate of this). The standard definition of the cohomological dimension $cd(X)$ ...
3
votes
0answers
84 views

Lie group cohomology with coefficients in Lie algebra

I'm looking for a reference, and basic results, about Lie algebra as modules over a Lie group (with the adjoint representation), from the point of view of cohomology. Links with the Lie algebra ...
1
vote
0answers
40 views

Independence of inverse system to define continuous cohomology for profinite groups

I have a problem concerning cohomology of profinite groups as it is defined e.g. in Gille's and Szamuely's "Central Simple Algebras and Galois Cohomology" on page 86. For a profinite group ...
2
votes
2answers
446 views

What is the $\mathbb Z/2$-cohomology of $\mathrm B^n(\mathbb Z/2)$?

I would like to know the cohomology groups $\mathrm H^\bullet(\mathrm B^n(\mathbb Z/2);\mathbb Z/2)$. I assume that this is a standard computation, but I'm not sure where to look up the answer (and, ...
2
votes
2answers
343 views

Lifting projective Galois representation with condition

Let $\bar{\rho}: G_K\to PGL_n(\mathbb{C})$ be projective representation of the absolute Galois group of a number field $K$ and $\varphi\in Aut(G_K)$. A theorem of Tate tells us that we can always ...
4
votes
0answers
58 views

Minimal rank of a permutation resolution of a $G$-lattice

Let $G$ be a finite group. By a $G$-lattice I mean a finitely generated free abelian group $L$ with an action of $G$. One says that $L$ is a permutation lattice if $L$ has a $\mathbb{Z}$-basis ...
9
votes
1answer
251 views

Nontrivial finite group with trivial cohomology in prescribed degree

For any non-trivial finite group $G$ there exists some $j > 0$ such that $H^{aj}(G) \neq 0$ for all $a = 1,2,3,\dots$, see e.g. this question: Non-vanishing of group cohomology in sufficiently high ...
5
votes
0answers
113 views

Example of a torsionfree group satisfying a cohomological condition

Let us call a finitely generated group $G$ cohomologically rich if for each $k \geq 0$, we can find a subgroup $G'$ and a prime $p$ such that $H^k(G';\mathbb F_p) \neq 0$. Examples which come to mind ...
1
vote
1answer
206 views

What is the cokernel of the map $H^2\big(\pi_1(X), \mathbb Z\big) \longrightarrow H^2(X,\mathbb Z).$

For a manifold $X$ (for simplicity, assumed to be compact), let $\pi_1(X)$ be the fundamental group of $X$. What is the cokernel of the map $$H^2\big(\pi_1(X), \mathbb Z\big) \longrightarrow ...
0
votes
0answers
146 views

Hochschild-Serre spectral sequence

The Hochschild-Serre spectral sequence says that for a short exact sequence $$1 \to G \to H \to K \to 1 \quad (1)$$ of (discrete) groups, we have a first quadrant spectral sequence with $E_2$ page ...
2
votes
0answers
47 views

How to compute group homology of Iwasawa algebra

Let $G$ be a $p$-adic Lie group, $H$ a subgroup of $G$. What is $H_1(H,\Lambda(G))$, where $\Lambda(G)$ is the Iwasawa algebra of $G$ over $\mathbb Z_p$? If it simplies the question, we may assume ...
1
vote
0answers
70 views

explicit zero 2-cocycle

Let $G$ be a group which acts linearly on a vector space of dimension $n$ over a field $k$. Denote by $\rho$ this representation and consider the associated adjoint representation $Ad\rho$ which is ...
9
votes
1answer
205 views

Dyer-Lashof operations and the homology of GL_n

For any ring R, $\bigsqcup_n {BGL}_n(R)$ is an $E_\infty$-space. Are there examples of rings where people have calculated $H_*(\bigsqcup_n {BGL}_n(R);\mathbb{Z}/2)$ and determined the Dyer-Lashof ...
9
votes
1answer
318 views

Identifying a Hopf algebra cohomology theory

Here is a cohomology theory for a Hopf algebra, which I am sure has appeared elsewhere. I met it in the van Est spectral sequence for Hopf algebras. Apologies for my being stupid here, but it would be ...
8
votes
2answers
448 views

Interpretation of the monomorphism $H^2(\pi_1(X),\mathbb{Z}) \rightarrow H^2(X,\mathbb{Z})$

Let $X$ be a nice topological space and denote by $\pi_1(X)$ its fundamental group. It is well-known that there is a well-defined map $$ 0 \rightarrow H^2(\pi_1(X),A) \rightarrow H^2(X,A),$$ where ...
5
votes
2answers
379 views

Universal coefficient theorem for group homology and cohomology

I've been looking for any kind of universal coefficient theorem for group homology and cohomology, including dual universal coefficient theorems. However, the only things I can find are ones where the ...
2
votes
1answer
177 views

When is finding an explicit inverse of an isomorphism not possible

My question is about Shapiro's lemma. Consider the isomorphism $\phi: H^n(G, Hom_{ZH}(ZG, A))\cong H^n(H,A)$ of shapiro's lemma. I would like to describe this via cochains. So the obvious map is ...
5
votes
1answer
168 views

Dimensions of a vector space akin to modular symbols

The group $\operatorname{SL}_2(\mathbb Z)$ acts on polynomials in two variables $\mathbb C[x,y]$ via $A\cdot f(x,y)\mapsto f(A^{-1}.(x,y))$ where $(x,y)$ is regarded as a column vector. There are two ...
1
vote
0answers
77 views

Cohomology of discrete group with compact support

This is closely related to a previous question on the topic, but hopefully adds some motivation. Let $G_{/\mathbf Q}$ be a semisimple group, $K\subset G(\mathbf R)$ a maximal compact subgroup, and ...
0
votes
0answers
58 views

question about Computations of gelfand-fuks cohomology, the cohomology of function spaces, and the cohomology of configuration spaces

In the paper Computations of gelfand-fuks cohomology, the cohomology of function spaces, and the cohomology of configuration spaces, F. R. Cohen, L. R. Taylor, Geometric Applications of Homotopy ...
1
vote
1answer
78 views

What if the low-degree cohomology of a $G$-module and all its restrictions vanish?

Let $G$ be a finite group. If $M$ is a free $\mathbf{Z}[G]$-module, then $H^1(G',M) = H^2(G',M) = 0$ for all subgroups $G' \subset G$. Are there any other modules, free of finite rank over ...
3
votes
1answer
192 views

Conjugation of group extensions

Let $H$ be a finite group. We write ${{\mathbb{C}}}^{*n}$ for the $n$-dimensional complex torus $({{\mathbb{C}}}^*)^n$. We have a short exact sequence $$ 0\to {{\mathbb{Z}}}^n\to ...
1
vote
0answers
100 views

Integral Cohomology of Symmetric Groups

Does anybody know a reference for the explicit description of the integral cohomology ring of $S_5$ and $S_6$. I can not find them anywhere in the internet. For $S_4$, I found C. B. Thomas's nice ...
6
votes
0answers
126 views

vanishing of Lie algebra cohomology with coefficients in an infinite-dimensional module

Let $G$ be a real semisimple Lie group, $K$ its maximal compact subgroup, $\mathfrak g, \mathfrak k$ the corresponding Lie algebras. Let $V$ be a locally convex, Hausdorff vector space, which is a ...
1
vote
1answer
162 views

Cohomology of lattice with coefficients in field of rational functions

In my research, I came across a 1-cocycle in the following group cohomology complex: Let $\Lambda_\mathbb{Z}$ be a lattice (i.e. isomorphic to $\mathbb{Z}^n)$; let $\Lambda_\mathbb{C} = ...
3
votes
1answer
217 views

For a cross section $\sigma\colon G/N\to G$, how is $\sigma(y)^{-1}\sigma(x)^{-1}\sigma(xy)$ called?

Let $G$ be a locally compact group, let $N$ be a closed normal subgroup of $G$, and let $\sigma\colon G/N\to G$ be a cross section. Let us define $\alpha\colon G/N\times G/N \to N$ by the formula $$ ...
2
votes
1answer
135 views

cohomology of orthogonal (or general linear) group over finite fields

Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let $$ O(\mathbb{Z}_2^{\oplus k})=\{A\mid A \text{ is a } k\times k \text{ - matrix with entries } 0,1, det(A)=\pm 1\} $$ What is $$ ...
1
vote
1answer
193 views

cohomology of orthogonal group of integers

Let $$ O(\mathbb{Z}^{\oplus k})=GL(\mathbb{Z}^{\oplus k})\cap O(k). $$ What is $$ H^*(BO(\mathbb{Z}^{\oplus k});\mathbb{Z})? $$ If it cannot be computed out, can we get $$ H^*(O(\mathbb{Z}^{\oplus ...
2
votes
1answer
309 views

cohomology ring of symmetric group of order $3$

Let $S_3$ be the symmetric group of order $3$. What is the cohomology ring $$ H^*(S_3;\mathbb{Z})?$$ My attempt: I want to use mathematical induction on $n$ for $S_n$. For $n=1$, $S_1$ is trivial. ...
3
votes
1answer
220 views

Centralizers in the universal central extensions of the alternating groups?

For $n \ge 8$ the Schur multiplier $H_2(BA_n, \mathbb{Z})$ (where $A_n$ denotes the alternating group) stabilizes to $\mathbb{Z}_2$, and hence there is a universal central extension $\widetilde{A}_n$ ...
3
votes
3answers
444 views

classifying space and cohomology of integer general linear group

I have obtained that the classifying space $$ BGL(\mathbb{R}^n)=BO(\mathbb{R}^n)=G_n(\mathbb{R}^\infty) $$ is the Grassmannian. I have also obtained that the mod 2 cohomology is the polynomial ...
6
votes
0answers
136 views

Proof that the second Borel cohomology group of $(\mathbb R, +)$ is trivial

Does anyone have a reference for a fairly direct proof that the second Borel cohomology group for $(\mathbb R, +)$ (with the trivial action on the circle group) is trivial? The motivation is to show ...
3
votes
2answers
362 views

symmetric 2-cocycle / many projective representations

Let $G$ be a finite group, $k$ the field of complex numbers. Are there (cohomologically nontrivial) group 2-cocycles $\sigma\in Z^2(G,k^\times)$ such that for all $g,h\in G$: ...
3
votes
1answer
189 views

Lyndon–Hochschild–Serre spectral sequence for not normal subgroup

Is there analog of Lyndon–Hochschild–Serre spectral sequence for not normal subgroup? What can you say about it? Can you describe $E^{p, q}_1$ ? What is about $E^{p, q}_2$? What is the best technique ...
5
votes
2answers
232 views

Embedding $G$ in a $Z(G)$ extension of $\operatorname{Aut}G$

This question follows up a question I asked on math.SE. This is a refinement and a reference request. For what groups $G$ does there exist a $Z(G)$-extension of $\operatorname{Aut}G$ (call it ...
3
votes
2answers
238 views

cohomology algebra of braid spaces, configuration spaces

In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, Chapter 5, 6, 7, 8, 9, 10, 11, the cohomology algebra $H^*(B(\mathbb{R}^{n+1},p),\mathbb{Z}_p)$, for ...
-1
votes
1answer
176 views

cohomology version of Cartan-Leray spectral sequence that deduces cup product

On page 338, A User's Guide to Spectral Sequences. 2nd Edition, by John McCleary, Theorem 8.9, there is a Cartan-Leray spectral sequence for homology: If $X$ is a connected pace on which the group ...
5
votes
1answer
156 views

Reference for Mod 2 cohomology of $BZ_{2r}$ in terms of Stiefel-Whitney Classes

I was hoping for an explicit reference to the description of the mod 2 cohomology of a cyclic group $C_{2r}=\langle t \rangle$ of even order in terms of Stiefel-Whitney classes, i.e., that ...
10
votes
0answers
168 views

Stable range of some classifying spaces and iterated loop spaces

Galatius (in his talk) has made very interesting remarks about the stable range of some classifying spaces of groups. To be more concrete, I will mention two examples to illustrate his (?) point of ...
5
votes
1answer
260 views

Relation between cohomology of ordered and unordered configuration spaces

Let $M$ be a manifold. Then $F(M,k)/\Sigma_k$, the unordered configuration space of $k$ points, is obtained as a quotient of $F(M,k)$, the ordered configuration space of $k$ points, by the group ...
0
votes
1answer
173 views

Two questions on the Schur multiplier of groups of order $p^4$

I tried to find a reference for the computation of the Schur multiplier of groups of order $p^4$. The case in which $p=2$ is well known, see e.g. Table 1 at ...
4
votes
1answer
105 views

Triviality of local system extension

Take a nice space $X$. Let us call a local system on $X$ a functor from the fundamental groupoid of $X$ to groups, so that $G$ is a local system on $X$ if for each $x \in X$ there is a group $G_x$ and ...
2
votes
0answers
105 views

Schur covering group [closed]

It is known that every finite group has a Schur covering group. I'm eager to know every finite group can be considered as a Schur covering group of a group. If it is not true in general, under what ...