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0
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0answers
122 views

Rational cohomological dimension of the lattices in $SL_n(\mathbb{Q}_p)$

What is the rational cohomological dimension of the lattices in $SL_n(\mathbb{Q}_p)$, where $n\geq 3$ ? A reference would be appreciated. For the definition of "cohomological dimension of a group ...
0
votes
0answers
63 views

Homology of product of two groups [duplicate]

There is well known formula for the homology of product of two groups with coefficient in integers, that is $0 \rightarrow \oplus_{p+q=n}H_p(G,\mathbb{Z}) \otimes H_q(H,\mathbb{Z}) \rightarrow H_n(G \...
4
votes
1answer
149 views

Obstruction for two subgroups to be conjugated by an automorphism

Altough this sounds as a very basic question, I didn't receive any answer on stack exchange and by people more knowledgeable than me Take $p$ a prime number and $P$ an abelian finite $p$-group. Let $...
8
votes
2answers
289 views

Groups with trivial rational homology and their finite index subgroups

For a short exact sequence $0 \to G \to H \to K \to 0$ of (discrete) groups with $K$ finite we have, as a consequence of the Hochschild-Serre spectral sequence, that $H^{\ast}(H;\mathbb Q) = H^{\ast}(...
7
votes
2answers
337 views

How to compute second homology of a group given by presentation with two relators

I am interested in calculating of $H_2(G,\mathbb{Z})$, where $G$ is a group given by presentation with two relators $\langle a,b| r_1 = r_2 = 1\rangle$. Moreover, I am interested in such ...
3
votes
2answers
288 views

mod p cohomology ring of alternating groups

Let $A_n$ be the alternating group of $\{1,2,\cdots,n\}$. (1). What is the cohomology ring $$ H^*(A_4;\mathbb{Z}/3) $$ and its Steenrod operation $P^i$'s? (2). Are there general results about the ...
3
votes
0answers
98 views

Schur Multiplier of Tarski Monsters

Is it known whether the Schur Multiplier of the Tarski monsters are finitely generated?
3
votes
2answers
209 views

Do quasi-isometric groups have the same rational cohomology?

Let $G_1$ and $G_2$ be two finitely generated groups which are quasi-isometric in the sense of geometric group theory. Are their rational cohomology rings $H^{\ast}(G_i; \mathbb Q)$ necessarily ...
4
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0answers
146 views

Is the Tensor/Exterior square $G\otimes G$ or $G\wedge G$ of infinite p-group also a p-group?

Let $G$ be an infinite countable p-group. Is it true that $G\otimes G$ or $G\wedge G$ are also p-groups? (where G acts on itself by conjugation). For simplicity, you can consider that $G=[G,G]$, and ...
7
votes
3answers
393 views

Computations in modular cohomology of finite groups

Let $k$ be an algebraically closed field of characteristic $p$, let $G$ be a finite group whose order is divisible by $p$, and let $H(G)$ be the commutative cohomology algebra of $G$ with coefficients ...
0
votes
1answer
111 views

cohomology ring of the fundamental group of unordered configuration space

From the lecture notes INTRODUCTION TO CONFIGURATION SPACES AND THEIR APPLICATIONS, p. 18, I find: Os it possible to derive the cohomology ring $H^*(Conf(S,k)/\Sigma_k;\mathbb{Z}_2)$ from the above ...
8
votes
1answer
152 views

Obstructions to Picard-graded groups of maps

Suppose $(C,\odot,\Bbb I)$ is an additive category with a compatible symmetric monoidal structure and $Pic(C)$ is the group of isomorphism classes of objects which have an inverse under $\odot$. For $\...
4
votes
1answer
139 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 ...
2
votes
1answer
184 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 $j\...
3
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0answers
85 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 $\mathbb{C}[[z]]_1$-...
3
votes
1answer
212 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 ...
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0answers
162 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
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1answer
301 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 $\text{Tor}...
4
votes
1answer
270 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
556 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
103 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
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0answers
47 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 $G=\...
2
votes
2answers
491 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
381 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
62 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
305 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
150 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
232 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 H^2(X,\...
0
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0answers
186 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
65 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 $G$...
1
vote
0answers
83 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
263 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
332 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
481 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 $...
6
votes
2answers
786 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 ...
3
votes
1answer
218 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
185 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
84 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 $...
1
vote
1answer
92 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 $\mathbf{...
3
votes
1answer
233 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 {{\mathbb{C}}}^n\to{{\...
1
vote
0answers
109 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
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0answers
138 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 $G$-...
1
vote
1answer
196 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} = \Lambda_\...
3
votes
1answer
235 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
156 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 $$ H^*(BO(\mathbb{Z}_2^{\...
1
vote
1answer
233 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
329 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
245 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
598 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 ...