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### Untwisting the Cohomology with Twisted Coefficients

This question is set on a finite $2$-group $G$ and a subgroup $H$ of index $2$ (but perhaps the question could be answered for arbitrary orders/indexes).
It was asked here on MO whether ...

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**1**answer

171 views

### What is the group cohomology of the mapping class group of a surface

Let $MCG_g$ be the mapping class group of genus $g$ closed surface.
(Say $MCG_1=SL(2,Z)$).
I would like to know what is the group cohomology of $MCG_g$ with coefficients in Z, such as ...

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votes

**1**answer

108 views

### In H_2 of Sp(2g,Z), why does Meyer's signature cocycle give 4 times a generator?

Fix some $g \geq 2$, let $\Gamma_g$ be the mapping class group of a genus $g$ surface, and let $\pi : \Gamma_g \rightarrow Sp(2g,\mathbb{Z})$ be the projection. In
Meyer, Werner
Die Signatur von ...

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236 views

### What is the status of the Friedlander-Milnor conjecture today?

For the purposes of this question, the Friedlander-Milnor (FM) conjecture asserts an equality of the group homology for algebraic groups, and their discretizations in the following sense:
Conjecture ...

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### Non split extension isomorphic (as a group) to a split extension

$\def\Z{\mathbb{Z}}$
Let $A$ be a finite abelian group and $G$ a finite group acting on $A$.
Then the extension $0 \to A \to E \to G \to 1$ is splits if and only if the corresponding $2$-cocycle is ...

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votes

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125 views

### Third cohomology of mapping class group

I would like to know the third cohomology with coefficients in $U(1)$ or $\mathbb{C}^\ast$ of the mapping class group of a surface of genus at least one. I found many results on the rational ...

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votes

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196 views

### Group extensions isomorphic as groups

Let $G$ be a group and $A$ a $G$-module. It well know that there is a group isomorphism between the second cohomologoy group $H^2(G,A)$ and the abelian group $OpExt(G,A)$ of classes of extension ...

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110 views

### Relations between the cohomology of discrete groups and of profinite groups

Let $G$ be a discrete group and $K$ be the profinite completion of $G$. Let $C_K$ denote the category of contionuous $K$-modules and ${C_K}'$ denotes category of finite continuous $K$-modules. Now for ...

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votes

**1**answer

349 views

### Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles?
Can you give me a text or an article?
For example:
Consider a vector bundle $E$ with fiber $V$ and base manifold $M$.
Consider ...

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votes

**1**answer

86 views

### Are finite-dimensional representations of groups of type $\text{FP}_{\infty}$?

Let $G$ be a group (possibly infinite) and $k$ be a field. A module $M$ over $k[G]$ is said to be of type $\text{FP}_{\infty}(k)$ if it has a projective resolution each of whose terms is finitely ...

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86 views

### Groups such that all finite-dim representations are finitely presented

Let $G$ be an infinite group. What sorts of finiteness properties can I put on $G$ to ensure the following holds for all $M$?
Let $M$ be a finite-dimensional vector space over $\mathbb{Q}$ upon ...

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**2**answers

286 views

### Group cohomology without G-modules (a.k.a. what does this bar construction compute?)

Without any prior exposure to the cohomology of groups, one might naively proceed by replacing a group by a sort of resolution.
For instance, let's take $G = \mathbb{Z}^2$, and "resolve":
$$ 0 \to ...

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votes

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76 views

### On restriction to centralizers of all elements in group cohomology

If $G$ is a finite group, what do we know of the natural «restriction» map $$H^\bullet(G,\mathbb Z)\to\left(\bigoplus_{g\in G}H^\bullet(Z(g),\mathbb Z)\right)^G,$$
with $Z(g) $ the centralizer of $g ...

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266 views

### n-cocycles of finite abelian groups from cohomology group

Question: Given a generic finite abelian group $G=\mathbb{Z}_{N^{(1)}} \times \cdots \times \mathbb{Z}_{N^{(k)}}$.
(1) What is the explicit forms of its cohomology group (see my definition) in a ...

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391 views

### A map from the coinvariants of the dual to the dual of the invariants for a G-module

Suppose $G$ is a group and $X$ is a $\mathbb{Z}[G]$-module. Recall that the augmentation ideal $I \subset \mathbb{Z}[G]$ is generated by elements of the form $g - 1$ for $g \in G$, the coinvariants ...

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234 views

### Group cochains invariant under the action of the symmetric group

Let $G$ be a finite group and $A$ an abelian group. Recall the cochain groups
$$
C^k = \{f: G^k \to A\}
$$
and the coboundary map
$$
\delta : C^k \to C^{k+1}
$$
$$
(\delta f)(g_1, \ldots, g_{k+1}) ...

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votes

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190 views

### Cohomology of the tangent sheaf of $\mathbb{P}(1,2,3)$

Using the exact sequence
$$0\mapsto\mathcal{O}_{\mathbb{P}^{2}}\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(1)^{\oplus 3}\rightarrow T_{\mathbb{P}^{2}}\mapsto 0$$
it is easy to compute ...

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votes

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183 views

### 3-cocycle representatives for the dihedral group $D_{2n}$?

I am looking for a reference for a complete list of 3-cocycle representatives for $H^3(D_{2n},\mathbb{C}^\times)$, where
$$
D_{2n}=\langle a, b\mid a^2=b^2=(ab)^n=e\rangle
$$
is the dihedral group of ...

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703 views

### central extensions of Diff(S^1) and of the semigroup of annuli

$\mathit{Diff}(S^1)$ refers to the group of orientation preserving diffeomorphisms of the circle. The semigroup of annuli $\mathcal A$ is its "complexification": the elements of $\mathcal A$ are ...

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vote

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239 views

### Cohomology after completion

I've been scouring google and asking friend about something I was certain must be absolutely the easiest thing to people who do homological algebra, and none seem to know the answer to this, so if ...

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### Differentials in the Lyndon-Hoschild-Serre Sequence for p=0

I'm interested in whether there is a simple description of the differentials in the first column of the LHS spectral sequence (the column with $E_2^{0,q}=H^0(BK,H^q(BG))=H^q(BG)^K$ for a short exact ...

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183 views

### Duality between orbifold and quasi-Hopf algebra (twisted quantum doubles)

A quick Question:
Is there some duality known between the quasi Hopf algebra
$D^\omega(H)$ of a finite group $H$ to an orbifold model (such as
SU(2)/$G$ or SO(3)/$G$ orbifold of some ...

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votes

**1**answer

246 views

### Calculations of nonabelian group cohomology of R^n

I am looking at $H^1(\mathbb{R}^n,G)$ where $G$ is a finite 2-group. I'm wondering if such things have been calculated. I'm afraid I can't say I know anything here, past the result that this ...

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**1**answer

615 views

### Double coset formulas for Orthogonal groups [Solved]

According to Madsen-Brumfiel "Evaluation of the Transfer and the Universal Surgery Classes" Inventiones mathematicae 32 (1976): 133-170 Theorem 3.11, we can compute
the composition
...

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votes

**1**answer

155 views

### one-cocycles over finite groups

Let $G$ be a finite group and let $S$ be a subgroup of $G$. Let
$\alpha$ be a one-cocycle $G\to M$ for some $G$-module $M$, and
suppose that $Res_{S}^{G}\alpha=0$. For $g\in G$, and $S^{g}=gSg^{-1}$,
...

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### Do there exist non-isomorphic groups with the same cohomology?

For any group $G$, cohomology can be viewed as a functor
$$
H^\ast(G,-): G{\sf\text{-}mod}\to {\sf GrAbGrp},
$$
where $G{\sf\text{-}mod}$ denotes the category of (left) $\mathbb{Z}[G]$-modules and ...

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### Obstruction to extension of non-abelian groups - finite example?

Let $G$ be a non-abelian group, let $\Pi$ be a group, and let $\eta: \Pi\rightarrow Out(G)$ be a homomorphism, where $Out(G)$ is the group of automorphisms of G modulo the normal subgroup of inner ...

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255 views

### Maps between classifying spaces

Let $G$ be a discrete group and let $BG \simeq K(G,1)$ be its classifying space. Let $H$ be a topological group with classifying space $BH$.
In case $H$ is also discrete, it was pointed out in the ...

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197 views

### Crossed modules and degree-3 group cohomology

It is well known (see e.g. K. Brown, "Cohomology of groups") that a degree-3 cohomology class of a group G with coefficients in a module A can be thought of as an equivalence class of crossed modules, ...

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### on the Galois cohomology of reductive groups

Let $G$ a simply connected group over an algebraically closed field.
$F=k((t))$ and $\mathcal{O}=k[[t]]$.
Let $\gamma\in G(\mathcal{O})\cap G(F)^{rs}$.
Let $E=k((t^{1/n}))$ with $n$ prime to the ...

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267 views

### Cohomology of Formal Groups

Lubin and Tate, in discussing moduli of 1-dimensional formal groups construct a cohomology theory of formal groups, at least in degrees 0,1 and 2. Does their result about deformations actually follow ...

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### The Teichmüller's algebraic interpretation of $H^3$ in group cohomology

In the book "Cohomology of Groups" of Kenneth S. Brown, it is told in the introduction that Teichmüller arrived to $H^3$ in an algebraic context, i.e. that Teichmüller worked with an algebraic ...

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188 views

### Interesting families of groups as group extensions

Let me start this question with an example that hopefully makes clear what I am looking for:
A discrete subgroup $G$ of the group of euclidean isometries of $\mathbb{R}^d$ is called a ...

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131 views

### Descent for group actions

Suppose I have a finite Galois extension of fields $K/k$, as well as a finite group $G$ with a surjection $f: G \rightarrow \mathrm{Gal}(K/k)$.
Finally, suppose I have an action $\sigma$ of $G$ on a ...

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319 views

### A question on some computation of group cohomologies

Let $G=H\times J$, where $H\cong J\cong C_2$ (cyclic group of order 2). Let $M \cong \mathbb{Z}$ be a $G$-module via "trivial $H$-action and negation $J$-action". My question is "What are the group ...

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489 views

### Second homology group of free nilpotent p-group

Let $F_n$ be a free group on $n$ generators. Fix a prime $p$. Let $\gamma_k^p(F_n)$ be the mod $p$ lower central series, i.e. the inductively defined series
$$\gamma_0^p(F_n) = F_n \quad \text{and} ...

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### non-Abelian inflation-restriction sequence?

Let $1 \to H \to G \to G/H \to 1$ be a group extension. The long exact sequence that arises from Hochschild-Serre spectral sequence for this extension relates objects that are clasified by cohomology ...

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### Albrecht Fröhlich's text `Groupoids, groupoid spaces and cohomology' (1965)

I am looking for Albrecht Fröhlich's unpublished text `Groupoids, groupoid spaces and cohomology' (1965). In this text Fröhlich defines cohomology of a group with coefficients in a groupoid (this was ...

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### Analysis of Eilenberg-MacLane Stacks

In a series of three papers from the fifties, Eilenberg and MacLane did a pretty exhaustive study of what we now call "Eilenberg-MacLane spaces" and used a lot of machinery to do it, e.g. Whitehead's ...

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### Lazard's $\Gamma_n(f)$ as cocycle

In Michel Lazard's "Commutative Formal Groups" Springer Lecture Notes, he defines an operator on a polynomial 3-cochain $f$ denoted $\Gamma_n(f)$, which defines as the $n^{th}$ homogeneous piece of ...

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### cohomology with coefficients in a $p$-adic representation

I am cross-posting this question which I posted in math.stackexchange.com since I realized that there are people in Mathoverflow who are not signed-up there.
Edit: For a topological group $G$ and a ...

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### “Symmetric” Polynomial 4-cocycles

It is an old theorem of Heaton's (based on work of Eilenberg and MacLane), that a polynomial 3-cocycle $f(x,y,z)$ which is "symmetric," in the sense that $f(x,y,z)-f(x,z,y)+f(z,x,y)=0$, is always a ...

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### Kuenneth-formula for group cohomology with nontrivial action on the coefficient

For a trivial action on the coefficient, we have the following Kuenneth formula
for group cohomology:
$$
H^n(G_1 \times G_2; M) \cong
[\oplus_{i= 0}^n H^i(G_1;M) \otimes_M H^{n-i}(G_2;M)]
\oplus ...

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### H*(braid group, irrep of symmetric group) = ?

As in the title, say $\lambda$ is some irrep of the symmetric group $S_n$, and $Br_n$ the braid group on $n$ strands,
What is $H^*(Br_n, \lambda)$?

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### A sort of “group-ring” construction on coefficient systems in group homology (+ special case involving GL(n,Z))

Let $G$ be a discrete group and $M$ be an $RG$-module for some ring $R$ (I'm happy to assume that $R = \mathbb{Q}$). Define $R[M]$ to be the set of $R$-linear combinations of formal symbols of the ...

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### Group cohomology in dimension $-1$

This may seem like a pie-in-the-sky speculation question, but I have good reasons for asking this.
Is there any sense in which $H^{-1}(G;M)$ is defined for a group $G$ and a $G$-module $M$?
The ...

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**1**answer

87 views

### Resolution of coefficient system in group homology

Let $G$ be a discrete group and let $M$ be a $G$-module. Assume that I have a resolution
$$\cdots \rightarrow M_1 \rightarrow M_0 \rightarrow M \rightarrow 0$$
of $M$ by $G$-modules (with no further ...

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votes

**3**answers

412 views

### Reference for Ring Structure on Group Cohomology

As a graded $\mathbb{Z}$-module, the structure of the group cohomology $H^{*}(\mathbb{Z}/n\mathbb{Z};\mathbb{Z})$ is extremely well-known. Yet, I am having difficulty finding a reference concerning ...

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### Cohomological dimension of knit products

Let $G$ be a group with complementary subgroups $A$ and $B$ (meaning $A\cap B=1$ and $AB=G$).
If $A$ and $B$ are both normal in $G$, then $G\cong A\times B$ is a direct product. If $A$ is normal, ...

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### $H^2(K, Q_p(1))$

In Tate's local duality theorem we find the isomorphism $H^2(K, Q_p(1)) \cong Q_p$ where $K$ is a finite extension of $Q_p$. I haven't found any reference where this isomorphism is given explicitly ...