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### Introductory text on Group Cohomology [closed]

What are good introductory textbooks available on Cohomology of Groups?
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### Cohomology of elementary Abelian p-group

Let $E=(\mathbb{Z}/p\mathbb{Z})^n$, an elementary Abelian p-group. Let $k$ be an algebraically closed field of characteristic 0. There is a good description of $H^*(E,F^{\times})$ where $F$ is a field ...
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### What are the low-degree 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 $H^2(MCG_g,Z)$...
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### 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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### 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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### 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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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 $$1\... 1answer 1k views ### Relations between the cohomology of discrete groups and of profinite groups Let G be a discrete group and K be the profinite completion of G. Let C_K denote the category of contionuous K-modules and {C_K}' denotes category of finite continuous K-modules. Now for ... 1answer 453 views ### Can group cohomology be used to study fiber bundles? Is (non-abelian) cohomology used to study vector and principal bundles? Can you give me a text or an article? For example: Consider a vector bundle E with fiber V and base manifold M. Consider ... 1answer 102 views ### Are finite-dimensional representations of groups of type \text{FP}_{\infty}? Let G be a group (possibly infinite) and k be a field. A module M over k[G] is said to be of type \text{FP}_{\infty}(k) if it has a projective resolution each of whose terms is finitely ... 0answers 91 views ### Groups such that all finite-dim representations are finitely presented Let G be an infinite group. What sorts of finiteness properties can I put on G to ensure the following holds for all M? Let M be a finite-dimensional vector space over \mathbb{Q} upon ... 2answers 445 views ### Group cohomology without G-modules (a.k.a. what does this bar construction compute?) Without any prior exposure to the cohomology of groups, one might naively proceed by replacing a group by a sort of resolution. For instance, let's take G = \mathbb{Z}^2, and "resolve":$$ 0 \to \...
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 $g$....