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Reference on calculation of 2nd cohomology group

Let $G$ be a finitely generated, infinite, countable discrete nonamenable group with zero first Betti number, I.e., $H^1(G, \ell^2(G))=0$, e.g., $G=F_2\times F_2$, the product of free groups of two ...
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Given a countable discrete group $G$ with Kazhdan's property (T), consider $\mathbb{C}G$ or $l^2(G)$ as a left $G$-module, then we can consider the group cohomology, Is it known that $H^n(G, ... 1answer 247 views Cohomology after completion I've been scouring google and asking friend about something I was certain must be absolutely the easiest thing to people who do homological algebra, and none seem to know the answer to this, so if ... 1answer 245 views Group cochains invariant under the action of the symmetric group Let$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 $$\delta : C^k \to C^{k+1}$$ $$(\delta f)(g_1, \ldots, g_{k+1}) ... 1answer 253 views Calculations of nonabelian group cohomology of R^n I am looking at H^1(\mathbb{R}^n,G) where G is a finite 2-group. I'm wondering if such things have been calculated. I'm afraid I can't say I know anything here, past the result that this ... 2answers 332 views A question on some computation of group cohomologies Let G=H\times J, where H\cong J\cong C_2 (cyclic group of order 2). Let M \cong \mathbb{Z} be a G-module via "trivial H-action and negation J-action". My question is "What are the group ... 1answer 95 views “Symmetric” Polynomial 4-cocycles It is an old theorem of Heaton's (based on work of Eilenberg and MacLane), that a polynomial 3-cocycle f(x,y,z) which is "symmetric," in the sense that f(x,y,z)-f(x,z,y)+f(z,x,y)=0, is always a ... 3answers 447 views Reference for Ring Structure on Group Cohomology As a graded \mathbb{Z}-module, the structure of the group cohomology H^{*}(\mathbb{Z}/n\mathbb{Z};\mathbb{Z}) is extremely well-known. Yet, I am having difficulty finding a reference concerning ... 3answers 446 views Does this subgroup of “even braids” have a name? The full braid group on n strands B_n admits a surjective homomorphism p\colon\thinspace B_n\to \Sigma_n onto the symmetric group on n letters, which takes a braid to the induced permutation ... 1answer 215 views Naturality of the transfer in group cohomology Let G be a (discrete) group and H\le G a subgroup of finite index. Then there is a transfer map$$tr\colon\thinspace H^\ast(H;M)\to H^\ast(G;M)$$in group cohomology, where$M$is any$G$-module ... 1answer 263 views Why is the transfer map Tate-dual to restriction ? In one of their papers (before Theorem 7.2), Benson and Carlson state that the transfer map is Tate-dual to the restriction homomorphisms (also see Remark 1.3 of this recent paper). More ... 6answers 1k views reference for (co)homology theories Hi everyone, Every now and then, I find myself dealing with such or such (co)homology theory, and I'm frustrated I don't feel more comfortable around it. I was wondering if someone could recommend a ... 0answers 86 views Isomorphisms of group extensions arising from antisymmetric forms Let$V,W$be topological vector spaces and fix continuous antisymmetric bilinear forms$\omega_1:V\times V\to \mathbb{R}$,$\omega_2:W\times W\to\mathbb{R}$. Since$\omega_1$is a 2-cocycle (in fact ... 2answers 747 views Proofs of the Stallings-Swan theorem It is a well-known and deep${}^\ast$theorem that if a group$G$has cohomological dimension one then it must be free. This was proved in the late 60's by Stallings (for finitely generated groups) and ... 1answer 286 views Transgressions commute with the Steenrod operations on the base and fiber in a central group extension? The following sentence is quoted from the paper ON THE COHOMOLOGY OF SPLIT EXTENSIONS by D. J. BENSON AND M. FESHBACH: In general, the differentials in the Lyndon-Hochschild-Serre spectral sequence ... 1answer 326 views Equivalence of central extensions of Abelian groups Background: For a projective representation of$G$on a Hilbert space there is a 2-cocycle$c:G\times G \to \mathbb T$where the cocycle condition$\delta c=0$reads$c(f,g)c(fg,k) =c(f,hk)c(h,k)$... 1answer 355 views Posets of cosets and contractibility For this question let$G$be a group, perhaps infinite, and let$H_i$for$i\in I$be a (finite) family of subgroups closed under taking intersections. I am interested in the coset poset ... 1answer 999 views Reference request: The first cohomology of SL(2,Z) with coefficients in homogeneous polynomials Let$H_k$be the vector space of degree$k$homogeneous polynomials in two variables.I'm looking for a reference for the fact that$H^1(SL(2,\mathbb Z);H_k)=M^0(k+2)\oplus\overline{M^0(k+2)}\oplus ...
I know my question is very imprecise. I am trying to understand Tate-Farrell cohomology of the infinite Lie group $S^1$ (say, with coefficients in $\mathbb C$). I would expect that the answer is ...
For an example I'm trying to understand, I need to calculate some cohomology group of some $\mathbb Z$-module with coefficients in some other $\mathbb Z$-module (with no interesting actions). (In ...