# Tag Info

## Hot answers tagged group-cohomology

31 votes
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### Where should I search for computations of group cohomology rings of not-too-complicated finite groups?

Simon King and David Green maintain a computer calculated computation of the mod p cohmology of many finite $p$-groups ('order at most 128, of all but 6 groups of order 243, and of some sporadic ...
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### $H^4$ of the Monster

In arXiv:1707.08388, I calculate that the cohomology class you described has order 24 and that it is not a characteristic class in the ordinary sense.
21 votes
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• 5,490
16 votes
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### Acyclic aspherical spaces with acyclic fundamental groups

Yes, such things exist. Take any finitely presented infinite acyclic group $G$, for example, Higman's group. It is a theorem by Kervaire (''Smooth homology spheres and their fundamental groups'') ...
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16 votes

### Do there exist acyclic simple groups of arbitrarily large cardinality?

I just realized this is indeed, as Neil Strickland and Tom Goodwillie predicted, not hard, thanks to the fact that a directed union of simple groups is simple. Since homology commutes with direct ...
15 votes

• 50k
14 votes
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• 7,984
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### Groups all of whose extensions are split

Proposition. Given a group $G$, this happens (every exact sequence $1\to G\to H\to H/G\to 1$ splits) iff $G$ has a trivial center and $1\to G\to \mathrm{Aut}(G)\to\mathrm{Out}(G)\to 1$ splits. Lemma: ...
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14 votes
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### Relation between the cohomology group of a curve and the cohomology group of its jacobian

$\def\Alb{\text{Alb}}\def\Pic{\text{Pic}}\def\CC{\mathbb{C}}\def\ZZ{\mathbb{Z}}\def\RR{\mathbb{R}}\def\cO{\mathcal{O}}$There are two abelian varieties associated to a smooth projective connected $n$-...
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14 votes
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### Explanation for $\chi(\operatorname{SL}_2(\mathbb{Z})) = -1/12$ with zeta function

(Expanding my comment into an answer) It is not a coincidence. Relating the Euler characteristic of certain arithmetic groups to the Zeta function is a theorem due to Harder [1] from 1971. It is ...
13 votes
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### The term $H^1(N,A)^{G/N}$ in the inflation-restriction exact sequence

You do not need LHS spectral sequence for this action. The functors $H^*(N,-)$ are the derived functors of $(-)^N:G\text{-mod}\rightarrow G/N\text{-mod}$, so they will carry a structure of $G/N$-...
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13 votes

### First Galois cohomology of Weil restriction of $\mathbb{G}_m$

One can do much better: it is not necessary to assume $L/K$ is Galois (merely separable is sufficient). And in fact one can formulate the result in a manner which works beyond that of fields, working ...
13 votes

One fairly general class of groups $G$ for which $\operatorname{cd}(G\times G)=2\operatorname{cd} (G)$ are the duality groups. A group $G$ is a duality group of dimension $n$ if there is a dualizing $... • 34.6k 13 votes Accepted ### Cohomological dimension of torsion-free groups and its subgroups This is Theorem 3.1, p. 190, in Brown, "Cohomology of groups". He also attributes it to Serre. As a remark, this is the reason that the virtual cohomological dimension (vcd) is well-defined. • 1,950 12 votes ### Yoga of six functors for group representations? Allow me to answer not in stacks but rather in spaces (which may be appropriate if the answer is about getting intuition; e.g. for$B\mathbb{Z}/2\mathbb{Z}=\mathbb{RP}^\infty\$ there is probably ...

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