The group-cohomology tag has no wiki summary.

**34**

votes

**9**answers

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### Intuition for Group Cohomology

I'm beginning to learn cohomology for cyclic groups in preparation for use in the proofs of global class field theory (using ideal-theoretic arguments). I've seen the proof of the long exact sequence ...

**15**

votes

**3**answers

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 ...

**8**

votes

**2**answers

521 views

### H^d[U(1)^n,U(1)] of the Borel cohomology and Chern-Simons theory

Firstly I apologize that I am a physicist, with a relatively unrigorous math training. My approach of the problem can be Feynman style. Below $Z$ is the integer $\mathbb{Z}$, and $U(1)$ Abelian group ...

**7**

votes

**1**answer

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 ...

**3**

votes

**1**answer

1k views

### 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 ...

**16**

votes

**2**answers

677 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 ...

**10**

votes

**2**answers

667 views

### Hilbert 90 for algebras

Let $L\diagup K$ be a Galois extension of fields satisfying $\left[L:K\right] < \infty$. Let $B$ be a finite-dimensional (as a $K$-vector space) $K$-algebra. Then, the Galois group $G$ of $L\diagup ...

**4**

votes

**2**answers

673 views

### Calculate the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2, Z_T]$

I would like to know what are the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2, Z_T]$, and/or how to calculate them.
It can be shown that $H^d[U(1), Z]$ is $Z$ for ...