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36
votes
9answers
7k views

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 ...
5
votes
1answer
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
1answer
1k 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 ...
27
votes
6answers
2k views

Why is the standard definition of cocycle the one that _always_ comes up??

This question might not have a good answer. It was something that occurred to me yesterday when I found myself in a pub, needing to do an explicit calculation with 2-cocycles but with no references ...
16
votes
3answers
769 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 ...
7
votes
1answer
252 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 ...
17
votes
2answers
829 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 ...
15
votes
7answers
2k views

universal cover of SL2(R): does it admit central extensions?

Is it true that the universal cover of SL2(ℝ) has no non-trivial central extensions... as an abstract group? (that's certainly true as a Lie group) Motivation: I have a projective action of ...
10
votes
2answers
719 views

(co)homology of symmetric groups

Let $S_n=\{\text{bijections }[n]\to[n]\}$ be the n-th symmetric group. Its (co)homology will be understood with trivial action. What are the $\mathbb{Z}$-modules $H_k(S_n;\mathbb{Z})$? Using GAP, we ...
8
votes
2answers
539 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 ...
10
votes
2answers
705 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 ...
5
votes
0answers
159 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 ...
5
votes
2answers
792 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 ...
3
votes
0answers
998 views

group cohomology and cohomology of classifying space [closed]

Let $G$ be a discrete group, and $BG$ is the classifying space. It is well-known that the group cohomology of $G$-module M, is the same as the cohomology on $BG$ with coefficient in $\tilde{M}$, which ...
2
votes
1answer
374 views

Why does the Cheeger--Chern--Simons class descend to H_3(G/B)?

Cheeger and Simons here defined a family of characteristic classes for principal $G$-bundles ($G$ a Lie group). I'm interested in the special case of $\hat c_2:H_3(SL(2,\mathbb C);\mathbb ...
6
votes
1answer
588 views

Classification of $p$-groups of order $p^n$ with rank $n-1$

Hello, i've been looking for a way to classify the non-trivial $p$-groups $G$ that live in an exact sequence of the form $ 0 \rightarrow \mathbb{Z}/p\mathbb{Z} \rightarrow G \rightarrow ...
9
votes
1answer
302 views

Second cohomology of group of $S_n$

Hello, Let $k$ be a field of characteristic different from $2$. Let $n\geq 1$ be an integer, and let $T$ be the maximal torus of the $k$-algebraic group $PGL_n$, namely the quotient of diagonal ...
5
votes
0answers
66 views

(Eichler-Shimura Isomorphism) Proving c(f) is not a boundary

I have seen a couple of questions related to the Eichler-Shimura Isomorphism, but almost all of them have to do with hodge theory (things I am unfamiliar with) and seem, to me, different/unrelated. ...
3
votes
3answers
365 views

classifying space and cohomology of integer general linear group

I have obtained that the classifying space $$ BGL(\mathbb{R}^n)=BO(\mathbb{R}^n)=G_n(\mathbb{R}^\infty) $$ is the Grassmannian. I have also obtained that the mod 2 cohomology is the polynomial ...
2
votes
1answer
292 views

vanishing higher cohomology group for property T group?

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, ...