All Questions
23
questions
6
votes
0
answers
111
views
Explicit representatives for Borel cohomology classes of a compact Lie group?
I'm looking for explicit representatives of $H^3_{Borel}(G, R/Z)$, i.e. a measurable function $G^3\to R/Z$ representing a generator of the cohomology group. (Here $G$ is a compact (perhaps simple) ...
- 12.7k
7
votes
2
answers
479
views
Injectivity of the cohomology map induced by some projection map
Given a (compact) Lie group $G$, persumably disconnected, there exists a short exact sequence
$$1\rightarrow G_c\rightarrow G\rightarrow G/G_c\rightarrow 1$$
where $G_c$ is the normal subgroup which ...
- 125
6
votes
1
answer
406
views
What is known about the discrete group cohomology $H^2(\mathrm{SL}_2(\mathbb C), \mathbb C^\times)$?
The cohomology ring of $\mathrm{SL}_2(\mathbb C)$ as a topological group is straightforward (it's generated by a Chern class), but what is known in the discrete case? I'm particularly interested in $H^...
- 2,493
5
votes
0
answers
294
views
What is the image of the diagonal map on the cohomology of Lie groups
Consider a simple Lie group $G$ and its mod $p$ cohomology $H^*(G, \mathbb{Z}_p)$.
A good reference is the book
Mimura, Mamoru; Toda, Hirosi, Topology of Lie groups, I and II. Transl. from the Jap. by ...
- 1,429
31
votes
2
answers
1k
views
Is Lie group cohomology determined by restriction to finite subgroups?
Consider the restriction of the group cohomology $H^*(BG,\mathbb{Z})$, where $G$ is a compact Lie group and $BG$ is its classifying space, to finite subgroups $F \le G$. If we consider the product of ...
- 413
5
votes
1
answer
162
views
Elementary $p$-subgroups of a compact Lie group
How to determine (say up to conjugacy) elementary $p$-subgroups of a compact Lie group $G$?
Of course there are the $p$-subgroups of a maximal torus, and in the case $G=\mathrm{PU}_p$, there is an ...
- 935
20
votes
0
answers
442
views
Which classes in $\mathrm{H}^4(B\mathrm{Exceptional}; \mathbb{Z})$ are classical characteristic classes?
Let $G$ be a compact connected Lie group. Recall that $\mathrm{H}^4(\mathrm{B}G;\mathbb{Z})$ is then a free abelian group of finite rank. Let us say that a class $c \in \mathrm{H}^4(\mathrm{B}G;\...
- 53.5k
3
votes
0
answers
76
views
When are there continuous families of pull-backs of a discrete cohomology class of a compact Lie group?
Let $\mathcal{G}$ be a compact Lie group. Then define $H^n(\mathcal{G},\mathrm{U}(1))$ to be the cohomology of measurable cochains $\mathcal{G}^{\times n} \to \mathrm{U}(1)$ with the usual coboundary ...
- 853
4
votes
1
answer
977
views
First homology group of the general linear group
The abelianization of the general linear group $GL(n,\mathbb{R})$, defined by $$GL(n,\mathbb{R})^{ab} := GL(n,\mathbb{R})/[GL(n,\mathbb{R}), GL(n,\mathbb{R})],$$ is isomorphic to $\mathbb{R}^{\times}$....
- 323
3
votes
2
answers
283
views
How many non-isomorphic extensions with kernel $S^1$ and quotient cyclic of order $p$?
I want to determine how many non-isomorphic extensions (as group they are non-isomorphic) are possible of the form $1 \to \mathbb{S}^1 \to G \to (\mathbb{Z}_p)^k \to 1$, where $G$ is a compact lie ...
- 123
11
votes
1
answer
219
views
Reference requests: Integral cohomology of $G_2$-homogeneous spaces
Do you know a place where the integral cohomology of $G_2$-homogeneous spaces is computed?
Great computational efforts using representation theory in order to determine the ...
- 2,620
5
votes
0
answers
101
views
Group cohomology of "twisted" projective SU(N) with various coefficients
Given a group
$$
G= PSU(N) \rtimes \mathbb{Z}_2,
$$
where $PSU(N)$ is a projective special unitary group. Say $a \in PSU(N)$, $c \in \mathbb{Z}_2$, then
$$
c a c= a^*,
$$
which $c$ flips $a$ to its ...
- 10.4k
35
votes
3
answers
1k
views
Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$
We fix $G=\mathrm{SL}_3(\mathbf{R})$.
Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b_2(\Gamma)=0$?
Here the second Betti number $b_2(\Gamma)$ is both the dimension of the ...
- 62.3k
2
votes
0
answers
69
views
Connected topological/Lie group $H$ and $Q$, inflate $Q$-cocycle to coboundary in $H$
I am interested in finding mathematical examples and criteria of inflating $Q$-cocycle to coboundary in $H$, under the requirement:
(1) Both $H$ and $Q$ are connected topological groups or Lie groups (...
- 2,147
3
votes
0
answers
119
views
Trivialize a cocycle of a continuous Lie group-cohomology to a coboundary
Someone recently asks a question $SO(3)$ 2-cocycle trivialized to a 2-coboundary in $SU(2)$? now inspires me to revisit an earlier general question to ask an example of 3-cocycle
$\omega_3^G$ of a ...
- 10.4k