37
votes
2answers
1k views

$H^4(BG,\mathbb Z)$ torsion free for $G$ a connected Lie group

Recently, prompted by considerations in conformal field theory, I was let to guess that for every compact connected Lie group $G$, the fourth cohomology group of it classifying space is torsion free. ...
8
votes
0answers
346 views

Homology of Lie groups

Let $G$ be a Lie group and $G^{\delta}$ the underlying group (with discrete topology). Obviously, we have a continuous map of groups $i:G^{\delta}\rightarrow G$ which induces a map between classifying ...
14
votes
1answer
872 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 ...
6
votes
1answer
902 views

Double coset formulas for Orthogonal groups [Solved]

According to Madsen-Brumfiel "Evaluation of the Transfer and the Universal Surgery Classes" Inventiones mathematicae 32 (1976): 133-170 Theorem 3.11, we can compute the composition ...
4
votes
2answers
544 views

Group cohomology of orthogonal groups with integer coefficient

I would like to know the group cohomology of orthogonal groups $SO(n)$, which is the topological cohomology of the classifying space of the group: $H^*(BSO(n);\mathbb{Z}) = $ ? (for example for ...
1
vote
0answers
159 views

Exotic Chains for Group Homology of a Complex Lie Group

Related Question: Exotic Chains for Group Cohomology of a Complex Lie Group Let's take the group homology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural ...
4
votes
0answers
149 views

Exotic Chains for Group Cohomology of a Complex Lie Group

Related Question: Exotic Chains for Group Homology of a Complex Lie Group Let's take the group cohomology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural ...
5
votes
3answers
594 views

Continuous cohomology of semi-simple Lie group.

Let $G$ be a real connected semi-simple Lie group. Let $M$ be a finite dimensional representation of it. Are there general criteria when the continuous cohomology groups $H_{cont}^q(G,M)$ vanish? A ...
3
votes
3answers
1k views

Group cohomology of compact Lie group with integer coeffient

It is known that group cohomology class $H^d[U(1),Z]$ is Z for even d and 0 for odd d. Do we know $H^d[G,Z]$ for $G=SO(3)$, $SU(2)$ and other compact Lie group? Also is the Borel-group-cohomology ...
5
votes
2answers
748 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 ...
8
votes
0answers
342 views

Is there a general dilogarithm formula for the Cheeger--Chern--Simons class?

I'm looking for a generalization of the calculation of the hyperbolic volume and Chern--Simons invariant for $\operatorname{SL}(2,\mathbb C)$ representations in terms of the Rogers dilogarithm. ...
11
votes
0answers
390 views

To what extent does (co)homology of groups made discrete depend on set theory?

There's a well-known paper by Milnor, "On the homology of Lie groups made discrete," that discusses the relation between the homology of a Lie group $G$ and the underlying discrete group $G^\delta$. ...