All Questions
7 questions
3
votes
0
answers
168
views
Obstruction to delooping
Let $G$ be a finite group. It can be think of as a $1$-category with one object and $|G|$ many morphisms. If $A$ happens to be abelian, then one can think of it to an $n$-category. Conversly, this ...
- 5,230
2
votes
1
answer
391
views
cohomology of orthogonal (or general linear) group over finite fields
Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let
$$
O(\mathbb{Z}_2^{\oplus k})=\{A\mid A \text{ is a } k\times k \text{ - matrix with entries } 0,1, det(A)=\pm 1\}
$$
What is $$
H^*(BO(\mathbb{Z}_2^{\...
- 1,990
1
vote
1
answer
614
views
cohomology of orthogonal group of integers
Let
$$
O(\mathbb{Z}^{\oplus k})=GL(\mathbb{Z}^{\oplus k})\cap O(k).
$$
What is $$
H^*(BO(\mathbb{Z}^{\oplus k});\mathbb{Z})?
$$
If it cannot be computed out, can we get
$$
H^*(O(\mathbb{Z}^{\oplus ...
- 1,990
8
votes
1
answer
378
views
What's a good example/reference for cohomology classes on Springer fibers that aren't restricted from the flag variety
As usual, by Springer fiber, I mean the fixed points $X^u$ of a unipotent element $u$ of the group $G$ on the flag variety $X=G/B$. It's a lovely theorem that when $G=SL_n$, the induced map on ...
- 44.7k
4
votes
1
answer
370
views
Cohomology of Projective Classical Lie Groups
Let $G$ be a compact, connected, simply-connected Lie group with centre $Z(G)$, and consider the Lie group $G/Z(G)$. I believe that for $G$ a classical group, the Lie group $G/Z(G)$ is sometimes ...
- 4,920
20
votes
2
answers
3k
views
Is there any "deep" relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory
First let's consider equivariant cohomology: if a compact Lie group $G$ acts on a compact manifold $M$. We have the equivariant cohomology $ H_G(M)$ defined as the cohomology of the cochain complex $((...
- 9,019
9
votes
3
answers
636
views
Group Extensions and Line Bundles on $BG$
I am sure the answer to this question is well-known, but
It is well known that the group cohomology $H^2(G,\mathbb Z)$ classifies group extensions $0\to \mathbb Z\to E\to G\to 1$ and that for a ...
- 2,283