All Questions
21
questions
7
votes
0
answers
216
views
Relation beween Chern-Simons and WZW levels, and transgression
3d Chern-Simons gauge theories based on a Lie group $G$ are classified by an element $k_{CS}\in H^4(BG,\mathbb{Z})$, its level. Via the CS/WZW correspondence the theory is related with a 2d non-linear ...
- 761
6
votes
0
answers
147
views
Cohomology of $\mathrm{BPSO}(2d)$ with $Z_2$ coefficients
$\DeclareMathOperator\BPSO{BPSO}\DeclareMathOperator\PSO{PSO}\DeclareMathOperator\SO{SO}\DeclareMathOperator\BSO{BSO}$The projective group is $\PSO(2d)=\SO(2d)/Z_2$.
$\BPSO(2d)$ is the classifying ...
- 487
4
votes
2
answers
303
views
Low dimensional integral cohomology of $BPSO(4n)$
Toda has calculated the $\mathbb{Z}/2$‐cohomology ring of $BPSO(4n+2)$, and also gave the simple exceptional calculation of the $\mathbb{Z}/2$‐cohomology of $BPSO(4)$, in
Hiroshi Toda, Cohomology of ...
- 34.8k
2
votes
0
answers
95
views
Compact $G$-ENR's and Euler characteristic computed with Alexander-Spanier cohomology with compact support
Let $(Z,A)$ a compact ENR pair, then
$$\chi(Z)=\chi_c(Z-A)+\chi(A)$$
where $\chi_c$ is the Euler characteristic taken in Alexander-Spanier cohomology with compact support (ENR means euclidean ...
- 795
8
votes
0
answers
265
views
Integral cohomology of compact Lie groups and their classifying spaces
Let $G$ be a compact Lie group and let $BG$ be its classifying space. Let $\gamma\colon \Sigma G \to BG$ be the adjoint map for a natural weak equivalence $G \xrightarrow{\sim} \Omega BG$. Taking ...
20
votes
3
answers
2k
views
Integral cohomology of $SU(n)$ - looking for constants
I am interested in explicit generators of the cohomology $H^\bullet(SU(n),\mathbb{Z})$. Let $\omega = g^{-1} dg$ be the Maurer-Cartan form on $SU(n)$. The forms $\alpha_3,\alpha_5,\dots,\alpha_{2n-1}$,...
- 633
5
votes
2
answers
383
views
The mod p cohomologies of classifying spaces of compact Lie groups
I want to do some computation which need the mod p cohomologies of classifying spaces of connected compact Lie groups as input. I need the table for both the simply connected case and the central ...
- 71
14
votes
1
answer
749
views
The fifth k-invariant of BSO(3)
From work of Pontryagin and Whitney, as I understand it, the homotopy 4-type of $BSO(3)$ is $K(\mathbb{Z}/2,2) \times_{K(\mathbb{Z}/4,4)} K(\mathbb{Z},4)$, where the pullback is along the maps $\...
- 34.8k
1
vote
1
answer
261
views
cohomology ring of homogenous manifold
Let $[d_1^{t_1}, \dotsc, d_s^{t_s}]$ be a partition of a positive integer $n$, i.e., $\sum d_r t_r = n$. I want to know the de Rahm cohomology ring of the following types of homogenous spaces :
$$
G/H ...
- 116
8
votes
1
answer
496
views
Cohomology ring of a fiberwise join
I am very interested in the cohomology ring of the following construction. Let $f: Y\to X$ be a map between (connected) topological spaces. Suppose that the image of the map $f^*:H^*(X) \to H^*(Y)$ is ...
1
vote
1
answer
610
views
cohomology of orthogonal group of integers
Let
$$
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 ...
- 1,980
3
votes
1
answer
379
views
A Comparison between $\pi_{1}$ of cohomology and cohomology of $\pi_{1}$
Let we have a complex of abelian topological or lie groups $$\ldots \to G_{n}\to G_{n+1}\to \ldots$$ such that the image of $G_{n}$ is a closed subgroup of $G_{n+1}$. Then we have a complex of ...
- 452
51
votes
2
answers
2k
views
$H^4(BG,\mathbb Z)$ torsion free for $G$ a connected Lie group
Recently, prompted by considerations in conformal field theory, I was lead to guess that for every compact connected Lie group $G$, the fourth cohomology group of it classifying space is torsion free.
...
- 42.7k
2
votes
0
answers
189
views
The transfer map $H_*(BSO(3))\rightarrow H_*(BO(2))$: reference request
All cohomology and homology will be $Z/2$ coefficient. The restriction map
$H^*(BSO(3))\rightarrow H^*(BO(2))$ is well-known to be the inclusion of
the Dickson invariant $Z/2[w_2,w_3]$ into the ...
- 3,051
4
votes
1
answer
365
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 ...
- 4,890