All Questions
15
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
4
votes
2
answers
383
views
Classifying space of a non-discrete group and relationship between group homology and topological homology of Lie groups
I have a very soft question which might be very standard in textbooks or literature but I haven't seen it.
To a fixed group $G$ we may attach different topologies to make it different topological ...
- 389
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
8
votes
1
answer
262
views
When does $BG \to BA$ loop to a homomorphism?
If I have a compact connected Lie group $G$ and a (relatively nice) simply-connected topological abelian group $A$, when is it the case that a given $f\colon BG \to BA$ loops to a (continuous) ...
- 34.8k
9
votes
1
answer
711
views
Sullivan conjecture for compact Lie groups
Let $G$ be a topological group, and $M$ a connected compact smooth manifold. I'm studying
$$ \pi_0 (map (BG,M)). $$
For $G$ a finite group, we know that this is just a point by the Sullivan ...
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
4
votes
0
answers
228
views
The homotopy type of the mapping space $Map_{B\rho}(BS^1,BG)$? for $G$ a compact Lie group
Given a homomorphism $\rho:S^1\rightarrow G$ with $G$ a compact Lie group there is an induced map of classifying spaces $B\rho:BS^1\rightarrow BG$. What is known about the homotopy type of the mapping ...
- 5,221
7
votes
0
answers
175
views
Manifold approximations to $BO(3)$
We know that $BO(1) =\mathbb{R}P^\infty$ has closed, finite-dimensional manifold approximations $\mathbb{R}P^1\subset \mathbb{R}P^2\subset\cdots.$
Similarly $BO(2)$ can be approximated by closed, ...
- 35.5k
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
9
votes
2
answers
460
views
Integral versus real (universal) characteristic classes
I'm pretty confused about the precise relation of the integral and the real cohomology of the classifying space $BG$ of a compact Lie group $G$. The natural map $H^n(BG;\mathbb{Z})\to H^n(BG;\mathbb{R}...
- 1,040
6
votes
1
answer
3k
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
$BO(1)^2\stackrel{...
- 3,051
8
votes
1
answer
336
views
The Image of the Mod 2 Homology of BSp in the Homology of BSO
I'm essentially trying to figure out exactly what the title asks for. I've been scouring old Seminaires Henri Cartan and books by Stong to try to see exactly how to do this, but the combination of ...
- 10.3k
23
votes
6
answers
5k
views
cohomology of BG, G compact Lie group
It has been stated in several papers that $H^{odd}(BG,\mathbb{R})=0$ for compact Lie group
$G$. However, I've still not found a proof of this. I believe that the proof is as follows:
--> $G$ compact ...
- 1,709
13
votes
3
answers
2k
views
Representations of \pi_1, G-bundles, Classifying Spaces
This question is inspired by a statement of Atiyah's in "Geometry and Physics of Knots" on page 24 (chapter 3 - Non-abelian moduli spaces).
Here he says that for a Riemann surface $\Sigma$ the first ...
- 2,694