All Questions
Tagged with topological-groups classifying-spaces
13 questions
3
votes
0
answers
90
views
Topological groups satisfying the Borel transgression theorem
I am using the Borel transgression theorem as given in Mimura and Toda's "Topology of Lie groups I and II", page 378, Theorem 2.7. I know that it applies when the fiber has the homotopy type ...
- 61
8
votes
1
answer
485
views
A question about cohomology of the classifying spaces of compact groups
Let $G$ be a compact group (maybe non-Lie group). Let $B_{G}$ denote the
classifying space of $G$. If $G$ contains a circle group $\mathbb{S}^{1}$,
then I think that $H^{\ast }( B_{G};\mathbb{Q}
)$ is ...
- 1,367
5
votes
0
answers
192
views
When is the classifying space of a group/H-space rationally equivalent to a product of Eilenberg-MacLane spaces?
The MO-question asks why the classifying space of a group is not necessarily rationally a product of Eilenberg–MacLane spaces.
I am looking for classes of examples of connected topological groups/...
- 1,174
10
votes
1
answer
233
views
Classifying space of centralizer
$\DeclareMathOperator\Map{Map}\newcommand{\B}{\mathrm{B}}\newcommand{\h}{\mathrm{h}}$Let $f:G\to H$ be a morphism of topological groups and let
$$H^{\h G}:=\Map_G(\mathrm{E}G, H)$$
be the homotopy ...
- 103
3
votes
1
answer
211
views
Defining the classifying space of a group acting on a set
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gal{Gal}$Recall that $n$-simplices of the classifying space (simplicial set) $BG$ of a group are orbits of the diagonal action by shifts
on $n+1$-...
- 513
7
votes
2
answers
335
views
If $G$ is a topological group that contains a torsion element, then the classifying space $BG$ is infinite-dimensional?
We know that if $G$ is a topological group that contains a torsion element and $G$ satisfies additional conditions such as $G$ discrete or $G$ finite-dimensional, then the classifying space $BG$ is ...
- 10.5k
9
votes
0
answers
367
views
Is every space a classifying space?
Despite a pretty thorough look (I think) I can’t find the answer to the following question: Is every (reasonable?) path connected space weakly equivalent to the classifying space of some topological ...
- 1,198
8
votes
1
answer
688
views
For which G is BLG weak homotopy equivalent to LBG?
Let $G$ be a (Edit: path-)connected topological group. Under what additional hypotheses on $G$ is it true that $LBG$ is a classifying space for $LG$? (or, I guess equivalently, when is $LBG \sim BLG$?)...
- 35.5k
1
vote
0
answers
142
views
Topological groups with homeomorphic underlying spaces, isomorphic abstract groups and homotopy equivalent classifying spaces
Define the classifying space $BG$ of a well-pointed topological group $G$ as the fat realization of the nerve of $G$.
Let $G$, $H$ be well-pointed topological groups. Assume that there is a ...
- 231
5
votes
1
answer
443
views
Classification of fibrations for classifying spaces $B^2\mathbb{Z}_2$ and $BSO(3)$ or $BO(3)$
I am interested in knowing what can we say about the classification of fibrations for classifying spaces $B^2M \equiv B^2\mathbb{Z}_2$ and $BG \equiv BSO(3)$ or $BO(3)$.
Here we can take either:
$B^...
- 2,147
4
votes
0
answers
205
views
Why is any $G$-resolution a principal $G$-bundle?
In the article The Cohomology of Classifying Spaces of H-Spaces by M. Rothenberg and N. Steenrod (https://projecteuclid.org/euclid.bams/1183527356) it is stated as a theorem that if $G$ is a ...
- 79
3
votes
0
answers
505
views
fibre sequence of classifying space
I read Steve Mitchell's Notes on principal bundles and classifying spaces (pdf).
There is a theorem: Let $G$ be any topological group, $H$ an admissible
normal subgroup. Then there is a homotopy-fibre ...
- 1,367
16
votes
2
answers
1k
views
rationalization of classifying spaces
This question is probably trivial for anyone who is more familiar with rational homotopy theory than me, but anyway:
Let $G$ be a simply-connected topological group. In particular, it is an $H$-...
- 7,637