# Tag Info

## Hot answers tagged classifying-spaces

27 votes
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• 18.5k
14 votes

### If $G$ is a topological group that contains a torsion element, then the classifying space $BG$ is infinite-dimensional?

Here is sort of a canonical example. Consider $GL(\mathbb H)$ the group of invertible operators on a Hilbert space. By Kuipers theorem it is contractible. But $GL(\mathbb H)$ acts freely and properly ...
• 7,185
13 votes
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### Equivariant classifying spaces from classifying spaces

Added Aug 2016: I've written this up, available at https://arxiv.org/abs/1608.02999 $\def\Hom{\mathrm{Hom}} \def\Map{\mathrm{Map}} \def\ad{\mathrm{ad}}$ I think this is true. I'll sketch a ...
• 26.5k
13 votes
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### When does $BG \to BA$ loop to a homomorphism?

If $G$ is a compact connected topological group and $A$ is a locally compact abelian topological group, then for any map $f:BG\to BA$ the looped map $\Omega f:\Omega BG\to \Omega BA$ is homotopically ...
• 34.8k
13 votes
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• 6,716
11 votes

### quotient space of Eilenberg-MacLane space

$K(\pi,1)/G$ is not necessarily $K(\pi\times G,1)$ for example take $G=Z$, $K(Z,1)=S^1$. $S^1=R/(t_1=x\rightarrow x+1)$ the quotient of $S^1$ by the group $Z/n$ generated by the transformation induced ...
• 3,644
11 votes
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• 50.3k
10 votes
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### Classifying space of a non-discrete group and relationship between group homology and topological homology of Lie groups

You may want to look at the classical paper of Jack Milnor, "On the homology of Lie groups made discrete." The Friedlander-Milnor conjecture states that the map $BG^\delta \to BG$ (where $G$ ...
• 9,361
10 votes

• 10.2k
9 votes
Accepted

### Interesting properties in $...\to K(\mathbb{Z}_4,1) \overset{f}{\to} K(\mathbb{Z}_2,1)\overset{g}{\to}K(\mathbb{Z}_2,2) \to ...$

Represent $p$ by the identity map $id: \mathbb{Z}_2 \to \mathbb{Z}_2$. Then $(p\cup p)(a,b) = p(a)p(b)$ is non-zero only on the 2-chain $(1,1)$. Namely, as a polynomial mod 2, $(p\cup p)(a,b) = ab$. ...
• 4,024
8 votes

### Classifying space as the geometric realization of the nerve of $G$ viewed as a small category

You should ignore simplicial objects at first, and just consider groupoids. In the following, you can let $G$ be a topological group such that $e\hookrightarrow G$ is a closed cofibration. All ...
• 33.1k
8 votes

### Cohomology of $BE_8$ and $BSU(2)$

I believe Appendix 1. in Finite H-spaces and Lie Groups" by Frank Adams shows that BE8 has 2,3 and 5-torsion. The letter from E8 at the end of this paper is also quite amusing: ....Be it therefore ...
• 5,241
8 votes

### Classification of fibrations for classifying spaces $B^2\mathbb{Z}_2$ and $BSO(3)$ or $BO(3)$

Firstly, yes, your examples are all correct. However, in example~(B) we just have $O(3)=\{\pm I\}\times SO(3)$ as groups, so your fibration is just the product of $$BSO(3)\xrightarrow{1}BSO(3)\to 1$$...
• 54.5k

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