Classifying spaces [related to Eilenberg–MacLane] for explicit group examples

Asked May 21 '18 at 14:52

Active May 22 '18 at 08:57

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I am interested in knowing the generalized classifying spaces (related to Eilenberg–MacLane space $K(G,n)$ when $G$ is discrete) for explicit group examples ($G=$ the entries given at the top row) given below. Can someone fill in the Table?

$$\begin{array}{c|c|c|c|c|c|c|} G & \mathbb{Z}^n & \mathbb{Z} & \mathbb{R} &\mathbb{R}/\mathbb{Z}=U(1) & \mathbb{Z}_2 & \mathbb{Z}_n \\ \hline BG & T^n & S^1 & & CP^{\infty} & RP^{\infty} & S^{\infty}/\mathbb{Z}_n \\ \hline B^2G & & CP^{\infty} & & & & \\ \hline B^3G & & & & & & \\ \hline \end{array}$$

Here we define $B^nG$ as the $\pi_{n}(B^nG)=G$, while other homotopy groups are zero, $\pi_{i}(B^nG)=0$, for $i \neq n$.

edited May 22 '18 at 08:57

asked May 21 '18 at 14:52

wonderich

6,059

What's a contractible space on which $\mathbb{R}$ acts freely? – Exit path May 21 '18 at 16:24
See also: Which BGs are also K(π,n)s? https://math.stackexchange.com/q/2348307/79069 – wonderich May 21 '18 at 18:09
and also a previous trial https://math.stackexchange.com/q/2789563/79069 – wonderich May 21 '18 at 18:10
I'm saying you should be able to figure out $BG$ for $G= \mathbb{R}$ – Exit path May 21 '18 at 18:18
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@ leibnewtz, it is a point. – wonderich May 21 '18 at 18:51
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"No" is a pretty good answer to your question. Most classifying spaces just don't have nice explicit descriptions, though there are a few more cells of your table that can be filled in nicely. – Eric Wofsey May 21 '18 at 19:25
@ leibnewtz, it is a point with cover R. – wonderich May 21 '18 at 20:18
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Your definition of $B^n G$ is still not correct if $G$ is not discrete. – Qiaochu Yuan May 21 '18 at 22:21
@ Qiaochu Yuan, +1, please feel free to modify my question/ and illuminate. Thanks! – wonderich May 21 '18 at 23:17

Classifying spaces [related to Eilenberg–MacLane] for explicit group examples

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