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Let $\pi$ be a group and $K(\pi,1)$ the Eilenberg-MacLane space. Let $G$ be a finite group acting on $K(\pi,1)$ such that the following is a covering map $$ K(\pi,1)\longrightarrow K(\pi,1)/G. $$ Question. Is $K(\pi,1)/G$ an Eilenberg-MacLane space? Does $$ K(\pi,1)/G=K(\pi \times G,1)? $$

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up vote 14 down vote accepted

Suppose a group $H$, not necessarily finite, acts on an Eilenberg-MacLane space $BN$. The homotopy quotient $BN/H$ (which agrees with the ordinary quotient if the action of $H$ is free) fits into a fiber sequence

$$BN \to BN/H \to BH$$

and the long exact sequence in homotopy shows that $BN/H$ has vanishing higher homotopy. Hence it is an Eilenberg-MacLane space $BG$ for a group $G$ fitting into a short exact sequence

$$1 \to N \to G \to H \to 1$$

(determined by the action). In other words, it's an extension of $H$ by $N$.

Every such extension arises in this way. Among them, semidirect products (extensions for which the above sequence splits) correspond to pointed actions of $H$ on $BN$, or equivalently actions of $H$ on $BN$ admitting a (homotopy) fixed point.

For example, take $BN$ to be the configuration space of $n$ ordered points in $\mathbb{R}^2$, so that $N = P_n$ is the pure braid group, and $H = S_n$ acts by permutations of the points. Then $BN/H = BG$ is the configuration space of $n$ unordered points in $\mathbb{R}^2$, so that $G = B_n$ is the usual braid group. The corresponding short exact sequence

$$1 \to P_n \to B_n \to S_n \to 1$$

does not split.

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Tsemo's example is the short exact sequence $1 \to \mathbb{Z} \xrightarrow{n} \mathbb{Z} \to \mathbb{Z}/n \to 1$ which also does not split. – Qiaochu Yuan Jan 13 at 5:03

$K(\pi,1)/G$ 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 by $x\rightarrow x+1/n$ is $S^1$.

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