What is the classifying space for $S^1$-bundle? Here, $S^1$-bundle means a fiber bundle which doesn't mean that it is principal $S^1$-bundle.


I know that for a space $F$, 


$\lbrace$the set of fiber bundles over $M$ whose fiber is $F\rbrace$ = $[M,B\operatorname{Homeo}(F)]$.


Therefore, my question can be rephrased as what is the homotopy type of $B\operatorname{Homeo}(S^1)$? 



Classifying space for S1-bundle?
As mentioned by Tom Goodwillie, $Homeo(S^1)$ is homotopy equivalent to $O(2)=\mathbb Z/2\ltimes S^1$. We have a (split) short exact sequence of groups
$$
S^1 \to \mathbb Z/2 \ltimes S^1 \to \mathbb Z/2
$$
which, upon applying applying the functor $B$, produces a (split) fibration sequence of classifying spaces
$$
BS^1 \to B(\mathbb Z/2 \ltimes S^1) \to B\mathbb Z/2.
$$
The latter can be rewritten as
$$
\mathbb CP^\infty \to BHomeo(S^1) \to \mathbb RP^\infty.
$$


The space $BHomeo(S^1)$ can be characterized by the facts that its $\pi\_1$ is $\mathbb Z/2$, its $\pi\_2$ is $\mathbb Z$, it has a non-trivial action of $\pi\_1$ on $\pi\_2$, and it has a trivial $k$-invariant in $H^3(\mathbb Z/2, \{ \mathbb Z \} )\cong\mathbb Z/2$. The last statement comes from the fact that the above fibration sequence is split.


