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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 $B\operatorname{Homeo}(S^1)$?

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up vote 7 down vote accepted

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.

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$Homeo(S^1)$ is homotopy equivalent to $O(2)$. This follows from the fact that the space of all homeomorphisms $S^1\to S^1$ having degree one and fixing a given point is contractible. In fact, the latter space is homeomorphic to the (convex) space of all homeomorphisms $[0,1]\to [0,1]$ that fix both endpoints.

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