$Homeo(S^1)$ 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 tand 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.

$Homeo(S^1)$ 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.