Does the space of homotopy equivalences of $S^1$ deformation retract onto the space of homeomorphisms of $S^1$? If so, does anyone have a reference?
I found that Kneser proved that $Homeo(S^1)$ deformation retracts onto $O(2)$ and $Homeo^+(S^1)$ deformation retracts onto $SO(2)$ (orientation preserving homeos deformation retracts onto rotations). I'd like the space $HE^+(S^1)$ of degree 1 homotopy equivlances of $S^1$ to deformation retract onto these. The space I'm calling $HE^+(S^1)$ may go by $HomEq(S^1)$ or $SG_n$ and seems to be of interest to homotopy theorists for higher $n$.