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 $n$.



space of homotopy equivalences of $S^1$
Put $$HE^+\_1(S^1)=\{f\in HE^+(S^1):f(1)=1\}. $$  

There is an evident homeomorphism $m:S^1\times HE\_1^+(S^1)\to HE^+(S^1)$ given by $m(z,f)(x)=z f(x)$, and this restricts to give a homeomorphism $S^1\times Homeo\_1^+(S^1)\to Homeo^+(S^1)$.


It will thus suffice to discuss $HE\_1^+(S^1)$ and $Homeo\_1^+(S^1)$. 


Next, define $e:\mathbb{R}\to S^1$ by $e(t)=\exp(2\pi i t)$. Let $X$ denote the space of maps $$ u:[0,1]\to\mathbb{R} $$
with $u(0)=0$ and $u(1)=1$, and let $Y$ be the subspace of strictly increasing maps. For any $u\in X$ there is a unique map $p(u):S^1\to S^1$ with $p(u)(e(t))=e(u(t))$ for all $t\in [0,1]$. This construction gives a homeomorphism $p:X\to HE^+\_1(S^1)$, and restricts to a homeomorphism $p:Y\to Homeo\_1^+(S^1)$. This is a fairly straightforward exercise with covering space theory and topologies on mapping spaces. Now $X$ and $Y$ are both convex, so they have obvious contractions to the identity map given by 
$$ h(t,u)(x) = tx+(1-t)u(x). $$


It is not hard to see that $Y$ is not closed in $X$, so it cannot be a retract, so $Homeo\_1^+(S^1)$ is not a retract of $HE\_1^+(S^1)$. 


