# Orientation preserving self-homotopy equivalences of the 2-sphere

According to this question, Hansen proved that the space $\mathrm{Aut}_0(\mathbb{S}^2)$ of orientation preserving self-homotopy equivalences of the 2-sphere is homotopy equivalent to $\mathrm{SO}(3)\times\mathbf{\Omega}$, where $\mathbf{\Omega}$ is the universal cover of the connected component of $\Omega^2(\mathbb{S}^2)$ containing the generator of $\pi_2(\mathbb{S}^2)$ (or any other component, they are all homotopy equivalent).

Let’s now consider the space $\mathrm{Aut}^{\mathrm{pr}}_0(\mathbb{S}^2)$ of pairs $(f,H)$ where $f$ is a self-homotopy equivalence of the 2-sphere and $H$ is an homotopy between f and the identity function a connected component of the space of homotopies between $f$ and the identity function. You can see $H$ as a "proof" that $f$ is orientation preserving and $\mathrm{Aut}^{\mathrm{pr}}_0(\mathbb{S}^2)$ is the space of orientation preserving self-homotopy equivalences $f$ of the 2-sphere where you do not throw away the proof that $f$ is orientation preserving (the "pr" stands for "proof relevant").

What is the homotopy type of $\mathrm{Aut}^{\mathrm{pr}}_0(\mathbb{S}^2)$?

One can reasonably expect that the answer is $\mathrm{SO}(3)$, the noise in $\mathrm{Aut}_0(\mathbb{S}^2)$ looks very similar to the missing homotopy, but I don’t know how to prove it.

Also, thanks to Springer I can’t find an online version of Hansen’s paper, so maybe this is already in his paper.

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Isn't this contractible? Just contract $f$ to the identity through $H$.
To answer this we just need to figure out what the set of connected components of the space of homotopies between $f$ and the identity function is. Clearly this is homotopy-invariant, so we may assume $f$ is the identity function. These are just homotopy classes of maps $S^1 \times S^2 \to S^2$ that are degree $1$ on the $S^n$ part. I don't know what that looks like, but note that it is a set with no topology, and the projection to $Aut_0(S^2)$ is locally an isomorphism, so the space is a union of connected components that are either $SO_3 \times \Omega$ or $SU_2 \times \Omega$. –  Will Sawin Mar 23 '13 at 6:01
Actually isn't this just the universal cover of $Aut_0(S^2)$, by the definition of the unniversal cover as pairs of a point $p$ and a homotopy class of paths from the base point to $p$? So it's $SU_2 \times \Omega$ –  Will Sawin Mar 23 '13 at 19:10