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$\DeclareMathOperator\Diff{Diff}$I am looking for the paper computing the homotopy type of the group $\Diff(\mathbb{R}P^3)$ of diffeomorphisms of the $3$-dimensional real projective space $\mathbb{R}P^3$. I know that the homotopy types of $\Diff(S^3)$ and $\Diff(S^2\times S^1)$ are computed by Allen Hatcher, see also explanations by Aramita Amabel.

It should be mentioned that $\pi_0\Diff(L_{p,q})$ for $p\geq2$ is described by Fransis Bonahon. The homotopy types of groups of diffeomorphisms of lens spaces $L_{p,q}$ with $p>2$ are also described in the book "Diffeomorphisms of Elliptic 3-Manifolds" by Sungbok Hong, John Kalliongis, Darryl McCullough, J. Hyam Rubinstein devoted to Smale Conjecture claiming that the inclusion $\operatorname{Isom}(M) \subset \Diff(M)$ is a homotopy equivalence.

However, for the remaining lens space $L_{2,1} = \mathbb{R}P^3$ I can not find a precise information about the homotopy type of $\Diff(L_{2,1})$.

Thus the question is whether the homotopy type of $\Diff(\mathbb{R}P^3)$ is computed? And if yes, could you please provide a reference to the corresponding paper?

Added after the comment below by Andy Putman: If the Smale Conjecture holds for $\mathbb{R}P^3$, then $\Diff(\mathbb{R}P^3) \simeq \mathrm{Isom}(\mathbb{R}P^3) = O(4)/\{I,-I\}$, see description of isometry groups, e.g. Theorem 2.3 in this paper:

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    $\begingroup$ It hasn't been computed. Roughly speaking, it's expected that adapting the techniques from Hatcher's Smale Conjecture paper should do the job. But it would be an enormous amount of work in an era where this kind of work does not get much recognition. So nobody has done it. $\endgroup$
    – Ryan Budney
    Commented Sep 25, 2022 at 17:52
  • $\begingroup$ Thank you very much Ryan. $\endgroup$
    – Sergiy Maksymenko
    Commented Sep 25, 2022 at 18:00
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    $\begingroup$ I think this was recently dealt with using Ricci flow by Bamler and Kleiner. See here: math.nyu.edu/%7Ebkleiner/psc.pdf $\endgroup$
    – Andy Putman
    Commented Sep 25, 2022 at 18:48
  • $\begingroup$ (I can post this as an answer later — I am typing on my phone on a plane about to take off) $\endgroup$
    – Andy Putman
    Commented Sep 25, 2022 at 18:49
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    $\begingroup$ @GSM: For example take the cited Rubinstein book. I believe they first wanted that to be a paper, but difficulty finding referees (as far as I understand) forced the publication mode to book. These are important results, but if there isn't a sufficiently large base for an area using specialized techniques, it can be difficult for editors to find people to vet even quality results. This is doubly-frustrating as people reading the result would like to know it has been checked in detail. Similarly for authors, you'd like to know someone (other than yourself) has read your work. $\endgroup$
    – Ryan Budney
    Commented Sep 25, 2022 at 22:14

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This was answered by Bamler and Kleiner, who proved more generally that the diffeomorphism group of any spherical space form deformation retracts to its isometry group. This in particular gives a new proof of Hatcher's resolution of the Smale conjecture for $S^3$. Their main tool is Ricci flow.

This result is in their paper "Ricci flow and contractibility of spaces of metrics", available here. It builds on their previous paper "Ricci flow and diffeomorphism groups of 3-manifolds", available here.

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    $\begingroup$ Thank you very much for references. Thus Diff(RP^3) ~ Isom(RP^3) = O(4)/{I,-I}. $\endgroup$
    – Sergiy Maksymenko
    Commented Sep 26, 2022 at 8:16
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    $\begingroup$ Note also that there is a surjective map $S^3\times S^3\to SO(\mathbb{H})=SO(4)$ given by $(u,v)\mapsto(x\mapsto ux\overline{v})$. This induces an isomorphism from $SO(3)\times SO(3)\simeq(S^3/\{\pm 1\})\times(S^3/\{\pm 1\})\simeq \mathbb{R}P^3\times\mathbb{R}P^3$ to $SO(4)/\{\pm I\}$. $\endgroup$
    – Neil Strickland
    Commented Sep 26, 2022 at 10:00

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