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In his paper "Concordance spaces, higher simple homotopy theory, and applications", Hatcher calcuates the smooth, PL, and topological mapping class groups of the $n$-torus $T^n$. This requires an understanding of the surgery structure set of the $n$-torus, as well as the space of self-equivalences (easy, because the torus is aspherical and a Lie group).

At least $\pi_0$ and $\pi_1$ of $\text{hAut}(\Bbb{CP}^n)$, the space of self-homotopy equivalences, are known: the former is $\Bbb Z/2$ because $[\Bbb{CP}^n, \Bbb{CP}^n] = [\Bbb{CP}^n, \Bbb{CP}^\infty] = \Bbb Z$, and the only invertible elements are $\pm 1$; further the homology of this space seems to have been calculated by Sasao, who gives that $\pi_1 \text{hAut} = \Bbb Z/(n+1)$.

The best references I know for the surgery structure set of $\Bbb{CP}^n$ are Wall's book on surgery and the Madsen-Milgram book on Top/PL cobordism groups "The classifying spaces for surgery and cobordism of manifolds", neither of which seem to give a completely explicit description.

From this, it's not immediately clear to me whether enough is known to run Hatcher's argument for $T^n$ on complex projective space.

Has anybody computed the various mapping class groups of $\Bbb{CP}^n$ when $n \geq 3$? Maybe in any specific examples, or when $n$ is very large? If not, is it out of reach with current technology?

(I would also be interested in other infinite families of high-dimensional manifolds, like real projective spaces and lens spaces, but I imagine those require strictly more work to understand, given the presence of non-trivial fundamental group without being aspherical like the torus is.)

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    $\begingroup$ There are spectral sequences developed by Schultz, and Becker-Schultz that converge to homotopy groups of the identity component of the space of self-maps of $CP^n$. See section 6 in arxiv.org/abs/0912.4874 and references therein. The computational difficulties are substantial, e.g., in the linked paper we only manage to get partial information on $\pi_7$. $\endgroup$
    – Igor Belegradek
    Commented Dec 29, 2019 at 17:02
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    $\begingroup$ Only small homotopy groups seem relevant in Hatcher's computation, though, so I hope that doesn't cause too much trouble. Thanks for the reference! $\endgroup$
    – mme
    Commented Dec 29, 2019 at 17:09
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    $\begingroup$ Kreck and Su have announced a paper containing the case n=3, see Remark 1.4 of arxiv.org/abs/1907.05693. You could try asking one of them. $\endgroup$
    – skupers
    Commented Dec 29, 2019 at 17:17
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    $\begingroup$ @skupers Kreck points out to me that the oriented smooth mapping class group $\pi_0 \text{Diff}^+(\Bbb{CP}^3)$ was calculated by Brumfiel to be $\Bbb Z/4$, all of whose elements are represnted by diffeomorphisms supported in a ball. I did not check but presumably $\pi_0 \text{Diff}$ is the dihedral group on 8 elements. $\endgroup$
    – mme
    Commented Jan 5, 2020 at 5:33
  • $\begingroup$ Cool, I wasn't aware of that paper! $\endgroup$
    – skupers
    Commented Jan 5, 2020 at 10:52

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