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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