# Is the mapping class group of $\Bbb{CP}^n$ known?

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.)

• 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 $\pi_7$. Dec 29 '19 at 17:02
• 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!
– mme
Dec 29 '19 at 17:09
• 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. Dec 29 '19 at 17:17
• @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.
– mme
Jan 5 '20 at 5:33
• Cool, I wasn't aware of that paper! Jan 5 '20 at 10:52