Short answer. Yes: for n ≥ 3 the mapping class group of CP^n (in each of the topological, PL and smooth categories) is Z/2, generated by complex conjugation. Equivalently:

• π0(Homeo(CP^n)) ≅ π0(PL(CP^n)) ≅ π0(Diff(CP^n)) ≅ Z/2.
• The nontrivial class is represented by complex conjugation [z0:…:zn] ↦ [z̄0:…:z̄n].
• Orientation: conjugation reverses orientation when n is odd and preserves it when n is even (since it multiplies the generator x ∈ H^2 by −1, hence x^n by (−1)^n). Thus:
  • π0(Diff^+(CP^{2m+1})) is trivial.
  • π0(Diff^+(CP^{2m})) ≅ Z/2, still generated by conjugation.
  • There is no orientation-reversing diffeomorphism of CP^{2m} (m ≥ 1), since the signature is 1 and would have to change sign under an orientation-reversing diffeomorphism.

What lies behind this. Two issues have to be addressed:

1. Realization of homotopy classes. The homotopy classes of self-homotopy equivalences of CP^n form Z/2, detected by the action on H^2(CP^n; Z) ≅ Z (sending the positive generator x to ±x). In dimensions ≥ 5 (i.e. n ≥ 3), surgery theory shows that every self-homotopy equivalence of CP^n is homotopic to a diffeomorphism/homeomorphism/PL-homeomorphism; in particular, the two homotopy classes are realized by the identity and complex conjugation.

2. Possible extra mapping classes in the kernel (homotopically trivial but not isotopic). For simply connected manifolds there can be a “concordance kernel,” produced by twisting inside a ball by elements of Θ_{2n+1} (the group of homotopy (2n+1)-spheres). For CP^n with n ≥ 3, that kernel is trivial: the (concordance) inertia group I_c(CP^n) vanishes. Consequently, a diffeomorphism/homeomorphism/PL-homeomorphism of CP^n that is homotopic to the identity is in fact isotopic to the identity.

Putting these together gives π0(Diff(CP^n)) ≅ Z/2 (and the same in PL and TOP).

References and pointers.

• Complex conjugation and the induced action on cohomology are classical; the computation π0 hAut(CP^n) ≅ Z/2 and π1 hAut(CP^n) ≅ Z/(n+1) goes back to Sasao and others.

• The vanishing of the concordance/inertia kernel for CP^n (and realization of the two homotopy classes by diffeomorphisms) is proved using surgery theory; see for example:

  • R. E. Schultz, On the inertia groups of complex projective spaces (and related papers on diffeomorphisms of CP^n). These works show I_c(CP^n) = 0 for n ≥ 3 and that any self-homotopy equivalence of CP^n is homotopic to a diffeomorphism.
  • C. T. C. Wall, Surgery on Compact Manifolds, for the general framework (structure sets, normal invariants, and realization of homotopy equivalences by diffeomorphisms in the simply connected, high-dimensional case).
  • I. Madsen and R. J. Milgram, The Classifying Spaces for Surgery and Cobordism of Manifolds, which gives explicit calculations of structure and normal invariant groups in examples including projective spaces and records the vanishing of the concordance inertia group in this case.

Remarks on low dimensions and orientation.

• n = 1: CP^1 = S^2, π0(Diff(S^2)) ≅ Z/2 by Smale; consistent with the above.
• n = 2 (real dimension 4) is outside the high-dimensional range for surgery, but the same conclusion for the mapping class group (Z/2 generated by conjugation) still holds; there is no orientation-reversing diffeomorphism because the signature is 1.
• For n even (real dimension 4m), the signature argument rules out any orientation-reversing diffeomorphism/homeomorphism; for n odd, complex conjugation provides an orientation-reversing diffeomorphism.

Comments on other families.

• For real projective spaces and lens spaces the situation is subtler because π1 is nontrivial and the mapping class group typically maps onto Out(π1); one also has to account for concordance phenomena. There is an extensive literature (e.g. results of Hatcher–McCullough for 3-manifolds, and surgery-theoretic computations for high-dimensional lens spaces) but they require more case-by-case analysis and do not admit a uniform “Z/2” answer as for CP^n.