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The conjugation map $$\mathrm{conj}\colon\mathbb{C}P^{n-1}\to \mathbb{C}P^{n-1}$$ extends to an endomorphism of $\mathbb CP^n\smallsetminus \mathring{D}^{2n}$ (unique up to homotopy), since $\mathbb CP^{n-1}$ is a deformation retract of $\mathbb CP^n\smallsetminus \mathring{D}^{2n}$.

I'm wondering whether an extension of the conjugate map is homotopic to an endomorphism that preserves the boundary pointwise, when $n\geq 4$ is even.

When $n$ is odd, it is impossible to make a boundary-preserving extension since it is impossible at the level of the rationalizations of these spaces (which can be proven algebraically, using algebraic models). However, when $n\geq 4$ is even, boundary-preserving extensions are possible rationally, and I am wondering whether this can be realized at the level of the non-rationalized spaces.

I'd be thankful for any suggestion of literature that may be relevant for this type of questions.

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    $\begingroup$ Sure, the map induced on the boundary sphere is the restriction of complex conjugation in $T_p \Bbb{CP}^n = \Bbb C^n$ to the unit sphere. This is orientation preserving precisely when $n$ is even, and orientation preserving diffeomorphisms of the sphere are isotopic. So start with your given map and add a small collar $[0,1] \times S^{2n-1}$ where you perform this isotopy. $\endgroup$
    – mme
    Commented Jan 29, 2020 at 17:17
  • $\begingroup$ @Mike Miller, thank you :-) $\endgroup$
    – Bashar Saleh
    Commented Jan 29, 2020 at 18:08

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