Fundamental group of the space $\Bbb CP^n-p(X)$

I don't know how to do this with algebra, but the space is diffeomorphic to the tangent bundle on $\mathbb RP^n$ (see my question and the answer here), and hence homotopic to $\mathbb RP^n$, so its fundamental group is $\mathbb Z/2$.

For a different approach: I think the suggestion in the comments is also good, since for a manifold of dimension at least three, you can remove a point from your space, and pass to affine coordinates, solving the questiion there (since the fundamental group will be unchanged, which is proven by seifert van kampen.)

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Algebraic Topology

Fundamental Groups

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