TITLE: topology of $\mathbb{P}_2 (\mathbb{C}) \setminus \mathbb{P}_2 (\mathbb{R})$?
QUESTION [8 upvotes]: What is the topology of $\mathbb{P}_2 (\mathbb{C}) \setminus \mathbb{P}_2 (\mathbb{R})$? For example what is the homology of this manifold with coefficients in $\mathbb{Z}$. I know that this is known but I can't find a good reference for it. Can anyone give me a reference?

REPLY [4 votes]: There is a beautiful (and elementary) paper by V. I. Arnold, which discusses this and generalizations.