topology of $\mathbb{P}_2 (\mathbb{C}) \setminus \mathbb{P}_2 (\mathbb{R})$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T22:19:25Z http://mathoverflow.net/feeds/question/93894 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93894/topology-of-mathbbp-2-mathbbc-setminus-mathbbp-2-mathbbr topology of $\mathbb{P}_2 (\mathbb{C}) \setminus \mathbb{P}_2 (\mathbb{R})$? unknown (yahoo) 2012-04-12T18:55:08Z 2012-06-07T12:33:04Z <p>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 cant find a good reference for it. Can anyone give me a reference? Thanks</p> http://mathoverflow.net/questions/93894/topology-of-mathbbp-2-mathbbc-setminus-mathbbp-2-mathbbr/93900#93900 Answer by Johannes Ebert for topology of $\mathbb{P}_2 (\mathbb{C}) \setminus \mathbb{P}_2 (\mathbb{R})$? Johannes Ebert 2012-04-12T19:59:43Z 2012-06-07T12:33:04Z <p>$H_i (CP^2 \setminus RP^2)\cong H^{4-i}(CP^2, RP^2)$ by Poincare-Alexander-Lefschetz duality (Bredon, Topology and Geometry, Theorem 8.3 on p. 351). The latter can be computed using the long exact sequence. $H^4 = Z$, $H^0=H^1=0$ is immediate. The piece </p> <p>$$0 \to H^2 (CP^2,RP^2) \to H^2 (CP^2) \to H^2 (RP^2) \to H^3 (CP^2 , RP^2) \to 0$$ </p> <p>needs an extra argument. The map $Z=H^2 (CP^2 ) \to H^2 (RP^2)=Z/2$ is onto because the tautological complex line bundle restricts to the complexification of the real tautological line bundle, whose first Chern class generates $H^2 (RP^2)$. Thus $H_2 (CP^2 \setminus RP^2)=Z$ and $H_1 (CP^2 \setminus RP^2)=0$.</p> <p>Because the first map in the above sequence is multiplication by $2$ (after identification with $Z$), it follows that the inclusion $H_2 (CP^2 \setminus RP^2)\to H_2 (CP^2)$ takes a generator to twice a generator. A generator of $H_2 (CP^2-RP^2)$ can be represented by an embedded sphere as follows. Take a quadric $Q \subset CP^2$ without real point, for example the one defined by the homogeneous equation $z_{0}^{2}+z_{1}^{2}+z_{2}^{2}=0$. By the degree genus formula, $Q$ has genus $0$, hence is a sphere. It lies in $CP^2 - RP^2$, and because its fundamental class is twice a generator of $H_2 (CP^2)$, it must represent a generator of $H_2 (CP^2 - RP^2)$. </p> <p>In fact, $CP^2-RP^2$ is diffeomorphic to the normal bundle of $Q$, see Tom's comment below.</p> http://mathoverflow.net/questions/93894/topology-of-mathbbp-2-mathbbc-setminus-mathbbp-2-mathbbr/93966#93966 Answer by Igor Rivin for topology of $\mathbb{P}_2 (\mathbb{C}) \setminus \mathbb{P}_2 (\mathbb{R})$? Igor Rivin 2012-04-13T15:29:08Z 2012-04-13T15:29:08Z <p>There is <a href="http://dl.dropbox.com/u/5188175/arnoldquadric.pdf" rel="nofollow">a beautiful (and elementary) paper by V. I. Arnold,</a> which discusses this and generalizations.</p>