$H_i (CP^2 \setminus RP^2)\cong H^{4-i}(CP^2, RP^2)$ 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. $H_0 H^4 = Z$, $H_4=H_3=0$ H^0=H^1=0$is immediate. The piece $$0 \to H^2 (CP^2,RP^2) \to H^2 (CP^2) \to H^2 (RP^2) \to H^3 (CP^2 , RP^2)$$ RP^2) \to 0$$needs an extra argument. The map Z=H^2 (CP^2 ) \to H^2 (RP^2)=Z/2 is onto because the tautological complex line bundle restricts to the complexification of the real tautological line bundle, whose first Chern class generates H^2 (RP^2). Thus H_2 (CP^2 \setminus RP^2)=Z and H_1 (RP^2)=0.CP^2 \setminus RP^2)=0. Because the first map in the above sequence is multiplication by 2 (after identification with Z), it follows that the inclusion H_2 (CP^2 \setminus RP^2)\to H_2 (CP^2) takes a generator to twice a generator. A generator of H_2 (CP^2-RP^2) can be represented by an embedded sphere as follows. Take a quadric Q \subset CP^2 without real point, for example the one defined by the homogeneous equation z_{0}^{2}+z_{1}^{2}+z_{2}^{2}=0. By the degree genus formula, Q has genus 0, hence is a sphere. It lies in CP^2 - RP^2, and because its fundamental class is twice a generator of H_2 (CP^2), it must represent a generator of H_2 (CP^2 - RP^2). I find it plausible to conjecture that Q is a strong deformation retract of CP^2 - RP^2 or even that In fact, CP^2-RP^2 is diffeomorphic to the normal bundle of Q, but I do not see how to prove itTom's comment below. 1 H_i (CP^2 \setminus RP^2)\cong H^{4-i}(CP^2, RP^2) by Poincare-Alexander-Lefschetz duality (Bredon, Topology and Geometry, Theorem 8.3 on p. 351). The latter can be computed using the long exact sequence. H_0 = Z, H_4=H_3=0 is immediate. The piece$$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)$$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$H_1 (RP^2)=0$. 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$H_2 (CP^2 - RP^2)$. I find it plausible to conjecture that$Q$is a strong deformation retract of$CP^2 - RP^2$or even that$CP^2-RP^2$is diffeomorphic to the normal bundle of$Q\$, but I do not see how to prove it.