Geodesic
On~homology classes determined by real points of a real algebraic variety
Izvestiya. Mathematics , Tome 38 (1992) no. 2, pp. 277-297

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For a nonsingular $n$-dimensional real projective algebraic variety $X$ the set $X(\mathbf R)$ of its real points is the union of connected components $X(\mathbf R)=X_1\cup\dots\cup X_m$. Those components give rise to homology classes $[X_1],\dots,[X_m]\in H_n(X(\mathbf C),\mathbf F_2)$. In this paper a bound on the number of relations between those homology classes is obtained. 
@article{IM2_1992_38_2_a2,
     author = {V. A. Krasnov},
     title = {On~homology classes determined by real points of a real algebraic variety},
     journal = {Izvestiya. Mathematics },
     pages = {277--297},
     publisher = {mathdoc},
     volume = {38},
     number = {2},
     year = {1992},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1992_38_2_a2/}
}
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V. A. Krasnov. On~homology classes determined by real points of a real algebraic variety. Izvestiya. Mathematics , Tome 38 (1992) no. 2, pp. 277-297. http://geodesic.mathdoc.fr/item/IM2_1992_38_2_a2/