On homology classes determined by real points of a real algebraic variety

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.