Geodesic
Algebraic cycles on a real algebraic GM-manifold and their applications
Izvestiya. Mathematics , Tome 43 (1994) no. 1, pp. 141-160

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For an algebraic cycle $Y\in A_k(X)$ on a real algebraic $\operatorname{GM}$-manifold $X$, the relationship between the homology classes $[Y(\mathbf C)]\in H_{2k}(X(\mathbf C),\mathbf Z)$ and $[Y(\mathbf R)]\in H_k(X(\mathbf R),\mathbf F_2)$ is studied. It is shown that similar relations hold for smooth cycles on a $\operatorname{GM}$-surface. The results are applied to prove congruences for the Euler characteristic of the set $X(\mathbf R)$. 
@article{IM2_1994_43_1_a8,
     author = {V. A. Krasnov},
     title = {Algebraic cycles on a real algebraic {GM-manifold} and their applications},
     journal = {Izvestiya. Mathematics },
     pages = {141--160},
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     volume = {43},
     number = {1},
     year = {1994},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1994_43_1_a8/}
}
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V. A. Krasnov. Algebraic cycles on a real algebraic GM-manifold and their applications. Izvestiya. Mathematics , Tome 43 (1994) no. 1, pp. 141-160. http://geodesic.mathdoc.fr/item/IM2_1994_43_1_a8/