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On homology classes determined by real points of a real algebraic variety. (English. Russian original) Zbl 0751.14036
Math. USSR, Izv. 38, No. 2, 277-297 (1992); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 55, No. 2, 282-302 (1991).
Connected components of a real $$n$$-dimensional non-singular algebraic variety $$A$$ define homology classes in $$H_ n(A(\mathbb{C}),\mathbb{F}_ 2)$$. The aim of this paper is to improve the author’s previous upper bounds [see Math. USSR, Izv. 22, 247-275 (1984); translation from Izv. Akad. SSSR, Ser. Mat. 47, No. 2, 268-297 (1983; Zbl 0537.14035)] on the number $$k$$ of relations between these classes. Here the author uses analogous computations with Galois-Grothendieck cohomology of $$A(\mathbb{C})$$ over $$\mathbb{F}_ 2$$ with respect to complex conjugation action. Among numerous upper bounds, generalizing the known ones $$(k\leq 1$$ for curves, $$k\leq 1+q$$ for surfaces, where $$q$$ is the irregularity) it should be underlined the estimate $$k\leq 1$$ for complete intersections.

##### MSC:
 14P25 Topology of real algebraic varieties 14F20 Étale and other Grothendieck topologies and (co)homologies
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