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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.

14P25 Topology of real algebraic varieties
14F20 √Čtale and other Grothendieck topologies and (co)homologies
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