Geodesic
Characteristic classes of vector bundles on a real algebraic variety
Izvestiya. Mathematics , Tome 39 (1992) no. 1, pp. 703-730.

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For a vector bundle $E$ on a real algebraic variety $X$, the author studies the connections between the characteristic classes $$ c_k(E(\mathbf C))\in H^{2k}(X(\mathbf C),\mathbf Z),\quad w_k(E(\mathbf R))\in H^k(X(\mathbf R),\mathbf F_2). $$ It is proved that for $M$-varieties the equality $w_k(E(\mathbf R))=0$ implies the congruence $c_k(E(\mathbf C))\equiv 0 \operatorname{mod}2$. Sufficient conditions are found also for the converse to hold; this requires the construction of new characteristic classes $cw_k(E(\mathbf C))\in H^{2k}(X(\mathbf C);G,\mathbf z(k))$. The results are applied to study the topology of $X(\mathbf R)$. 
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V. A. Krasnov. Characteristic classes of vector bundles on a real algebraic variety. Izvestiya. Mathematics , Tome 39 (1992) no. 1, pp. 703-730. http://geodesic.mathdoc.fr/item/IM2_1992_39_1_a1/

[1] Krasnov V. A., “Neravenstva Garnaka–Toma dlya otobrazhenii veschestvennykh algebraicheskikh mnogoobrazii”, Izv. AN SSSR. Ser. matem., 47:2 (1983), 268–297 | MR | Zbl

[2] Krasnov V. A., “O klassakh gomologii, opredelennykh veschestvennymi tochkami”, Izv. AN SSSR. Ser. matem., 55:2 (1991), 282–302 | MR | Zbl

[3] Grotendik A., O nekotorykh voprosakh gomologicheskoi algebry, IL, M., 1961

[4] Atya M., “$K$-teoriya i veschestvennost”, prilozhenie k knige: M. Atya, Lektsii po $K$-teorii, Mir, M., 1967

[5] Sommese A. J., “Real algebraic spaces”, Ann ali Scu. Norm. Sup. Pisa. Ser. 4, 4:4 (1977), 599–612 | MR | Zbl

[6] Krasnov V. A., “Orientiruemost veschestvennykh algebraicheskikh mnogoobrazii”, Konstruktivnaya algebraicheskaya geometriya, Yaroslavl, 1981

[7] Lojasiewicz S., “Triangulation of semi-analitic sets”, Annali Scu. Norm. Sup. Pisa. Ser. 3, 18:4 (1964), 449–474 | MR | Zbl

[8] Khyuzmoller D., Rassloennye prostranstva, Mir, M., 1970

[9] Arnold V. N., “O raspolozhenii ovalov veschestvennykh ploskikh algebraicheskikh krivykh, involyutsiyakh chetyrekhmernykh gladkikh mnogoobrazii i arifmetike tselochislennykh kvadratichnykh form”, Funkts. analiz i ego prilozh., 5:3 (1971), 1–9 | MR

[10] Rokhlin V. A., “Sravneniya po modulyu 16 v shestnadtsatoi probleme Gilberta”, Funkts. analiz i ego prilozh., 6:4 (1972), 58–64 ; 7:2 (1973), 91–92 | MR | Zbl | Zbl

[11] Rokhlin V. A., “Novye rezultaty v teorii chetyrekhmernykh mnogoobrazii”, DAN SSSR, 84 (1952), 221–224 | Zbl

[12] Borel A., Haefliger A., “La classe d'homologie fundamentale d'une espace analytique”, Bull. Soc. Math. France, 89 (1961), 461–513 | MR | Zbl

[13] Fulton U., Teoriya peresechenii, Mir, M., 1989 | MR