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 $X(\mathbf R)$. 
@article{IM2_1992_39_1_a1,
     author = {V. A. Krasnov},
     title = {Characteristic classes of vector bundles on a real algebraic variety},
     journal = {Izvestiya. Mathematics },
     pages = {703--730},
     publisher = {mathdoc},
     volume = {39},
     number = {1},
     year = {1992},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1992_39_1_a1/}
}
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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/