Geodesic
Euler classes of stably equivalent vector bundles
Izvestiya. Mathematics , Tome 8 (1974) no. 1, pp. 113-131

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The possible values of the Euler class of a $k$-dimensional bundle stably equivalent to a fixed $k$-dimensional real vector bundle are studied. Complete information is obtained in the case where the dimension of the bundle is more than half of the dimension of the base. In the case where $k$ is an arbitrary even number it is shown that if the original bundle possesses a vector field, then for every element of the corresponding cohomology group whose square is equal to zero there exists an element divisible by it realized as the Euler class of a bundle stably equivalent to the original. 
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     author = {B. D. Malyi},
     title = {Euler classes of stably equivalent vector bundles},
     journal = {Izvestiya. Mathematics },
     pages = {113--131},
     publisher = {mathdoc},
     volume = {8},
     number = {1},
     year = {1974},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1974_8_1_a6/}
}
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B. D. Malyi. Euler classes of stably equivalent vector bundles. Izvestiya. Mathematics , Tome 8 (1974) no. 1, pp. 113-131. http://geodesic.mathdoc.fr/item/IM2_1974_8_1_a6/