Geodesic
Orientability of bundles: obstruction theory and applications to $K$-theory
Sbornik. Mathematics, Tome 68 (1991) no. 2, pp. 429-451

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An obstruction theory is constructed for orientability of vector, piecewise-linear, and topological $\mathbf R^n$-bundles and homotopy sphere bundles (spherical fibrations) in generalized cohomology theories. The results are applied to study the orientability of bundles in complex $K$-theory. In particular, it turns out that the problem of $K$-orientability for each of the four classes of bundles mentioned above is to be solved differently. Bibliography: 20 titles 
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     author = {Yu. B. Rudyak},
     title = {Orientability of bundles: obstruction theory and applications to $K$-theory},
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Yu. B. Rudyak. Orientability of bundles: obstruction theory and applications to $K$-theory. Sbornik. Mathematics, Tome 68 (1991) no. 2, pp. 429-451. http://geodesic.mathdoc.fr/item/SM_1991_68_2_a6/