Abstract
An obstruction theory is constructed for orientability of vector, piecewise-linear, and topological Rn-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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