Orientability of bundles: obstruction theory and applications to $K$-theory

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.
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