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On the orientability of spherical, topological, and piecewise-linear fibrations in complex K-theory. (English. Russian original) Zbl 0695.55002
Sov. Math., Dokl. 37, No. 1, 283-286 (1988); translation from Dokl. Akad. Nauk SSSR 298, No. 6, 1338-1341 (1988).
The conditions of the orientability of a complex vector bundle in complex K-theory are well known [M. F. Atiyah, R. Bott and A. Shapiro, Topology 3, Suppl. 1, 3-38 (1964; Zbl 0146.190)]. The author asks if they are sufficient (the necessity is obvious) for the E- orientability of a spherical (in particular, linear, piecewise-linear, or topological) fibration over a cell complex, where E is a connected ring spectrum of finite type. More precisely, assuming that $$\xi$$ is $$E_ 0$$- orientable and taking the respective orientation v and the nth Postnikov invariant $$\theta_ n$$, the author proves that the fibration $$\xi$$ is E- orientable if and only if $$\emptyset =\theta_ n(v)$$ for all n. Another equivalent condition is also obtained. The proofs are sketched or omitted.
Reviewer: J.Kubarski

##### MSC:
 55N15 Topological $$K$$-theory 55R10 Fiber bundles in algebraic topology 55S35 Obstruction theory in algebraic topology