Abstract
The classical theory of Thom isomorphisms is extended to nonorientable vector bundles. The properties of orientation sheaves of bundles and of the Thom and Euler classes τ and e with respect to projections, fiber maps, Cartesian products, and Whitney sums of bundles are studied. The validity of standard constructions used in the applications of the classes τ and e is confirmed. It is shown that the Thom isomorphisms, together with their form, are consequences of the Poincaré duality.
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 4, pp. 55–103, 2003.
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Sklyarenko, E.G. The Thom isomorphism for nonorientable bundles. J Math Sci 136, 4166–4200 (2006). https://doi.org/10.1007/s10958-006-0226-3
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DOI: https://doi.org/10.1007/s10958-006-0226-3