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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