The Thom isomorphism for nonorientable bundles

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 $\tau$ 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 $\tau$ and $e$ is confirmed. It is shown that the Thom isomorphisms, together with their form, are consequences of the Poincaré duality.