Geodesic
The Thom isomorphism for nonorientable bundles
Fundamentalʹnaâ i prikladnaâ matematika, Tome 9 (2003) no. 4, pp. 55-103
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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. 
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E. G. Sklyarenko. The Thom isomorphism for nonorientable bundles. Fundamentalʹnaâ i prikladnaâ matematika, Tome 9 (2003) no. 4, pp. 55-103. http://geodesic.mathdoc.fr/item/FPM_2003_9_4_a2/

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