The top Stiefel-Whitney class of an orientable manifold is the Euler characteristic mod 2, since it is the mod 2 reduction of the Euler class. Does this still hold for an unorientable manifold?
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Every manifold is $Z_2$-orientable, so the Euler class (with $Z_2$-coefficients) is defined and coincides with the top Stiefel-Whitney class. Back to your question: Yes, the top Stiefel-Whitney class evaluated on the fundamental class of the manifold is equal to the Euler characteristic mod 2 - regardless of the $Z$-orientability of the manifold. This is Corollary 11.12 in Milnor, Stasheff: Characteristic Classes. |
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