A Compact Manifold with odd Euler characteristic whose tangent bundle admits a field of lines – mathoverflow.net

A Compact Manifold with odd Euler characteristic whose tangent bundle admits a field of lines – mathoverflow.net

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I understand that the top Stiefel Whitney class is an obstruction for the tangent bundle of a manifold to have a trivial line sub-bundle. I am looking for a counterexample when removing the word "...

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