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For an $n$-dimensional manifold $M$ and a ring $R$ one can define $R$-orientability, e.g. by choosing local orientations, i.e. generators of $H_n(M, M - \{x\}; R)$ for every point $x$ in the manifold, consistently. For the ring $R = \mathbb Z$ one has the familiar geometric picture of ordinary orientations. For the field $\mathbb{F}_2$ one always obtains orientability.

Can one get a geometric picture for other rings $R$, for example finite fields? Which other choices of $R$ are common that have a geometric interpretation?

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Hatcher has a good description on page 235 –  Tim kinsella Jun 11 '13 at 4:51

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