I am confused about the different notions for a closed smooth manifold to be oriented. In my mind there are several equivalent ones:

1) Coherent pointwise orientation of the tangent spaces.

2) Choice of nonvanishing top dimensional form.

3) Orientation of the top dimensional de Rham cohomology group as a real vector space.

I understand the equivalence of these above notions. There are also some similar definitions for an orientation of a topological manifold:

1) Coherent choice of generators for the pointwise homology groups $H_n(M, M \setminus \{x\}; \mathbb{Z})$.

2) Choice of generator ("fundamental class") for the top homology group $H_n(M; \mathbb{Z})$.

My question is the following: how does a choice of smooth orientation induce a choice of topological orientation?


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    $\begingroup$ Potentially helpful: map.mpim-bonn.mpg.de/Orientation_of_manifolds $\endgroup$ – Michael Albanese Apr 5 '16 at 18:40
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    $\begingroup$ You should ask yourself when you consider two orientations to be the same. It's not true that a top form induces a fundamental class in integral (co)homology, but it is true up to scale in an appropriate sense. $\endgroup$ – Qiaochu Yuan Apr 5 '16 at 19:15

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