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Yes, you can always choose an orientation of the path but not necessarily of the manifold along the path. I understand what he was saying now and misunderstood since it is not the usual way one talks about specifying an orientation at a point and in fact had imagined he was specifically specifying a "big" circle, like a nontrivial loop on the torus. You'd more typically talk about choosing a basis of the tangent space at a point - presumably ColinWright wanted to avoid involving extra definitions like tangent space and so chose a visual definition that wasn't quite as precise. I don't think there's any further confusion.

Cool, I think we're on the same page. I know what you mean by "orientation of the manifold along the path" but this isn't precisely the standard usage. (My vague memory is that you can define an orientation without needing a tangent space, but I can't think of an example, so could be wrong here.)

My background is in riemannian geometry so I never had to worry about lack of a tangent space. Certainly you can define orientability of some topological spaces that aren't even manifolds. I'd forgotten but you can even define a local orientation at a point for a general topological manifold in terms of it's top-dimensional homology. I think that's the most general situation it makes sense in, you need a well defined dimension to consider this.

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