Everybody seems to just use these maps but there is never a formal definition given. I looked at this question: Orientation preserving homeomorphisms but no answer is given and I can't make sense of the comments.

I've never seen a formal definition of orientability either, our professor just drew a circle-arrow on the blackboard. So I'm going with the definition I found on Wikipedia at the moment: A surface has an orientation if there is a triangulation such that I can draw a circle-arrow on each triangle such that on neighboring triangles the arrows on the circle 'point in the same direction', i.e. I could continuously move one into the other. Now going from this rather intuitive intuition, what does it mean for a self-homeomorphism to be orientation-preserving? I would suspect it means something like, if I describe my circle-arrow as embeddings $\gamma:S^1\rightarrow X$ which 'grow in the same direction as the arrow on the circle points' and where X is my space, then if the image $\phi\circ\gamma$ lies in some triangle, it again 'grows in the same direction' as the arrow on the circle in that triangle points. Is this correct?

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    $\begingroup$ If these are smooth surfaces, the best (IMO) definition is in terms of the derivative and its determinant. $\endgroup$ – Randall Aug 29 '17 at 16:59
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    $\begingroup$ Can you give me a reference? $\endgroup$ – azureai Aug 29 '17 at 16:59
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    $\begingroup$ Guillemin and Pollack's "Differential Topology" is pretty readable. Maybe someone can give a surface-oriented source (no pun intended). $\endgroup$ – Randall Aug 29 '17 at 17:00
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    $\begingroup$ What's your definition of surface? $\endgroup$ – user98602 Aug 29 '17 at 17:36
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    $\begingroup$ OK. When you work with smooth maps it's a lot easier because you can talk about derivatives; usually when people work with topological surfaces they define orientations with homology. There is a more down-to-earth approach that I'll write as an answer later today probably. $\endgroup$ – user98602 Aug 29 '17 at 17:44

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