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Mar2-11, 06:14 PM
Sci Advisor
P: 5,366
Quote Quote by PAllen View Post
This was very interesting, but I didn't find any answer to my question my key question: how is it winding number of closed curve computed / defined against the definition of a differentiable manifold?
Yes, not a single word.

Quote Quote by PAllen View Post
If it can change while the manifold is considered identical, then it must be computed in a way that is not invariant.
I don't agree. If the Dehn twist is a global diffeomorphism (and if we agree that in 2 dimensions homeomorphic manifolds are diffeomorphic and vice versa - which does not hold in higher dimensions) then the two manifolds before and after the twist are identical - there is no way to distinguish them. Now suppose we cannot compute the winding numbers (m,n) but only their change under twists. Then this change is not a property of the manifold but of the diffeomorphism. So we don't need a way to compute the winding numbers from the manifold but a way to compute their change from the diffeomorphism (this is similar to large gauge transformations where the structure is encoded in the gauge group, not in the base manifold). I think for large diffeomorphisms there is some similar concept.