187:. Despite this, between these three articles, there is still no coherent definition or discussion of an atlas of local, smooth frames on a generic vector bundle. The smooth frame article only deals with smooth frames on the tangent bundle, and never mentions atlases. The article on Atlases is OK, but hand-waves around in a section called "more structure". This article seems to have been written to highlight those things at aren't actually smooth (E8, etc.) We need somehing that covers all the bases, not just some of them...
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reflected in the linked page "atlas", and is not a function that can be composed in this way. I'm not sure what the author intended here: did they mean this should hold for each *chart* in the atlas? Even then, there are some issues with that, so at least a reference should be added for this definition.
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There seems to be no elaboration on equivalence of smooth structures in
Knowledge. As it stands, the section here is not very illuminating, and the notational liberties taken seem unmotivated. Is anyone with expertise with the subject (including required references in the back pocket) ready to expand
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Composition or pullback of charts (both sets and the mappings) is well defined and I agree that those definitions should be mentioned here. What I find worse is the word "diffeomorphism" in the same definition - it is not clear with respect to what structures is it meant. If w.r.t. \mu and \nu, then
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There's rather large technical problem in this article under "Equivalence of Smooth
Structures". It currently defines maximal atlases \mu and \nu. It then says a function f: M \to M is an equivalence of smooth structures iff you compose \mu with f and get \nu. An atlas is a set of ordered pairs (as
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being diffeo. is exactly the condition about compositions. So I would either write only "homeomorphism M to M" or I would specify that the diffeo is meant to \mu and \nu and I would delete the other condition.
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this? As a suggestion, it seems like there should be a neat category theoretical formulation (adjoint functors, perhaps) of this definition.
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Ive changed this article a lot, any comments/corrections welcome, since this is my fist major edit.
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on
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