Changes between Initial Version and Version 1 of Ticket #31241, comment 26


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Timestamp:
06/02/21 09:17:57 (6 months ago)
Author:
tscrim
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  • Ticket #31241, comment 26

    initial v1  
    55> Mathematically this is wrong. As soon as you apply the forgetful functor from differentiable manifolds to topological manifolds, the maximal atlas changes accordingly. In particular, the maximal atlas now consists of all charts that are continuously compatible w.r.t. to all charts in the differentiable structure.
    66
    7 You're wrong. Otherwise the category of differentiable manifolds is not a subcategory of topological manifolds. Alternatively, you have a covering of your manifold by charts with maps to open subsets of R^n^ that are diffeomorphic. The fact this is a covering forces any other chart to also be a diffeomorphism by the transition functions. Your maximal atlas cannot change under the forgetful functor.
     7You're wrong. Otherwise the category of differentiable manifolds is not a subcategory of topological manifolds. So there would be no forgetful functor. Alternatively, you have a covering of your manifold by charts with maps to open subsets of R^n^ that are diffeomorphic. The fact this is a covering forces any other chart to also be a diffeomorphism by the transition functions. Your maximal atlas cannot change under the forgetful functor.
    88
    99> That is exactly the punchline: adding a non-compatible chart (in the differentiable sense) which is still countinously compatible would corrupt the differentiable structure, but keep the maximal atlas of the topological manifold intact.