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I have proved that given an atlas for a topological space $M$ that a maximal atlas containing $M$ is unique. But my proof would fail to generalise to the statement that a maximal atlas conatining a chart is unique. Is this true?

Also given a map $f : M \mapsto N$. Then I have seen it written that the smoothness of $f$ is independant of the choice of chart. (Definition give here http://www.cis.upenn.edu/~cis610/cis61005sl7.pdf on page326, definition 6.15)

I can't see how this is independent from the way we 'chart' the topological spaces $M$ and $N$.

I would appreciate if someone could clear this up for me! Thanks!

joe ibbs
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You are correct that there is not a unique maximal atlas containing a given chart. It is possible to take a system of charts (atlas) on a manifold and adjust some of them to get an incompatible system of charts (i.e. they extend to different maximal atlases), but both systems will have many charts in common. If you'd like, I can go into more detail about how to do this.

Regarding your second question, the smoothness of $f$ is independent of the choice of chart in a fixed atlas. In the definition, they pick a pair of charts $(U,\phi)$ and $(V,\psi)$ around $p$ and $q$, and test for $C^{k}$ differentiability there. However, if it works for one pair of charts, it works for any pair, so you can be sure that if you take $(U',\phi')$ and $(V',\psi')$ around $p$ and $q$ that you will get a $C^k$ map as well. You can prove this by using the chain rule and noting that the transition functions between charts are $C^k$ maps.

Elchanan Solomon
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  • Thank you for clearing that up Isaac! Very helpful indeed! Your initial fact is very intriguing. If you find you have some spare time, then I would welcome you to explain this! But feel no obligation. Once again, thanks! – joe ibbs Oct 28 '13 at 13:12
  • The thing I am having issue with now, is that transition maps are only defined on the intersection of two charts. So I can only show that $\psi'\circ f \circ \phi'^{-1}$ is smooth, on $\phi'(U'\cap U)$. – joe ibbs Oct 28 '13 at 14:49