Let $a$ be a regular value of a smooth function on a closed manifold and $\{f=a\}$ a corresponding level submanifold. It is known that any such function can be approximated by a Morse function $g$. Assume that $a$ is also a regular value for function $g$. How different level submanifolds $\{f=a\}$ and $\{g=a\}$ can be?
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$\begingroup$ How do you want to measure the difference of the two submanifolds? Hausdorff distance or something else? $\endgroup$– Joonas IlmavirtaCommented Jan 18, 2015 at 12:08
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$\begingroup$ Different in what sense? They are certainly diffeomorphic as submanifolds (assuming, of course, that $a$ remains a regular value all the way during the perturbation); actually, they are even diffeotopic. As long as you are speaking about regular values, you do not need to bother about other singularities. $\endgroup$– Alex DegtyarevCommented Jan 18, 2015 at 12:09
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$\begingroup$ @JoonasIlmavirta, I am interested in a topological difference, homology or something. especcialy in a low-dimensional cases 1,2 and 3. $\endgroup$– GaussCommented Jan 18, 2015 at 12:12
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$\begingroup$ @AlexDegtyarev, you mean that morsifications of a singular points can be provided in a sufficiently small neighbourhoods and this doesn't effect the regualr submanifolds? $\endgroup$– GaussCommented Jan 18, 2015 at 12:14
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$\begingroup$ Yes, of course. Morse functions constitute a dense subset. Although, of course, it is not true that there is a single Morsification not affecting any of the regular values: some will be affected (a singular point will break into several Morse ones). But you can always keep one favorite regular value. $\endgroup$– Alex DegtyarevCommented Jan 18, 2015 at 12:23
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