Let's say $f$ and $g$ are two Morse functions on same manifold $M$, and have same domain, same range, same critical points, same critical values at those points and same index at those critical points. Then is there a relation between $f$ and $g$? Are those two related in any way? Any intuition, reference will be helpful.
Difference between 'similar looking' Morse functions
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differential-topology
smooth-manifolds
low-dimensional-topology
morse-theory
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1Their stable/unstable manifolds can be still quite different. In particular, you could get non-isotopic handle decompositions. – 2017-02-11
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1@MoisheCohen And it would be a great answer if you could have added what is a complete invariant for Morse functions :) – 2017-02-11
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0I see what you're pointing at @MoisheCohen, thank you. – 2017-02-11