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Two Morse functions $f$ and $g$ are called topologicaly equivalent if there are diffeomorphism $h$ of the source and orientational-preserving diffeomorphism $k$ of the target such that $f=k\circ g\circ h$.

In the case of 1 and 2-dimensional compact closed orientable sources the complete invariants are known. For a given number of critical points there is a finite number of equivalence classes and calculations were made in the case of spheres (Arnold-Nicolaescu).

What machinery from algebraic and differential topology can be used to study this problem for a general source-manifold?

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  • $\begingroup$ As a wild guess, this equivalence is not the same as the equivalence (in the obvious sense) of corresponding handle decompositions? $\endgroup$
    – Alex Degtyarev
    Feb 5, 2015 at 21:24
  • $\begingroup$ @AlexDegtyarev If by "obvious" equivalence you mean attachment of the correspondent handles by isotopic mappings, I think this is not true... It may be sufficient but not necessary. $\endgroup$
    – Gauss
    Feb 5, 2015 at 21:55
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    $\begingroup$ I think it is true, but I wouldn't bet my house on that. Anyway, it is definitely necessary, which makes your problem very difficult. Say, in dimension $3$, only the self-indexing Morse functions are pretty much the same as Heegard splittings, and it's not easy to describe all Heegard splittings of a given manifold. $\endgroup$
    – Alex Degtyarev
    Feb 5, 2015 at 22:31

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