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?