One can cut a manifold up along the critical levels of a Morse function and deduce something about the topology. In particular the critical points (and the connecting gradient flowlines) define a chain complex which can be used to compute homology.
A minimal Morse function on a compact manifold is a Morse function which has exactly enough critical points to generate the homology (e.g. perfect if the homology is torsion free). The existence of a minimal Morse function on a simply-connected h-cobordism between simply-connected manifolds of high enough dimension is equivalent to the h-cobordism theorem.
Two Morse functions are called 'homotopy equivalent' if there is a diffeomorphism isotopic to the identity sending critical levels to critical levels. I don't know if this is the best terminology, but it seems to be the one people use.
Matsumoto proved that on a simply-connected manifold any two minimal Morse functions which are homotopy equivalent are isotopic through Morse functions. This seems to be a beefed up version of the argument which proves the h-cobordism theorem.
Of course, in the non-simply-connected case one expects something different. On page 45 of the English translation of Sharko's book 'Functions on manifolds: algebraic and topological aspects' (Translations of Mathematical Monographs, AMS 1993), just after Corollary 2.3 he claims that there are examples of homotopy equivalent but nonisotopic Morse functions on non-simply-connected manifolds.
My question is: can anyone give me an example of a pair of homotopy equivalent but nonisotopic Morse functions on a non-simply-connected manifold? What about minimal ones?
The reference Sharko gives there seems to have nothing to do with this (it points to a paper of Heller, 'Homological resolutions of complexes with operators', Ann. Math. 1954) so I assume the bibliographic numbering is flawed. If I check the adjacent and promising-looking reference (Hatcher and Wagoner's 'Pseudoisotopies of compact manifolds', Asterisque Volume 6, 1973) I find that I would need to know more Cerf theory to work out what this example might look like.
Any ideas would be very welcome!