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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!

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It sounds like the examples have to occur for simple-homotopy reasons, so my first guess would be 3-dimensional lens spaces would be fertile hunting grounds for examples you are interested in. –  Ryan Budney Feb 28 '12 at 0:17

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I no longer think 3-dimensional lens spaces are a productive strategy. What you need is to have a manifold $M$ as a level-set of the Morse function and you need a non-trivial diffeomorphism of $M$ to be pseudo-isotopic to the identity.

The idea is that roughly, between any two consecutive critical levels of your Morse function (modulo the degeneracies on the ends) your flow is giving you a pseudo-isotopy diffeomorphism of $M$ -- by that I mean a diffeomorphism of $M \times [0,1]$ which is the identity on $M \times \{0\}$.

So provided $M$ admits non-trivial pseudo-isotopy diffeomorphisms you've got a head-start. Cerf's theorem says pseudo-isotopy implies isotopy for simply connected manifolds. So I suppose that's a tool Matsumoto uses. In particular, 2-manifolds don't admit any non-trivial pseudo-isotopy diffeomorphisms so that's likely not a productive route to go.

I think there's sort of a tautological answer to your question. Let $M$ be compact manifold. The manifold I'm going to give you is $M \times [0,1]$. And I'm going to give you two Morse functions on it with no critical points. The first one is $f(p,t) = t$, where $(p,t) \in M \times [0,1]$. Let $\phi : M \times [0,1] \to M \times [0,1]$ be a non-trivial pseudo-isotopy diffeomorphism, then the 2nd function is $f \circ \phi$. This is sort of a cheezy non-constructive "example".

By design, these are homotopy-equivalent but non-isotopic Morse functions. The critical point set of both is empty. That such functions $\phi$ exist when $M$ is not simply-connected, I'm not sure who first noticed this. But I think there are examples in the Hatcher-Wagoner reference you cite, when $M = (S^1)^n$ for $n \geq 7$. Hatcher's The Second Obstruction for pseudo-isotopies (1972) mentions that $\pi_0$ of the pseudo-isotopy group for $M$ is non-trivial whenever $M$ is not simply connected and the dimension is at least seven.

I think if you want an example on a closed manifold you'll have to work a fair bit more but I suspect capping off $(S^1)^n \times [0,1]$ with copies of $D^2 \times (S^1)^{n-1}$ on both sides should work with the Hatcher-Wagoner classes.

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Thanks, this is a great answer. –  Jonny Evans Feb 28 '12 at 7:04

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