Classical Morse theory gives a handle decomposition of a finite dimensional manifold $M$ for every choice of Morse function $f: M \to \mathbb R$. Is there always a Morse function that induces a "minimal" decomposition? In the sense that the number of handles of each index is minimal among all possible choices of handle decompositions of $M$.

For example, the minimal decomposition on the sphere $S^{n}$ would be given by the height function on the standard embedding as the unit sphere into $\mathbb R^{n+1}$.

I would also be interested to know if such a decomposition is necessarily unique, possibly up to some natural notion of equivalence.

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