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Let $(M^3,g)$ be a compact Riemannian 3-manifold and let $f\in C^{\infty}(M)$ be a smooth function. Does there exist a constant $k>0$ (possibly depending on $M$ and $g$) such that $f$ can be $C^2$-approximated by Morse functions whose number of critical points is at most $k$?

In other words, if we define $\mathrm{Morse}_k(M,g)\subset C^{\infty}(M)$ to be the subset of Morse functions which have at most $k$ critical points, then can $k$ be chosen large enough so that $\mathrm{Morse}_k(M,g)$ is dense in $C^2(M)$?

If such a $k$ exists, then the Morse inequalities give a necessary lower bound for $k$ in terms of the total Betti number, namely $$k \geq b_0(M)+b_1(M)+b_2(M)+b_3(M).$$ I am, however, asking about the existence of an upper bound for $k$.

Any references would be appreciated.

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  • $\begingroup$ As far as I know, even a seemingly easier question is open: given a smooth function f defined on an open set U of the Euclidean space (dimension 2 is hard enough) with a unique critical point x, is there a sequence of Morse functions on U converging to f in the C^1 topology and having a uniform upper bound on the number of their critical points? This was asked in a 1969 paper of Gromoll and Meyer ("On differentiable functions with isolated critical points"), and I am not aware of any progress since. $\endgroup$ Aug 27, 2021 at 12:41

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Any $C^2$ function close enough to $f$ in the $C^2$ distance has the same number of critical points of each index as $f$, just by the IFT. And any $C^1$ function close enough in the $C^0$ distance, has at least as many critical points as $f.$

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