# Morse approximation with bounded number of critical points

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.

• 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. Aug 27, 2021 at 12:41

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 $$f.$$