The Morse functions are dense in $C^\infty(M)$, and you can ask if a 1-parameter family of smooth functions between two given Morse functions will be a homotopy through Morse functions. Well, Cerf Theory shows that there is a codimension-one stratum $\mathcal{F}^1$ in $C^\infty(M)$, and so any path between two Morse functions will transversely intersect $\mathcal{F}^1$ a finite number of times (in a few specific ways).

**Can we somehow 'measure' how far away a given function is from $\mathcal{F}^1$?**

(Cerf's $\mathcal{F}^1$ consists of two components: the set of smooth functions that have exactly one birth point and distinct critical values, and the set of Morse functions where exactly two distinct critical points have the same value. A birth/death point of a smooth function on an $n$-manifold is a point with local neighborhood $-x_1^2-\cdots-x^2_i+x_{i+1}^2+\cdots+x^2_n+x^3_{n+1}$. The codimension-zero stratum $\mathcal{F}^0$ is essentially the space of Morse functions, but only the ones with distinct critical values. This separation is slightly weird to me.)

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