Due to Cerf, there's exist a certain homotopy between two Morse functions. It is said that this homotopy is "generic". What is a precise definition of the property to be "generic" in this case?



What does it mean that homotopy is generic?
Intuitively, for the homotopy to be generic has the same meaning as for the Morse function itself to be generic: the set of points where it fails to be a submersion is as simple as possible. Practically, this means that for all but a finite number of times (values of the $t$ parameter) the function $H(x,t)$ is a Morse function of the variable $x$ and the Morse singularities trace out smooth curves transverse to the $x$ direction; and for each of those exceptional times $t$ the function $H(x,t)$ is Morse at all values of $x$ except for a single value $x$ (I'm assuming compactness here) where it undergoes one of a certain class of very special singularities obtained by collapsing two Morse singularities, called "birth-death" singularities. Just as with Morse singularities themselves, which are described locally by specific functions of $x$ in some coordinate system, birth-death singularities are described locally by specific functions of $x,t$ in some coordinate system. 


The function $H(x,t) = x^3 - tx$ at $t=0$ is an example of a birth-death singularity in one dimension: for $t>0$ there are no critical points; and for $t<0$ there is one maximum and one minimum.


