Cerf theory

In mathematics, at the junction of singularity theory and differential topology, Cerf theory is the study of families of smooth real-valued functions

on a smooth manifold ${\displaystyle M}$, their generic singularities and the topology of the subspaces these singularities define, as subspaces of the function space.

Marston Morse proved that, provided ${\displaystyle M}$ is compact, any smooth function

could be approximated by a Morse function. So for many purposes, one can replace arbitrary functions on ${\displaystyle M}$ by Morse functions.

As a next step, one could ask, 'if you have a 1-parameter family of functions which start and end at Morse functions, can you assume the whole family is Morse?' In general the answer is no. Consider, for example, the family:

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