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  "body": "\n## Summary\n\nCerf theory studies one-parameter families of smooth real-valued functions on a manifold and the ways in which they fail to be [[Morse functions|KPQ96640ZJ]].\nIn a generic family, the function is a Morse function with distinct critical values at all but finitely many times.\nAt those times either a pair of critical points is born or dies, or two critical values cross.\nJean Cerf developed it to study diffeomorphisms of manifolds, in particular to prove pseudo-isotopy results.\nThrough the correspondence between Morse functions and [[handle decompositions|9V11N8VVTP]], it describes how any two handle decompositions of a manifold are related.\n\n## Explanations\n\n### L1 human\n\nMorse theory looks at one function on a manifold; Cerf theory looks at a film in which the function changes with time.\nMost frames of the film show a Morse function, with clean peaks, pits and saddles.\nAt a few moments something happens: a peak and a saddle melt together and disappear, or appear out of nothing, or two of them swap heights.\nFor example, $f_t(x) = x^3 - t x$ on the line has no critical points for $t \u003c 0$ and two for $t \u003e 0$, which are born at $t = 0$.\n",
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