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  "body": "\n## Definition\n\nLet $M$ be a compact [[smooth manifold|F45VQ47GPV]] of dimension $n$ without boundary.\n*Cerf theory* is the study of smooth one-parameter families $f_t \\colon M \\to \\R$, $t \\in [0, 1]$, of smooth functions, with $(x, t) \\mapsto f_t(x)$ smooth.\nIts basic notion is a *generic path*, also called a *Cerf path*.\nA family $(f_t)$ is a generic path if it satisfies the following conditions.\n\n1. For all but finitely many $t$, $f_t$ is a [[Morse function|KPQ96640ZJ]] with distinct critical values.\n2. At each remaining time $t_0$, one of two cases occurs.\n3. In the first case, $f_{t_0}$ is Morse, exactly two [[critical points|0YTCMFB4HC]] share a critical value, and these two values cross transversally at $t_0$.\n4. In the second case, $f_{t_0}$ has distinct critical values and exactly one degenerate critical point $p_0$.\n5. In the second case, near $p_0$ there are local coordinates $(x_1, \\dots, x_n)$ and a local time parameter $s$ with $s = 0$ at $t_0$ with the following form.\n   For some $0 \\le \\lambda \\le n - 1$ and a smooth function $c$, one has $f_t = c(s) + x_1^3 + s x_1 - x_2^2 - \\cdots - x_{\\lambda+1}^2 + x_{\\lambda+2}^2 + \\cdots + x_n^2$.\n\n## Remarks\n\nIn item 5, for $s \u003c 0$ there are two critical points near the origin, of indices $\\lambda$ and $\\lambda + 1$, and for $s \u003e 0$ there are none; this is a *birth–death* singularity.\nCerf's theorem says that generic paths are dense in the space of smooth paths between two given endpoints.\nHere the endpoints are Morse functions with distinct critical values.\nThe *Cerf graphic* of a generic path is the subset $\\set{(t, f_t(p)) : p \\text{ critical for } f_t}$ of $[0, 1] \\times \\R$; it shows births, deaths and crossings as cusps and crossings of curves.\nExample, in a chart of a $1$-dimensional $M$: $f_t(x) = x^3 - (t - \\tfrac{1}{2}) x$ satisfies item 5 at $t_0 = \\tfrac{1}{2}$ with $s = -(t - \\tfrac{1}{2})$ and $\\lambda = 0$.\nNon-example: the constant path $f_t(x) = x^3$ fails item 1, because the degenerate critical point $0$ occurs at every time.\n",
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