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  "body": "\n## Definition\n\nLet $f \\colon M \\to N$ be a smooth map between [[smooth manifolds|F45VQ47GPV]].\nFor an integer $i \\ge 0$ let\n$$\\Sigma^{i}(f) = \\set{p \\in M : \\dim \\ker df_p = i},$$\nwhere $df_p$ is the [[differential|6P9PXQDSEN]] of $f$ at $p$.\nInductively, let $I = (i_1, \\ldots, i_k)$ and suppose that $\\Sigma^{I}(f)$ is a submanifold of $M$.\nFor $j \\ge 0$ let $\\Sigma^{i_1, \\ldots, i_k, j}(f)$ be the set of points of $\\Sigma^{I}(f)$ where the restriction $f|_{\\Sigma^I(f)}$ has kernel of dimension $j$, that is, $\\Sigma^{j}(f|_{\\Sigma^I(f)})$.\nA point $p$ has *Thom–Boardman symbol* $I$ when $p \\in \\Sigma^I(f)$, whenever these sets are defined.\n\n## Remarks\n\nSince $\\Sigma^{i_1, \\ldots, i_k}(f) \\subseteq \\Sigma^{i_1, \\ldots, i_{k-1}}(f)$, the kernel of $f$ restricted to the smaller set lies in the kernel of $f$ restricted to the larger one.\nSo the symbols are non-increasing: $i_1 \\ge i_2 \\ge \\cdots \\ge i_k$.\nBoardman constructed submanifolds $\\Sigma^I$ of the [[jet space|1QE0Z0756D]] $J^k(M, N)$, depending only on $I$.\nSuppose the jet extension $j^k f$ is [[transverse|A2PDZ8GA9C]] to all of them.\nThen each $\\Sigma^I(f) = (j^k f)^{-1}(\\Sigma^I)$ is a submanifold and agrees with the inductive definition.\nBy Thom's transversality theorem, this holds for a dense set of maps.\nIf $\\dim M = m$ and $\\dim N = n$, the subset $\\Sigma^i$ of $J^1(M, N)$ has codimension $i(n - m + i)$.\nExample: for $f(x, y) = (x, y^3 + xy)$ on $\\R^2$, $\\Sigma^1(f)$ is the curve $x = -3y^2$.\nParametrising it by $y \\mapsto (-3y^2, y)$, the restriction of $f$ is $y \\mapsto (-3y^2, -2y^3)$, whose derivative vanishes only at $y = 0$.\nSo the origin lies in $\\Sigma^{1,1}(f)$, and the other points of the curve lie in $\\Sigma^{1,0}(f)$.\nNon-example: for $f(x, y) = (x, y^3 - x^2 y)$, $\\Sigma^1(f)$ is the set $3y^2 = x^2$, a pair of crossing lines.\nIt is not a submanifold at the origin, so the inductive definition breaks down there; the jet of $f$ is not transverse to $\\Sigma^1$.\n\n## Explanations\n\n### L1 human\n\nThe symbol of a singular point records how badly a map degenerates, one layer at a time.\nThe first entry counts the directions crushed by the map.\nThe next entry counts the directions crushed by the map along the set of points that share the first entry.\nThe construction is clean only for generic maps, where all these sets are smooth.\n",
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