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  "body": "\n## Statement\n\nLet $X$ and $Y$ be [[smooth manifolds|F45VQ47GPV]], and let $s \\ge 1$ and $k \\ge 0$ be integers.\nLet $X^{(s)} = \\set{(x_1, \\dots, x_s) \\in X^s : x_i \\ne x_j \\text{ for } i \\ne j}$.\nLet $J^k(X, Y)$ be the [[bundle of $k$-jets|HC3VBBCJQP]] of smooth maps from $X$ to $Y$, with source map $\\alpha \\colon J^k(X, Y) \\to X$.\nThe $s$-fold *multijet bundle* is\n$$J^k_s(X, Y) = \\set{(\\sigma_1, \\dots, \\sigma_s) \\in J^k(X, Y)^s : (\\alpha(\\sigma_1), \\dots, \\alpha(\\sigma_s)) \\in X^{(s)}}.$$\nFor a smooth map $f \\colon X \\to Y$, its multijet extension is $j^k_s f \\colon X^{(s)} \\to J^k_s(X, Y)$, $(x_1, \\dots, x_s) \\mapsto (j^k f(x_1), \\dots, j^k f(x_s))$, where $j^k f(x)$ is the [[$k$-jet|1QE0Z0756D]] of $f$ at $x$.\nLet $W$ be a [[submanifold|VKFW27WXEZ]] of $J^k_s(X, Y)$, and give $C^\\infty(X, Y)$ the [[Whitney $C^\\infty$ topology|7PN9MD1PQB]].\nThen the set of $f \\in C^\\infty(X, Y)$ such that $j^k_s f$ is [[transverse|A2PDZ8GA9C]] to $W$ is a [[residual set|GRH8MCVN3N]] in $C^\\infty(X, Y)$.\n\n## Remarks\n\nFor $s = 1$ this is the [[Thom transversality theorem|KWNRS3RA09]].\nThe proof follows the same pattern: one perturbs $f$ near the $s$ distinct points $x_1, \\dots, x_s$ independently.\nThe perturbations are polynomial maps of degree at most $k$, multiplied by bump functions with disjoint supports.\nThe family of perturbed multijets is a submersion onto the fibres, so the parametric transversality theorem applies, and a countable covering argument gives a residual set.\nThe points must be distinct, since the jets of $f$ at the same point cannot be perturbed independently.\n\nExample: let $s = 2$, $k = 0$, and let $W = \\set{((x_1, y_1), (x_2, y_2)) : y_1 = y_2}$, which has codimension $\\dim Y$ in $J^0_2(X, Y) = X^{(2)} \\times Y \\times Y$.\nIf $\\dim Y \u003e 2 \\dim X = \\dim X^{(2)}$, transversality to $W$ means that $j^0_2 f$ misses $W$, that is, $f$ is injective.\nSo injective maps are generic when $\\dim Y \u003e 2 \\dim X$.\nCombined with the genericity of immersions, this gives the easy form of the Whitney embedding theorem for compact $X$.\n\n## Explanations\n\n### L1 human\n\nThom's theorem says that a generic map behaves well at each single point.\nThe multijet version says that a generic map also behaves well at several points at once.\nFor example, two different points are not sent to the same place in a degenerate way.\n\nThis controls self-intersections of maps, not only their local shape.\n\n### L2 human\n\nThe theorem is the main tool for global properties of generic maps.\nIt shows that generic immersions have only transverse self-intersections, that is, normal crossings, and it is a basic tool in the theory of stable maps.\n",
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