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  "body": "\n## Definition\n\nLet $n \\ge 1$ and $1 \\le k \\le n$.\nLet $f \\colon (\\R^n, 0) \\to (\\R^n, 0)$ be a smooth map [[germ|VZXF7RNFKS]].\nIt is a *Morin singularity of order $k$* if there are germs of [[diffeomorphisms|0KZZ70ZHXP]] $\\varphi$ and $\\psi$ of $(\\R^n, 0)$ with this property.\nThe germ $\\psi \\circ f \\circ \\varphi^{-1}$ is the germ at $0$ of\n$$(u_1, \\ldots, u_{n-1}, t) \\mapsto \\left(u_1, \\ldots, u_{n-1}, \\ t^{k+1} + \\sum_{i=1}^{k-1} u_i t^i\\right).$$\n\n## Remarks\n\nFor $k = 1$ the normal form is $(u, t) \\mapsto (u, t^2)$, the fold; for $k = 2$ it is $(u_1, \\ldots, t) \\mapsto (u_1, \\ldots, t^3 + u_1 t)$, the cusp; for $k = 3$ it is the swallowtail.\nExample: $f(x, y) = (x, y^3 + xy)$ is a Morin singularity of order $2$ with $\\varphi$ and $\\psi$ the identity.\nThe local algebra $\\mathcal{E}_n / f^*(\\mathfrak{m}_n) \\mathcal{E}_n$ of a Morin singularity of order $k$ is $\\R[t]/(t^{k+1})$, of dimension $k + 1$; this is an invariant of the germ up to the equivalence above.\nNon-example: $f(x, y) = (x, y^4)$ has local algebra $\\R[y]/(y^4)$ of dimension $4$, so it could only be of order $3$, which needs $n \\ge 3$; hence it is not a Morin singularity.\nEvery Morin singularity is a stable germ of corank one, and its Thom–Boardman symbol is $(1, \\ldots, 1, 0)$ with $k$ ones.\nMorin proved that a germ with this symbol whose jet is transverse to the Boardman strata is a Morin singularity.\nThere are analogous normal forms for maps $\\R^m \\to \\R^n$ with $m \\ne n$, with extra terms in the remaining variables.\n",
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