Morin singularities

Remarks

For $k = 1$ 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. Example: $f(x, y) = (x, y^3 + xy)$ is a Morin singularity of order $2$ with $\varphi$ and $\psi$ the identity. The 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. Non-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. Every Morin singularity is a stable germ of corank one, and its Thom–Boardman symbol is $(1, \ldots, 1, 0)$ with $k$ ones. Morin proved that a germ with this symbol whose jet is transverse to the Boardman strata is a Morin singularity. There are analogous normal forms for maps $\R^m \to \R^n$ with $m \ne n$, with extra terms in the remaining variables.