This paper is a natural continuation of the previous paper [MZ] where we
studied a class of objects called Goursat distributions certain 2-plane fields in
n-space using Cartan’s method of prolongation. The class of Goursat germs have
interesting singularities which get exponentially deeper and more complicated with
increasing n. In that paper we constructed a sequence of circle bundles called the
“Monster tower” such that any Goursat singularity in dimension n can be found
in the tower at the same dimension. After writing that paper it became clear that
their must be a dictionary between singularities of Legendrian curves (dimension
3), Goursat singularities, and points of the Monster tower (any dimension). The
current paper develops this dictionary and uses it to prove a host of new classi-
fication results concerning Goursat singularities. Simultaneously we develop the
geometry of the Monster tower and use it for resolving singularities of plane and
Legendrian curves by prolonging them to the Monster.
Richard Montgomery
Michail Zhitomirskii
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