Preface

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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