Curve Singularities and Monster / Semple Towers

TL;DR

This paper extends the concept of the Monster tower from planar to n-dimensional space, constructing the Semple Tower and exploring the classification of its points under diffeomorphism actions, linking to curve singularities.

Contribution

It generalizes the Monster tower to n-space, introduces the Semple Tower, and initiates classification of points under diffeomorphism groups, connecting to Arnold's singularity list.

Findings

Construction of the Semple Tower for n-space.

Initial steps in classifying points under diffeomorphism actions.

Connection to Arnold's list of simple curve singularities.

Abstract

In earlier work, we introduced the `Monster tower', a tower of fibrations associated to planar curves. We constructed an algorithm for classifying its points with respect to the equivalence relation generated by the action of the contact pseudogroup on the tower. Here, we construct the analogous tower for curves in $n$-space. (This tower is known as the Semple Bundle in Algebraic Geometry.) The pseudo-group of diffeomorphisms of $n$-space acts on each level of the extended tower. We take initial steps toward classifying points of this extended Monster tower under this pseudogroup action. Arnol'd's list of stable simple curve singularities plays a central role in these initial steps. We end with a list of open problems.