The structural invariants of Goursat distributions

TL;DR

The paper develops a systematic framework for local invariants of Goursat distributions by embedding curves and distributions into the monster tower over a surface. It introduces RVT code words, Kumpera–Ruiz coordinates, and a dual front-end/back-end recursion to compute Puiseux characteristics, multiplicity sequences, and vertical orders directly from combinatorial data. The central results establish the universality of the monster tower for rank-2 Goursat germs and connect structural invariants to traditional curve invariants via Nash-like modifications and prolonged data. This provides a recursive, computable approach to link local distributional invariants with classical singularity theory concepts, setting the stage for the companion paper on small growth invariants.

Abstract

This is the first of a pair of papers devoted to the local invariants of Goursat distributions. The study of these distributions naturally leads to a tower of spaces over an arbitrary surface, called the monster tower, and thence to connections with the topic of singularities of curves on surfaces. Here we study those invariants of Goursat distributions akin to those of curves on surfaces, which we call structural invariants. In the subsequent paper we will relate these structural invariants to the small growth invariants.

Figures (6)

• Figure 1: The invariants listed in the top three boxes are invariants of points on monster spaces (as defined in Section \ref{['MT']}) and of focal curve germs (as defined in Section \ref{['fcg']}), akin to invariants of curves on surfaces; the others are invariants of germs of Goursat distributions (as defined in Section \ref{['Goursatdef']}). The invariants listed in the bottom box are invariants obtained by considering the small growth sequence (as defined in Section \ref{['sgs']}).
• Figure 2: An example of the invariants of Figure \ref{['diaginv']}. The dashed arrows indicate compatible front-end recursions.
• Figure 3: A ramphoid cusp.
• Figure 4: A proximity diagram.
• Figure 5: Left portion of the proximity diagram in case (B). There may be additional edges going rightward from the rightmost vertex labeled $T$, but they are irrelevant to the argument.

Theorems & Definitions (42)

• Lemma 1: Sandwich Lemma of MR1841129
• proof
• Definition 2
• Lemma 3
• proof
• Lemma 4
• Example 5
• Example 6
• Example 7
• Example 8