A Short Note on the Thom-Boardman Symbols of Differentiable Maps

TL;DR

This paper proves that any non-increasing sequence of nonnegative integers can be realized as the Thom-Boardman symbol of some differentiable map-germ, confirming a natural conjecture in singularity theory.

Contribution

It establishes that all non-increasing sequences of nonnegative integers correspond to Thom-Boardman symbols of differentiable maps, completing the realization problem.

Findings

Any non-increasing sequence of nonnegative integers can be realized as a Thom-Boardman symbol.

The result confirms the completeness of the Thom-Boardman symbol classification.

The paper provides a constructive proof for the realization of these sequences.

Abstract

It is well known that Thom-Boardman symbols are realized by non-increasing sequences of nonnegative integers. A natural question is whether the converse is also true. In this paper we answer this question affirmatively, that is, for any non-increasing sequence of nonnegative integers, there is a map-germ with the prescribed sequence as its Thom-Boardman symbol.