Lawrence-Sullivan models for the interval

TL;DR

This paper proves that two different Lie models of the interval, constructed by Lawrence and Sullivan using distinct methods, are in fact identical, confirming a conjecture in the field.

Contribution

It establishes the equivalence of two previously distinct Lie models of the interval, confirming a conjecture by Lawrence and Sullivan.

Findings

The two models are mathematically identical.

The proof confirms the conjecture about their equivalence.

The models are constructed via different methods, yet coincide.

Abstract

Two constructions of a Lie model of the interval were performed by R. Lawrence and D. Sullivan. The first model uses an inductive process and the second one comes directly from solving a differential equation. They conjectured that these two models are the same. We prove this conjecture here.