Jiggling: an h-principle without homotopical assumptions

TL;DR

This paper generalizes Thurston's jiggling lemma to show that piecewise smooth solutions to certain differential relations can be constructed through a process called jiggling, without relying on homotopical assumptions, thus establishing an h-principle.

Contribution

It introduces a new h-principle for piecewise smooth solutions of differential relations that does not depend on homotopical assumptions, extending Thurston's original lemma.

Findings

Piecewise smooth solutions can be constructed via jiggling for a broad class of differential relations.

The result holds in parametric and relative settings, broadening its applicability.

Establishes an h-principle without homotopical assumptions for first-order differential relations.

Abstract

The jiggling lemma of Thurston shows that any triangulation can be jiggled (read: subdivided and then perturbed) to be in general position with respect to a distribution. Our main result is a generalization of Thurston's lemma. It states that piecewise smooth solutions of a given open and fiberwise dense differential relation $\mathcal{R} \subset J^1(E)$ of first order can be constructed by jiggling arbitrary sections of $E$. Our statement also holds in parametric and relative form. We understand this as an h-principle without homotopical assumptions for piecewise smooth solutions of $\mathcal{R}$.