The space of tight contact structures on ${\mathbb R}^3$ is contractible

TL;DR

This paper proves that the space of tight contact structures on a0^3, fixed at a point, is contractible, extending previous results on the 3-sphere and confirming a longstanding claim.

Contribution

It provides a rigorous proof that the space of tight contact structures on a0^3 is contractible, generalizing earlier results and confirming a prior unproven claim.

Findings

The space of tight contact structures on a0^3 is contractible.

The result extends previous work on the 3-sphere.

It confirms a longstanding claim about the structure of tight contact structures.

Abstract

It was proven in the first author's paper "Contact 3-manifolds twenty years since J. Martinet's work" (Ann. Inst. Fourier, 42(1992), 165--192) that any tight contact structure on the 3-sphere is diffeomorphic to the standard one. It was also claimed there without a proof that similar methods could be used to prove a multi-parametric version: the space of tight contact structures on $S^3$, fixed at a point, is contractible. We prove this result in the current paper.