Tight contact structures on hyperbolic homology 3-spheres

TL;DR

This paper constructs numerous hyperbolic homology 3-spheres with many distinct tight contact structures, including within the same homotopy class, and explores their behavior under geometric limits.

Contribution

It introduces methods to produce hyperbolic homology 3-spheres with multiple tight contact structures and studies their limits, advancing understanding of contact topology in hyperbolic geometry.

Findings

Existence of hyperbolic homology 3-spheres with arbitrarily many tight contact structures.

Construction of tight contact structures within the same homotopy class.

Analysis of the behavior of contact structures under geometric limits.

Abstract

We produce a large class of hyperbolic homology 3-spheres admitting arbitrarily many distinct tight contact structures. We also produce a sub-class admitting arbitrarily many distinct tight contact structures within the same homotopy class of oriented plane distributions. As a corollary, we give a recipe to construct hyperbolic L-spaces admitting arbitrarily many distinct tight contact structures. We also introduce a notion of geometric limits of contact structures compatible with geometric limits of hyperbolic manifolds and study the behavior of the tight contact structures we construct under geometric limits.