Dodecahedral L-spaces and hyperbolic 4-manifolds

TL;DR

This paper classifies certain hyperbolic 3-manifolds as L-spaces, constructs explicit hyperbolic 4-manifolds with separating L-spaces, and addresses a question about their invariants, using computational and theoretical methods.

Contribution

It identifies all L-spaces among specific hyperbolic 3-manifolds and constructs hyperbolic 4-manifolds with separating L-spaces, answering a previously open question.

Findings

6 out of 29 rational homology 3-spheres are L-spaces

Explicit examples of hyperbolic 4-manifolds with separating L-spaces

Vanishing Seiberg-Witten invariants in constructed manifolds

Abstract

We prove that exactly 6 out of the 29 rational homology 3-spheres tessellated by four or less right-angled hyperbolic dodecahedra are L-spaces. The algorithm used is based on the L-space census provided by Dunfield in arXiv:1904.04628, and relies on a result by Rasmussen-Rasmussen arXiv:1508.05900. We use the existence of these manifolds together with a result of Martelli arXiv:1510.06325 to construct explicit examples of hyperbolic 4-manifolds containing separating L-spaces, and therefore having vanishing Seiberg-Witten invariants. This answers a question asked by Agol and Lin in arXiv:1812.06536.