Dodecahedral L-spaces and hyperbolic 4-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 [Dun20], and relies on a result by Rasmussen-Rasmussen [RR17]. We use the existence of these manifolds together with a result of Martelli [Mar16a] 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 [AL20].

A small cusped hyperbolic 4‐manifold

“…For a quick description of the computation without proofs, one may consult [18, section 4.5]. Up to isometry of ℍ 4 in its hyperboloid model, the bounding hyperplanes of the polytope 𝑄 ⊂ ℍ 4 (coherently oriented) are dually represented by these spacelike vectors of ℝ 1,4 :…”

“…The manifold 𝑀 is commensurable with a Coxeter polytope 𝑄 belonging to a continuous family of hyperbolic 4-polytopes discovered in 2010 by Kerckhoff and Storm [10]. Notably, classes (1) and (3) are represented by other Coxeter polytopes of the family, and the examples in (4) are hybrids of manifolds in (3) and (2). Our method applies to two additional Coxeter polytopes of the family, but giving less interesting examples from the viewpoint of this paper (see Subsection 2.4).…”

“…As a first step, we 'kill' in a natural way the 2𝜋∕3 angles by gluing in pairs some facets of five copies of 𝑃, to get a particularly symmetric hyperbolic manifold 𝑋 with right-angled corners. These objects have been fruitfully used in four-dimensional hyperbolic geometry in the very last years [1,16,19,20].…”