# Unravelling the Dodecahedral Spaces

## TL;DR

The paper investigates the structure of hyperbolic and spherical dodecahedral spaces, revealing new covers and cubulations that simplify their hierarchical and geometric properties.

## Contribution

It introduces specific covers of the dodecahedral spaces with simplified hierarchies and special cubulations, advancing understanding of their geometric structures.

## Key findings

- Existence of a 6-sheeted irregular cover with a short hierarchy.
- Description of a 60-sheeted cover with a special cubulation.
- Analysis of natural cubulations and covers of the Poincaré homology sphere.

## Abstract

The hyperbolic dodecahedral space of Weber and Seifert has a natural non-positively curved cubulation obtained by subdividing the dodecahedron into cubes. We show that the hyperbolic dodecahedral space has a 6-sheeted irregular cover with the property that the canonical hypersurfaces made up of the mid-cubes give a very short hierarchy. Moreover, we describe a 60-sheeted cover in which the associated cubulation is special. We also describe the natural cubulation and covers of the spherical dodecahedral space (aka Poincar\'e homology sphere).