Orientable hyperbolic 4-manifolds over the 120-cell

TL;DR

This paper constructs explicit orientable hyperbolic 4-manifolds of small volume using small cover theory, overcoming the lack of a higher-dimensional hyperbolic Dehn filling theorem.

Contribution

It classifies all orientable four-dimensional small covers over the right-angled 120-cell, providing explicit examples of hyperbolic 4-manifolds with even intersection forms.

Findings

Constructed hyperbolic 4-manifolds with volume 34π²/3 * 16

Classified all orientable small covers over the 120-cell

All classified covers have even intersection forms

Abstract

Since there is no hyperbolic Dehn filling theorem for higher dimensions, it is challenging to construct explicit hyperbolic manifolds of small volume in dimension at least four. Here, we build up closed hyperbolic 4-manifolds of volume $\frac{34\pi^2}{3}\cdot 16$ by using the small cover theory. In particular, we classify all of the orientable four-dimensional small covers over the right-angled 120-cell up to homeomorphism; these are all with even intersection forms.