Cusps of Hyperbolic 4-Manifolds and Rational Homology Spheres

In the present paper, we construct a cusped hyperbolic 4-manifold with all cusp sections homeomorphic to the Hantzsche-Wendt manifold, which is a rational homology sphere. By a result of Golenia and Moroianu, the Laplacian on 2-forms on such a manifold has purely discrete spectrum. This answers in the affirmative a question by Golenia and Moroianu from 2008 and, moreover, provides a counterexample to a theorem by Mazzeo and Phillips from 1990 about Laplacian spectral of conformally cusped manifolds. We also correct the incomplete classification of compact orientable flat 3-manifolds arising from cube colourings provided earlier by Kolpakov and Slavich.