Cusps of hyperbolic 4‐manifolds and rational homology spheres. Issue 6 (20th September 2021)

• Record Type: Journal Article
• Authors: Ferrari, Leonardo; Kolpakov, Alexander; Slavich, Leone
• Is Part Of: Proceedings of the London Mathematical Society. Volume 123:Issue 6(2021)
• Page Start: 636
• Page End: 648
• Publication Date: 2021-09-20
• Subjects: 57N16 (primary) -- 57M50 -- 52B10 -- 52B11 (secondary); Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1112/plms.12421
• Languages: English
• ISSNs: 0024-6115

Abstract

Abstract: 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 Golénia and Moroianu, the Laplacian on 2‐forms on such a manifold has purely discrete spectrum. This shows that one of the main results of Mazzeo and Phillips from 1990 cannot hold without additional assumptions on the homology of the cusps. This also answers a question by Golénia and Moroianu from 2012. We also correct and refine the incomplete classification of compact orientable flat 3‐manifolds arising from cube colourings provided earlier by the last two authors.