2020/09/30 by Leonardo Ferrari, Alexander Kolpakov, Leone Slavich · 8 citations
Mathematics · #Advanced Operator Algebra Research #Cube (algebra) #Cusp (singularity) #Geometric and Algebraic Topology #Homology (biology) #Laplace operator #Manifold (fluid mechanics) #Mathematical Dynamics and Fractals #SPHERES #math.GT #msc:52B10 #msc:52B11 #msc:57M50 #msc:57N16
paper · pdf · open access · doi:10.1112/plms.12421
published in Proceedings of the London Mathematical Society 123(6), 636-648 (Wiley) · 15 pages, 1 figure, 1 table; SageMath worksheets available at https://github.com/sashakolpakov/24-cell-colouring
openalex created_date 2020/09/25 · arxiv created 2021/06/11 · openalex publication_date 2021/09/20 · arxiv updated 2022/03/07 · openalex updated_date 2026/08/06
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.