Cusps of hyperbolic 4‐manifolds and rational homology spheres
| Autor: | Leonardo Ferrari, Alexander Kolpakov, Leone Slavich |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Cusp (singularity)
Pure mathematics General Mathematics 010102 general mathematics Geometric Topology (math.GT) 57N16 57M50 52B10 52B11 Homology (mathematics) 16. Peace & justice Mathematics::Geometric Topology 01 natural sciences Homology sphere Manifold Discrete spectrum Mathematics - Geometric Topology 010104 statistics & probability FOS: Mathematics SPHERES Mathematics::Differential Geometry 0101 mathematics Cube Mathematics::Symplectic Geometry Laplace operator Mathematics |
| Zdroj: | Proceedings of the London Mathematical Society. 123:636-648 |
| ISSN: | 1460-244X 0024-6115 |
| Popis: | 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 Gol\'enia 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 Gol\'enia 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. Comment: 15 pages, 1 figure, 1 table; SageMath worksheets available at https://github.com/sashakolpakov/24-cell-colouring |
| Databáze: | OpenAIRE |
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