Compact hyperbolic manifolds without spin structures
| Autor: | Bruno Martelli, Leone Slavich, Stefano Riolo |
|---|---|
| Přispěvatelé: | Martelli B., Riolo S., Slavich L. |
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
$120$–cell compact nonspin Spin structure 01 natural sciences Mathematics - Geometric Topology hyperbolic Genus (mathematics) 0103 physical sciences FOS: Mathematics 57R15 Intersection form 57N16 0101 mathematics Mathematics::Symplectic Geometry Mathematics - General Topology Spin-½ Mathematics manifold nonspin compact hyperbolic manifold 120-cell 010102 general mathematics General Topology (math.GN) Geometric Topology (math.GT) Surface (topology) Mathematics::Geometric Topology 57M50 Manifold Bundle Mathematics::Differential Geometry 010307 mathematical physics Geometry and Topology 120-cell |
| Zdroj: | Geom. Topol. 24, no. 5 (2020), 2647-2674 |
| ISSN: | 1364-0380 1465-3060 |
| DOI: | 10.2140/gt.2020.24.2647 |
| Popis: | We exhibit the first examples of compact orientable hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions $n \geq 4$. The core of the argument is the construction of a compact orientable hyperbolic $4$-manifold $M$ that contains a surface $S$ of genus $3$ with self intersection $1$. The $4$-manifold $M$ has an odd intersection form and is hence not spin. It is built by carefully assembling some right angled $120$-cells along a pattern inspired by the minimum trisection of $\mathbb{C}\mathbb{P}^2$. The manifold $M$ is also the first example of a compact orientable hyperbolic $4$-manifold satisfying any of these conditions: 1) $H_2(M,\mathbb{Z})$ is not generated by geodesically immersed surfaces. 2) There is a covering $\tilde{M}$ that is a non-trivial bundle over a compact surface. 23 pages, 16 figures |
| Databáze: | OpenAIRE |
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