Compact hyperbolic manifolds without spin structures

Martelli, Bruno¹ ; Riolo, Stefano² ; Slavich, Leone¹

¹ Dipartimento di Matematica, Università di Pisa, Pisa, Italy
² Institut de mathématiques, Université de Neuchâtel, Neuchâtel, Switzerland

Geometry & topology, Tome 24 (2020) no. 5, pp. 2647-2674.

Résumé

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 $n\ge 4$.

The core of the argument is the construction of a compact, oriented, 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 ${ℂ\mathbb{P}}^2$.

The manifold $M$ is also the first example of a compact, orientable, hyperbolic $4$–manifold satisfying either of these conditions:

Détail

DOI : 10.2140/gt.2020.24.2647

Classification : 57M50, 57N16, 57R15
Keywords: nonspin, compact, hyperbolic, manifold, $120$–cell

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