Geom. Topol. 24 (5), 2647-2674, (2020) DOI: 10.2140/gt.2020.24.2647
Bruno Martelli, Stefano Riolo, Leone Slavich
KEYWORDS: nonspin, compact, Hyperbolic, Manifold, $120$–cell, 57M50, 57N16, 57R15
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 .
The core of the argument is the construction of a compact, oriented, hyperbolic –manifold that contains a surface of genus with self-intersection . The –manifold has an odd intersection form and is hence not spin. It is built by carefully assembling some right-angled –cells along a pattern inspired by the minimum trisection of .
The manifold is also the first example of a compact, orientable, hyperbolic –manifold satisfying either of these conditions:
is not generated by geodesically immersed surfaces.
There is a covering that is a nontrivial bundle over a compact surface.