Compact hyperbolic manifolds without spin structures

Abstract

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 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.

arXiv comment23 pages, 16 figures

References in corpus (4)

• Some hyperbolic three-manifolds that bound geometrically
• Embedding arithmetic hyperbolic manifolds
• Noncoherence of some lattices in Isom(Hn)
• Telescopic actions

Cited by in corpus (2)

• Orientable hyperbolic 4-manifolds over the 120-cell
• A small cusped hyperbolic 4-manifold