  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.

Geometric Topology;General Topology