Compact hyperbolic manifolds without spin structures

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 ≥ 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 CP2 . The manifold M is also the first example of a compact, orientable, hyperbolic 4–manifold satisfying either of these conditions: • H2(M, Z) is not generated by geodesically immersed surfaces. • There is a covering M that is a nontrivial bundle over a compact surface.