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 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 CP2. The manifold M is also the first
example of a compact orientable hyperbolic 4-manifold satisfying any of these
conditions: 1) H2(M,Z) is not generated by geodesically immersed
surfaces. 2) There is a covering M~ that is a non-trivial bundle over
a compact surface.Comment: 23 pages, 16 figure