# Compact hyperbolic non-spin manifolds

Recently a preprint was posted on the arXiv ( arXiv:1904.12720 ) claiming to have constructed the first examples of compact orientable hyperbolic non-spin manifolds (in every dimension at least 4).

I was a bit surprised by this, since I thought that such examples should be already known. Apparently not …

One reason why constructing such examples is hard, is the following result of Deligne-Sullivan from the late 70s: every compact hyperbolic manifold is virtually stably parallelizable, i.e., every compact hyperbolic manifold admits a finite-sheeted covering whose tangent bundle becomes trivial after taking the direct sum with a trivial bundle.

Note that being stably parallelizable implies that all Stiefel-Whitney classes vanish, which is much stronger than being spin (which just needs the first two Stiefel-Whitney classes to vanish). So especially, every compact hyperbolic manifold is virtually spinnable.