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
.
The core of the argument is the construction of a compact, oriented, hyperbolic
–manifold
that contains
a surface
of
genus
with
self-intersection
.
The
–manifold
has an odd
intersection form and is hence not spin. It is built by carefully assembling some right-angled
–cells
along a pattern inspired by the minimum trisection of
.
The manifold
is also the first example of a compact, orientable, hyperbolic
–manifold
satisfying either of these conditions:
is not generated by geodesically immersed surfaces.
There is a covering
that is a nontrivial bundle over a compact surface.
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