  A common problem in lattice QCD simulations on the torus is the extremely
long autocorrelation time of the topological charge, when one approaches the
continuum limit. The reason is the suppressed tunneling between topological
sectors. The problem can be circumvented by replacing the torus with a
different manifold, so that the connectivity of the configuration space is
changed. This can be achieved by using open boundary conditions on the fields,
as proposed earlier. It has the side effect of breaking translational
invariance strongly. Here we propose to use a non-orientable manifold, and show
how to define and simulate lattice QCD on it. We demonstrate in quenched
simulations that this leads to a drastic reduction of the autocorrelation time.
A feature of the new proposal is, that translational invariance is preserved up
to exponentially small corrections. A Dirac-fermion on a non-orientable
manifold poses a challenge to numerical simulations: the fermion determinant
becomes complex. We propose two approaches to circumvent this problem.
