On spectral action over Bieberbach manifolds

We compute the leading terms of the spectral action for orientable three dimensional Bieberbach manifolds first, using two different methods: the Poisson summation formula and the perturbative expansion. Assuming that the cut-off function is not necessarily symmetric we find that that the scale invariant part of the perturbative expansion might differ from the spectral action of the flat three-torus by the eta invariant.