Abstract
The Inoue surfaces are certain non-Kaehler complex surfaces that have the
structure of a $T^3$ bundle over the circle. We study the Inoue surfaces $S_M$
with the Tricerri metric and the canonical spin$^c$ structure, and the
corresponding chiral Dirac operators twisted by a flat $\mathbb
C^*$--connection. The twisting connection is determined by $z \in \mathbb C^*$,
and the points for which the twisted Dirac operators $\mathcal D^{\pm}_z$ are
not invertible are called spectral points. We show that there are no spectral
points inside the annulus $\alpha^{-1/4} < |z| < \alpha^{1/4}$, where $\alpha
>1$ is the only real eigenvalue of the matrix $M$ that determines $S_M$, and
find the spectral points on its boundary. Via Taubes' theory of end-periodic
operators, this implies that the corresponding Dirac operators are Fredholm on
any end-periodic manifold whose end is modeled on $S_M$.