  This paper studies Dirac operators on end-periodic spin manifolds of
dimension at least 4. We provide a sufficient condition for such an operator to
be Fredholm for a generic end-periodic metric; this condition is shown to be
necessary in dimension 4. We make use of end-periodic Dirac operators to give
an analytical interpretation of an invariant of non-orientable smooth
4-manifolds due to Cappell and Shaneson. From this interpretation we show that
some exotic non-orientable 4-manifolds do not admit a metric of positive scalar
curvature.
