  We extend the Atiyah, Patodi, and Singer index theorem for first order
differential operators from the context of manifolds with cylindrical ends to
manifolds with periodic ends. This theorem provides a natural complement to
Taubes' Fredholm theory for general end-periodic operators. Our index theorem
is expressed in terms of a new periodic eta-invariant that equals the
Atiyah-Patodi-Singer eta-invariant in the cylindrical setting. We apply this
periodic eta-invariant to the study of moduli spaces of Riemannian metrics of
positive scalar curvature.
