  We deduce an index jump formula for first order elliptic complexes over
end-periodic manifolds, which generalizes the corresponding result for the
DeRham complex. In the case of the anti-self-dual DeRham complex, we define the
periodic rho invariant for a class of $4$-manifolds, and identify it with the
periodic spectral flow of this complex. As an application, we prove the
equivalence (under a mild homological assumption) of two signatures invariants
defined by means of Yang-Mills theory and geometric topology respectively for
essentially embedded tori in homology $S^1 \times S^3$. We also prove a surgery
formula for the singular Furuta-Ohta invariant, which corresponds to a
potential exact triangle of singular instanton homology for knots.
