An index theorem for end-periodic operators (Q2797475)

\textit{M.F. Atiyah} et al. [Math. Proc. Camb. Philos. Soc. 77, 43--69 (1975; Zbl. 0297.58008); ibid. 78, 405--432 (1975; Zbl. 0314.58016); ibid. 79, 71--99 (1976; Zbl. 0325.58015)] studied the index theory of first-order elliptic differential operators, with nonlocal boundary conditions, on compact oriented manifolds with boundary. They related this subject to index theory on manifolds with cylindrical ends. The authors of the paper under review generalize these results to the setting of Fredholm end-periodic operators of Dirac type, possibly on weighted Sobolev spaces, on end-periodic manifolds. Their proof is based on a method introduced for cylindrical ends by \textit{R. B. Melrose} [The Atiyah-Patodi-Singer index theorem. Wellesley, MA: A. K. Peters, Ltd. (1993; Zbl 0796.58050)]. The periodic \(\widetilde{\xi}\)-invariants that arise in this index theory have implications for the number of connected components in moduli spaces of positive-scalar-curvature metrics. End-periodic manifolds are manifolds with ends modeled on an infinite cyclic cover of a compact manifold. Results of \textit{C. Taubes} [J. Differ. Geom. 25, 363--430 (1987; Zbl. 0615.57009)] drew attention to the index theory of end-periodic elliptic operators on such manifolds.