Atiyah–Bott–Lefschetz Formula on Manifolds with Periodic Ends

Abstract

The classical Atiyah–Bott formula (1967) expresses the Lefschetz number of a geometric endomorphism of an elliptic complex of (pseudo)differential operators on a closed smooth manifold as the sum of contributions of fixed points of the corresponding diffeomorphism, assuming that all fixed points are nondegenerate. These contributions explicitly depend only on the endomorphism itself but not on the operators forming the complex. In 1999, one of the authors, together with B.-W. Schulze, B. Yu. Sternin, and V. E. Shatalov, generalized this formula to the case of manifolds with conical singularities (or, equivalently, with cylindrical ends), where the contributions of interior fixed points are supplemented by those of fixed cylindrical ends (which already depend on the operators of the complex themselves). In recent decades, a number of papers have appeared in the literature devoted to elliptic theory on manifolds with periodic ends. The present paper gives a Lefschetz formula on such manifolds; in the special case of manifolds with cylindrical ends, it strengthens previously obtained results.