An Atiyah-Bott-Lefschetz-Type Formula and Its Applications to Some Nonlocal Problems

This talk presents an Atiyah-Bott-Lefschetz-type formula for endomorphisms of elliptic complexes of general continuous linear operators acting in spaces of smooth sections of vector bundles. Certain conditions on the wavefronts of their Schwarz kernels are imposed on the operators under consideration. Next, two applications of the resulting formula are considered. First, complexes of pseudodifferential operators associated with bundles are considered. For such complexes, ellipticity conditions are formulated that ensure the Fredholm property in Sobolev spaces. Next, we consider geometric endomorphisms for fiberwise diffeo-morphisms of total fiber bundle spaces and give a Lefschetz formula expressing the Lefschetz number of the corresponding endomorphisms in terms of fixed points. Second, we consider operators associated with a compact Lie group. It is assumed that the operators are invariant under the group action, and the group action is semifree. In the case where the coincidence points of the diffeomorphism and the group action are nondegenerate, a Lefschetz formula is obtained.