Secondary large-scale index theory and positive scalar curvature


We develop a theory of secondary invariants associated to complete Riemannian metrics of uniformly positive scalar curvature outside a prescribed subset on a spin manifold. We work in the context of large-scale (or “coarse”) index theory. These invariants can be used to distinguish such Riemannian metrics up to concordance relative to the prescribed subset. We exhibit a general external product formula for partial secondary invariants, from which we deduce product formulas for the Rho-invariant of a metric with uniformly positive scalar curvature as well as for the coarse index difference of two metrics with uniformly positive scalar curvature. Our methods yield a new conceptual proof of the secondary partitioned manifold index theorem and a refined version of the delocalized APS-index theorem of Piazza–Schick for the spinor Dirac operator in all dimensions. We establish a partitioned manifold index theorem for the coarse index difference. Moreover, we reprove the existence of a transformation from the positive scalar curvature sequence of Stolz to the analytic surgery sequence of Higson–Roe for real K-theory. As applications of our theory, we construct several complete metrics of uniformly positive scalar curvature on non-compact spin manifolds which can be distinguished up to concordance relative to certain subsets. Moreover, we establish variants of obstructions to existence and concordance of positive scalar curvature metrics via index invariants on submanifolds. From a technical standpoint, the central novelty of this thesis is that we use Yu’s localization algebras in combination with the description of K-theory for graded C*-algebras due to Trout. This formalism allows direct definitions of all the invariants we consider in terms of the functional calculus of the Dirac operator and enables us to give concise proofs of the product formulas. It also allows us to consistently work in the setting of real K-theory.

Doctoral dissertation, University of Göttingen