Codimension two index obstructions to positive scalar curvature

Abstract

We derive a general obstruction to the existence of Riemannian metrics of positive scalar curvature on closed spin manifolds in terms of hyper-surfaces of codimension two. The proof is based on coarse index theory for Dirac operators that are twisted with Hilbert C -module bundles. Along the way we give a complete and self-contained proof that the minimal closure of a Dirac type operator twisted with a Hilbert C -module bundle on a complete Riemannian manifold is a regular and self-adjoint operator on the Hilbert C -module of L-2-sections of this bundle. Moreover, we give a new proof of Roe's vanishing theorem for the coarse index of the Dirac operator on a complete non-compact Riemannian manifold whose scalar curvature is uniformly positive outside of a compact subset. This proof immediately generalizes to Dirac operators twisted with flat Hilbert C -module bundles.