Non-negative scalar curvature on spin surgeries and Novikov conjecture

Abstract

Let $M$ be a closed aspherical manifold. Assume that the rational strong Novikov conjecture holds for $\pi_1(M)$. We show that on any spin surgery of $M$ along a region whose induced homomorphism on the fundamental group is trivial, every complete metric with non-negative scalar curvature is Ricci-flat. In particular, on the connected sum of $M$ with a spin manifold, any complete metric with non-negative scalar curvature is Ricci-flat.