Non-negative scalar curvature on spin surgeries and Novikov conjecture

TL;DR

This paper proves that for certain spin surgeries on aspherical manifolds satisfying the Novikov conjecture, any complete non-negative scalar curvature metric must be Ricci-flat, revealing geometric constraints related to topology.

Contribution

It establishes a new link between spin surgeries, scalar curvature, and the Novikov conjecture, showing that non-negative scalar curvature metrics become Ricci-flat under these conditions.

Findings

Complete metrics with non-negative scalar curvature are Ricci-flat after specific spin surgeries.

Connected sums with spin manifolds inherit Ricci-flatness for non-negative scalar curvature.

Results depend on the validity of the rational strong Novikov conjecture for the fundamental group.

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.