Positive scalar curvature and connected sums

TL;DR

This paper explores conditions under which connected sums of certain manifolds admit positive scalar curvature, extending classical results to nonspin cases using index theory for Dirac operators.

Contribution

It generalizes Gromov-Lawson's theorem to nonspin manifolds, providing new insights into scalar curvature obstructions via index theory.

Findings

Connected sum of a spin manifold with an enlargeable manifold admits no positive scalar curvature metric.

Potential extension of scalar curvature obstructions to nonspin manifolds.

Use of Dirac operator index theory to prove nonexistence results.

Abstract

Let $N$ be a closed enlargeable manifold in the sense of Gromov-Lawson and $M$ a closed spin manifold of equal dimension, a famous theorem of Gromov-Lawson states that the connected sum $M\# N$ admits no metric of positive scalar curvature. We present a potential generalization of this result to the case where $M$ is nonspin. We use index theory for Dirac operators to prove our result.