A Counterexample to a Conjecture about Positive Scalar Curvature

TL;DR

This paper provides a counterexample to Stanley Chang's conjecture that certain non-spin manifolds admit positive scalar curvature metrics if specific homological conditions are met, challenging previous assumptions in differential geometry.

Contribution

It constructs a counterexample to Chang's conjecture using a known counterexample to the unstable Gromov-Lawson-Rosenberg conjecture, advancing understanding of scalar curvature.

Findings

Counterexample disproves Chang's conjecture

Supports the complexity of scalar curvature conditions

Links to the unstable Gromov-Lawson-Rosenberg conjecture

Abstract

Conjecture 1 of Stanley Chang: "Positive scalar curvature of totally nonspin manifolds" asserts that a closed smooth manifold M with non-spin universal covering admits a metric of positive scalar curvature if and only if a certain homological condition is satisfied. We present a counterexample to this conjecture, based on the counterexample to the unstable Gromov-Lawson-Rosenberg conjecture given in Schick: "A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture".