Infinite connected sums, K-area and positive scalar curvature

TL;DR

This paper extends Whyte's work by using a variant of Gromov's infinite K-area to obstruct positive scalar curvature metrics on certain infinite connected sums, linking index theory and geometric analysis.

Contribution

It introduces a new approach employing Gromov's infinite K-area to obstruct positive scalar curvature, complementing previous index theory methods.

Findings

Infinite connected sums with certain properties cannot admit positive scalar curvature metrics.

A variant of Gromov's infinite K-area effectively obstructs positive scalar curvature.

The method provides a new perspective on scalar curvature obstructions in non-compact manifolds.

Abstract

Whyte used the index theory of Dirac operators and Block-Weiberger uniformly finite homology to show that certain infinite connected sums do not carry a metric with nonnegative scalar curvature in their bounded geometry class. His proof uses a coarse version of the $\hat{A}$-class to obstruct such metrics. In this note we prove a version of Whyte's result where a variant of the notion of infinite $K$-area, originally due to Gromov, is used to obstruct metrics with positive scalar curvature.