On a relation between the $\mathrm{K}$-cowaist and the $\hat{\mathsf{A}}$-cowaist

TL;DR

This paper explores the relationship between the K-cowaist and the -cowaist invariants on manifolds, providing a detailed proof of an inequality linking these invariants, which are related to positive scalar curvature metrics.

Contribution

It offers a detailed proof of Gromov's inequality connecting the K-cowaist and -cowaist invariants on manifolds.

Findings

Established the inequality -cowaist -cw  M  c  K-cw  M with a detailed proof.

Confirmed the invariants' relation to the existence of positive scalar curvature metrics.

Provided a dimensional constant c in the inequality.

Abstract

The $\mathrm{K}$-cowaist $\text{K-cw}_2 (M)$ and the $\hat{\mathsf{A}}$-cowaist $\hat{\mathrm{A}}$-$\mathrm{cw}_2 (M)$ are two interesting invariants on a manifold $M$, which are closely related to the existence of the positive scalar curvature metric on $M$. In this note, we give a detailed proof of the following inequality due to Gromov: $\text{K-cw}_2 (M) \le c \hat{\mathrm{A}}$-$\mathrm{cw}_2 (M)$, where $c$ is a dimensional constant.