Geography of $4$-manifolds with positive scalar curvature

TL;DR

This paper surveys the landscape of closed oriented 4-manifolds with positive scalar curvature, highlighting recent results, extensions, and the relationship between scalar curvature and manifold essentialness in four dimensions.

Contribution

It provides a comprehensive survey of the geography problem for 4-manifolds with positive scalar curvature, including recent extensions and strengthened results.

Findings

Extension of Carr's result to non-orientable manifolds

Survey of mathematical approaches to Gromov's observation

Identification of key properties influencing scalar curvature in 4-manifolds

Abstract

We discuss the geography problem of closed oriented 4-manifolds that admit a Riemannian metric of positive scalar curvature, and use it to survey mathematical work employed to address Gromov's observation that manifolds with positive scalar curvature tend to be inessential by focusing on the four-dimensional case. We also point out an strengthening of a result of Carr and its extension to the non-orientable realm.