Positive scalar curvature -- constructions and obstructions

TL;DR

This survey reviews the current understanding of which high-dimensional closed manifolds admit metrics with positive scalar curvature, highlighting topological obstructions, geometric constructions, and connections to stable homotopy theory.

Contribution

It summarizes known obstructions, construction methods, and the role of bordism and stable homotopy theory in classifying manifolds with positive scalar curvature.

Findings

Topological obstructions include Dirac operator and minimal hypersurfaces.

Surgery and bordism methods determine existence based on bordism classes.

Complete solutions in certain cases like simply connected manifolds.

Abstract

This is a survey of the current state of the question "Which closed connected manifolds of dimension $n\ge 5$ admit Riemannian metrics whose scalar curvature function is everywhere positive?" The introduction gives a brief overview of these results, while the body of the paper discusses the methods used in the proofs of these results. We mention the two flavors of topological obstructions to the existence of \pscm s: one is a consequence of the Weizenb\"ock formula for the Dirac operator, the other is obtained by considering stable minimal hypersurfaces. We talk about geometric constructions of \pscm s (the surgery/bordism theorem), which shows that the answer to the question above depends only the bordism class of the manifold in a suitable bordism group. Via the Pontryagin-Thom construction this can be translated into stable homotopy theory, and solved completely in some cases, in particular for simply connected manifolds, or manifolds with very special fundamental groups. The last section discusses some open questions.