A smooth variation of Baas-Sullivan Theory and Positive Scalar Curvature

TL;DR

This paper provides a self-contained proof, after inverting 2, that certain manifolds admit positive scalar curvature metrics based on their algebraic topological invariants, extending classical results in differential geometry.

Contribution

It offers a complete proof, with 2 inverted, of the relation between algebraic invariants and positive scalar curvature on spin and non-spin manifolds.

Findings

Established a proof for the existence of positive scalar curvature metrics based on algebraic invariants.

Extended classical results by providing a self-contained proof after inverting 2.

Clarified the topological conditions under which positive scalar curvature can be achieved.

Abstract

Let $M$ be a smooth closed spin (resp. oriented and totally non-spin) manifold of dimension $n\geq 5$ with fundamental group $\pi$. It is stated, e.g. in [RS95], that $M$ admits a metric of positive scalar curvature (pscm) if its orientation class in $ko_n(B\pi)$ (resp. $H_n(B\pi;\Z)$) lies in the subgroup consisting of elements which contain pscm representatives. This is 2-locally verified loc. cit. and in [Sto94]. After inverting 2 it was announced that a proof would be carried out in [Jun], but this work has never appeared in print. The purpose of our paper is to present a self-contained proof of the statement with 2 inverted.