Positive scalar curvature and homology cobordism invariants

TL;DR

This paper characterizes the Seiberg-Witten Floer stable homotopy type of certain 3-spheres in positive scalar curvature 4-manifolds, providing new obstructions and alternative proofs for existing invariants.

Contribution

It introduces a relative Bauer-Furuta-type invariant for periodic-end 4-manifolds and uses it to identify local equivalence classes and obstructions to positive scalar curvature.

Findings

Determines the local equivalence class of the Floer homotopy type.

Provides obstructions to positive scalar curvature metrics.

Offers alternative proofs for known invariants.

Abstract

We determine the local equivalence class of the Seiberg-Witten Floer stable homotopy type of a spin rational homology 3-sphere $Y$ embedded into a spin rational homology $S^{1} \times S^{3}$ with a positive scalar curvature metric so that $Y$ generates the third homology. The main tool of the proof is a relative Bauer-Furuta-type invariant on a periodic-end 4-manifold. As a consequence, we give obstructions to positive scalar curvature metrics on spin rational homology $S^{1} \times S^{3}$, typically described as the coincidence of various Fr{\o}yshov-type invariants. This coincidence also yields alternative proofs of two known obstructions by Jianfeng Lin and by the authors for the same class of 4-manifolds.