Positive scalar curvature and higher-dimensional families of Seiberg-Witten equations

TL;DR

This paper introduces a new invariant for 4-manifolds based on families of Seiberg-Witten equations, extending previous work and applying it to study positive scalar curvature metrics and homotopy groups.

Contribution

It generalizes Ruberman's invariant to higher-dimensional families and applies this to analyze the topology of positive scalar curvature metrics on 4-manifolds.

Findings

Invariant detects nontrivial elements in homotopy groups

Extension problem for families of 4-manifolds addressed

New tools for studying scalar curvature metrics

Abstract

We introduce an invariant of tuples of commutative diffeomorphisms on a 4-manifold using families of Seiberg-Witten equations. This is a generalization of Ruberman's invariant of diffeomorphisms defined using 1-parameter families of Seiberg-Witten equations. Our invariant yields an application to the homotopy groups of the space of positive scalar curvature metrics on a 4-manifold. We also study the extension problem for families of 4-manifolds using our invariant.