Positive scalar curvature and higher‐dimensional families of Seiberg–Witten equations

We introduce an invariant of tuples of commuting 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.