Equivariant Seiberg-Witten theory

Abstract

We introduce and study equivariant Seiberg-Witten invariants for $4$-manifolds equipped with a smooth action of a finite group $G$. Our invariants come in two types: cohomological, valued in the group cohomology of $G$ and $K$-theoretic, valued in the representation ring of $G$. We establish basic properties of the invariants such as wall-crossing and vanishing of the invariants for $G$-invariant positive scalar curvature metrics. We establish a relation between the equivariant Seiberg-Witten invariants and families Seiberg-Witten invariants. Sufficient conditions are found under which equivariant transversality can be achieved leading to smooth moduli spaces on which $G$ acts. In the zero-dimensional case this yields a further invariant of the $G$-action valued in a refinement of the Burnside ring of $G$. We prove localisation formulas in cohomology and $K$-theory, relating the equivariant Seiberg-Witten invariants to moduli spaces of $G$-invariant solutions. We give an explicit formula for the invariants for holomorphic group actions on K\"ahler surfaces. We also prove a gluing formula for the invariants of equivariant connected sums. Various applications and consequences of the theory are considered.