Finite group actions on the moduli space of self-dual connections. I

Abstract

Let M M be a smooth simply connected closed 4 4 -manifold with positive definite intersection form. Suppose a finite group G G acts smoothly on M M . Let π : E → M \pi :E \to M be the instanton number one quaternion line bundle over M M with a smooth G G -action such that π \pi is an equivariant map. We first show that there exists a Baire set in the G G -invariant metrics on M M such that the moduli space M ∗ G \mathcal {M}_ * ^G of G G -invariant irreducible self-dual connections is a manifold. By utilizing the G G -transversality theory of T \text {T} . Petrie, we then identify cohomology obstructions to globally perturbing the full space M ∗ {\mathcal {M}_ * } of irreducible self-dual connections to a G G -manifold when G = Z 2 G = {{\mathbf {Z}}_2} and the fixed point set of the Z 2 {\mathbf {Z}}_2 action on M M is a nonempty collection of isolated points and Riemann surfaces.