Perturbation of equivariant moduli spaces (Q810948)

Let \(X^ 4\) be a smooth, oriented closed 4-manifold with a smooth action of a finite group \(\pi\), preserving the orientation. If \(P\to X\) is an SU(2)-bundle, then the space \({\mathcal A}/{\mathcal G}\) of gauge equivalence classes of connections on P inherits a \(\pi\)-action. We choose a \(\pi\)- invariant Riemannian metric on X. Then the Yang-Mills functional associated to this metric is invariant with respect to the group action on \({\mathcal A}/{\mathcal G}\), and hence the moduli spaces of self-dual or anti- self-dual connections, up to gauge equivalence, also have a \(\pi\)-action. In this paper, an equivariant perturbation (\({\mathcal M},\pi)\) of this moduli space is constructed using the method of \textit{E. Bierstone} [Trans. Am. Math. Soc. 234, 447-466 (1977; Zbl 0318.57044)]. We give two applications of this moduli space to smooth finite group actions on positive definite four-manifolds. The first (Theorem B), where we determine the rotation numbers of a cyclic group of odd order acting smoothly on \(P^ 2(C)\), was earlier proved by \textit{A. Edmonds} and \textit{J. Ewing} [Topology 28, 211-223 (1989; Zbl 0682.57021)] using the G- signature theorem, number theory and a formidable computer assisted calculation. Our argument relies on the Whitney stratification of our equivariant moduli space to show that certain fixed sets are smooth submanifolds with linear normal bundles.