Smooth Group Actions on 4-Manifolds and the Seiberg-Witten Theory: I

摘要

The main purpose of this paper is to give some applications of the Seiberg-Wittentheory to the study of smooth group action on spin 4-manifold. Following Furuta, the authors' strategy is to use the 'Finite dimensional approximation' to the unperturbed Sieberg-Wittin equations. Let X be a spin, closed Riemannian 4-manifold. Let c be the spin structure on X. Given a finite group G action on X preserving the spin structure c and the isometries, there is a lifted action to the spinor bundles W(sup +) and W(sup -). Sometimes the lifted action of W(sup + or -) must have higher order. But an extension of G by Z(sub 2) can always be lifted. This implies that for an odd order group G, the action on X lifts to a G action on W(sup + or -).