Smooth Group Actions on 4-Manifolds and the Seiberg-Witten Invariants: II

摘要

In this paper the authors study the Seiberg-Witten invariants of 4-manifold with a finite group (or a compact Lie group) acting on. Among other things, the authors will prove the following result: Let X be a smooth closed 4-manifold. Suppose H(sub 1)(X, R) = 0 and b(sup + sub 2) > or = 2, where h(sup +)(sub 2) is the rank of H(sup 2)(sub +). Let C be a Spin(sup c)-structure on X. Assume that C is equivariant with respect to an Z(sub p) action on X, where p is a prime. If Z(sub p) acts on the space H(sup 2)(sub +)(X, R) of harmonic self duel 2-forms trivially. Then the Seiberg-Witten invariant SW(C) = 0(modp) if k(sub i) r(W(sup -)) is the Dirac operator corresponding to C and an equivariant connection A on detW(sup +).