Mod r Vanishing Theorem of Seiberg-Witten Invariant for 4-Manifolds acted by Cyclic Group Z_r

In this paper, a vanishing theorem is stated and proved. If a 4-manifold $M$ admits a smooth action by a cyclic group ${\mathbb{Z}}_r$, then given an ${\mathbb{Z}}_r$-equivariant $Spi{n}^c$-structure $\mathcal{C}$ on $M$, the Seiberg-Witten invariant $SW(\mathcal{C})$ is zero modulo $r$ under some slight assumptions. Here $r$ can be any positive integer. This theorem is a generalization of the mod p vanishing theorem when ${\mathbb{Z}}_p$ is prime order cyclic proved by F.Fang.