Mod 2 Seiberg-Witten invariants of real algebraic surfaces with the opposite orientation

Abstract

Let $X$ be a closed oriented smooth 4-manifold of simple type with $b_{1}(X)=0$ and $b_{+}(X)\geq 2$, and let $\tau : X \longrightarrow X$ generate an involution preserving a spin^c structure $c$. Under certain topological conditions we show in this paper that the Seiberg-Witten invariant $SW(X, c)$ is zero modulo 2. This then enables us to investigate the mod 2 Seiberg-Witten invariants of real algebraic surfaces with the opposite orientation, which is motivated by the Kotschick’s conjecture. The basic strategy is to use the new interpretation of the Seiberg-Witten invariants as a certain equivariant degree of a map constructed from the Seiberg-Witten equa- tions and the generalization of the results of Fang.