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) >= 2, and let tau : X -> 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 equations and the generalization of the results of Fang.