$\Bbb Z_p$-equivariant Spin$^c$-structures

Abstract

Let $X$ be a closed, oriented, Riemannian 4-manifold with $b_2^+(X)>1$ and of simple type. Suppose that $\sigma : X\to X$ is an involution preserving orientation with an oriented, connected, compact 2-dimensional submanifold $\Sigma$ as a fixed point set with $\Sigma\cdot\Sigma \ge 0$ and $[\Sigma]\ne 0 \in H_2 (X ; \Bbb Z).$ We show that if $\chi (\Sigma)+\Sigma\cdot\Sigma \ne 0$ then the Spin$^c$ bundle $\tilde P$ is not $\Bbb Z_2$-equivariant, where $\det\tilde P =L$ is a basic class with $c_1 (L)[\Sigma]=0$.