Spin structures on the Seiberg-Witten moduli spaces

Let $M$ be an oriented closed 4-manifold and $\text{cL}$ be a $spi{n}^c$ structure on $M$. In this paper we prove that under a suitable condition the Seiberg-Witten moduli space has a canonical spin structure and its spin bordism class is an invariant for $M$. We show that the invariant for M=#_{j=1}^l M_j is not zero, where each $M_j$ is a $K3$ surface or a product of two oriented closed surfaces with odd genus and $l$ is 2 or 3. As a corollary, we obtain the adjunction inequality for $M$. Moreover we show that M # N does not admit Einstein metric for some $N$ with $b^{+}(N)=0$.