Spin Structures on Seiberg-Witten Moduli Spaces
Vol. 13 (2006), No. 3, Page 347--363.
Sasahira, Hirofumi
Spin Structures on Seiberg-Witten Moduli Spaces
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Abstract:
Let $M$ be an oriented closed $4$-manifold with a spin$^c$ structure $\cL$. In this paper we prove that under a suitable condition for $(M,\cL)$ the Seiberg-Witten moduli space has a canonical spin structure and its spin bordism class is an invariant of $M$. We show that the invariant of $M=\#_{j=1}^l M_j$ is non-trivial for some spin$^c$ structure when $l$ is $2$ or $3$ and each $M_j$ is a $K3$ surface or a product of two oriented closed surfaces of odd genus. As a corollary, we obtain the adjunction inequality for the $4$-manifold. Moreover we calculate the Yamabe invariant of $M \# N_1$ for some negative definite $4$-manifold $N_1$. We also show that $M \# N_2$ does not admit an Einstein metric for some negative definite $4$-manifold $N_2$.
Mathematics Subject Classification (2000): Primary 57R57; Secondary 53C25.
Mathematical Reviews Number: MR2284407
Received: 2005-11-17
