Seiberg-Witten invariants on 3-manifolds with an orientation reversing involution
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[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] This work is devoted to the study of Seiberg-Witten theory for three dimensional manifolds in the presence of involutions. G. Tian and S.Wang explored real Seiberg-Witten invariants under the real structure of a Kahler manifold, which is conjugate linear in a proper sense. Their real structure, however, does not preserve spinc structure. a three-dimensional manifold, there is no such real structure. However, an orientation reversing involution on a three-dimensional spin manifold gives the powerful technique, like the real structure from G. Tian and S. Wang. Inspired by D. Freed, we know that if an orientation reversing involution preserves a pin structure, there always exists a self-adjoint lifting on complex spinor bundle satisfying certain conditions. We call the extension of lifting on spinc structure to a real structure on a three-dimensional spin manifold. My thesis is centered around the issues of Seiberg-Witten equations, the dimension and the orientability of the equivariant Seiberg-Witten moduli space with a real structure in our sense. Moreover, integer valued equivariant Seiberg-Witten invariants can be defined on a three-dimensional manifold with a dividing fixed point set.
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