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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Finite group actions on the moduli space of self-dual connections. I


Author: Yong Seung Cho
Journal: Trans. Amer. Math. Soc. 323 (1991), 233-261
MSC: Primary 58D15; Secondary 53C05, 57S17, 58B20, 58G10
MathSciNet review: 1010409
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Abstract: Let $ M$ be a smooth simply connected closed $ 4$-manifold with positive definite intersection form. Suppose a finite group $ G$ acts smoothly on $ M$. Let $ \pi :E \to M$ be the instanton number one quaternion line bundle over $ M$ with a smooth $ G$-action such that $ \pi$ is an equivariant map. We first show that there exists a Baire set in the $ G$-invariant metrics on $ M$ such that the moduli space $ \mathcal{M}_ * ^G$ of $ G$-invariant irreducible self-dual connections is a manifold. By utilizing the $ G$-transversality theory of T. Petrie, we then identify cohomology obstructions to globally perturbing the full space $ {\mathcal{M}_ * }$ of irreducible self-dual connections to a $ G$-manifold when $ G = {{\mathbf{Z}}_2}$ and the fixed point set of the $ {\mathbf{Z}}_2$ action on $ M$ is a nonempty collection of isolated points and Riemann surfaces.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1010409-2
PII: S 0002-9947(1991)1010409-2
Keywords: Group action, self-dual connection, moduli space, generic metric, Atiyah-Singer $ G$-index, obstruction class, $ G$-equivariant perturbation
Article copyright: © Copyright 1991 American Mathematical Society