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Finite group actions on the moduli space of self-dual connections. I. (English) Zbl 0724.57013
Let M be a simply-connected, smooth 4-manifold with a positive definite intersection form and $${\mathcal M}$$ be the moduli space of SU(2) instantons over M with Chern number 1. It is known that for generic Riemannian metrics on M the subspace $${\mathcal M}^*\subset {\mathcal M}$$ representing irreducible connections is a smooth 5-manifold, and itself is obtained by adding a finite number of singular points, around which $${\mathcal M}$$ is modelled on a cone over $$CP^ 2$$. The author extends the theory to the case when a finite group G acts on M. For a G-invariant metric on M there is a natural induced G action on $${\mathcal M}$$, with fixed point set $${\mathcal M}^ G$$ representing invariant connections. It is shown first that for generic G-invariant metrics $${\mathcal M}^ G$$ is a manifold except for a finite number of cone singularities. The G-index theorem is used to compute the dimension of $${\mathcal M}^ G$$. The other main topic is to seek G-invariant perturbations of the instanton equations which deform $${\mathcal M}$$ into a G-space with standard singularities. It is shown that these exist if a certain topological obstruction in $$H^ 3({\mathcal M}^*,{\mathcal M}^*_ 0)$$ vanishes, where $${\mathcal M}^*_ 0$$ is the complement of a large compact set in $${\mathcal M}^*$$.

MSC:
 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010)
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References:
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