Finite Group Actions on Smooth 4-Manifolds with Indefinite Intersection Form

Abstract

<p>In this thesis finite cyclic group actions on S² x S² and its moduli space of anti-self-dual connections will be investigated. In the first step the equivariant version of the Donaldson gluing construction of anti-self-dual connections will be developed. We obtain an equivariant obstruction map which provides an equivariant local model of the anti-self-dual moduli space. Then we investigate the special case when we glue the product connection on a trivial SU(2)-bundle over S² x S² with two concentrated anti-instantons. We can achieve transversality of the obstruction map by an equivariant perturbation of the conformal class. We obtain a 10-dimensional equivariant local model which is diffeomorphic to R⁸ x R X S¹. The action on R⁸ is the direct sum of the isotropy representations. The action on the circle depends on the rotation numbers and the self-dual harmonic form. Moreover, there exists an equivariant perturbation of the conformal class so that there are no reducible anti-self-dual connections over S² x S² besides the trivial product connection. These results can be used to show that under certain assumptions the rotation numbers of the isotropy representations of a finite cyclic, smooth action on S² x S² coincide with those of some linear action.</p>