We solve the problem posed by Boyer and Galicki about the existence of
K-contact simply connected manifolds with no Sasakian structure. Although the
result lies in the framework of metric contact geometry, our methods come from
contact and symplectic geometry and are based on the method of fat bundles
developed by Sternberg, Weinstein and Lerman
ABSTRACT. We construct smooth circle actions on symplectic manifolds with non-symplectic fixed point sets or cyclic isotropy sets. All such actions are not compatible with any symplectic form on the manifold in question. In order to cover the case of non-symplectic fixed point sets, we use two smooth 4-manifolds (one symplectic and one non-symplectic) which become diffeomorphic after taking the products with the 2-sphere. The second type of actions is obtained by constructing smooth circle actions on spheres with non-symplectic cyclic isotropy sets, which (by the equivariant connected sum construction) we carry over from the spheres on products of 2-spheres. Moreover, by using the mapping torus construction, we show that periodic diffeomorphisms (isotopic to symplectomorphisms) of symplectic manifolds can provide examples of smooth fixed point free circle actions on symplectic manifolds with non-symplectic cyclic isotropy sets.
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