### Local homology of groups of volume-preserving diffeomorphisms, II.

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We discuss conditions under which a symplectic 4-manifold has a compatible Kähler structure. The theory of $J$-holomorphic embedded spheres is extended to the immersed case. As a consequence, it is shown that a symplectic 4-manifold which has two different minimal reductions must be the blow-up of a rational or ruled surface.

We apply Gromov’s method of convex integration to problems related to the existence and uniqueness of symplectic and contact structures

Necessary conditions are found for a Cantor subset of the circle to be minimal for some ${C}^{1}$-diffeomorphism. These conditions are not satisfied by the usual ternary Cantor set.

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