We give several criteria on a closed, oriented 3 3 -manifold that will imply that it is the boundary of a (simply connected) 4 4 -manifold that admits infinitely many distinct smooth structures. We also show that any weakly fillable contact 3 3 -manifold, or contact 3 3 -manifold with non-vanishing Heegaard Floer invariant, is the boundary of a simply connected 4 4 -manifold that admits infinitely many distinct smooth structures each of which supports a symplectic structure with concave boundary, that is there are infinitely many exotic caps for any such contact manifold.

In this paper we will show how to classify Legendrian and transverse knots in the knot type of "sufficiently positive" cables of a knot in terms of the classification of the underlying knot. We will also completely explain the phenomena of "Legendrian large" cables. These are Legendrian representatives of cables that have Thurston-Bennequin invariant larger that the framing coming from the cabling torus. Such examples have only recently, and unexpectedly, been found. We will also give criteria that determines the classification of Legendrian and transverse knots the the knot type of negative cables.

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