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We give several criteria on a closed, oriented 3-manifold that will implythat it is the boundary of a (simply connected) 4-manifold that admitsinfinitely many distinct smooth structures. We also show that any weaklyfillable contact 3-manifold, or contact 3-manifolds with non-vanishing HeegaardFloer invariant, is the boundary of a simply connected 4-manifolds that admitsinfinitely many distinct smooth structures each of which supports a symplecticstructure with concave boundary, that is there are infinitely many exotic capsfor any such contact manifold.

On 3-manifolds that are boundaries of exotic 4-manifolds