Cork twists and automorphisms of $3$-manifolds

Here we study two interesting smooth contractible manifolds, whose boundaries have non-trivial mapping class groups. The first one is a non-Stein contractible manifold, such that every self diffeomorphism of its boundary extends inside; implying that this manifold can not be a loose cork. The second example is a Stein contractible manifold which is a cork, with an interesting cork automorphism $f:\partial W \to \partial W$. By \cite{am} we know that any homotopy $4$-sphere is obtained gluing together two contractible Stein manifolds along their common boundaries by a diffeomorphism. We use the homotopy sphere $\Sigma = -W\smile_{f}W$ as a test case to investigate if it is $S^4$? We show that $\Sigma$ is a Gluck twisted $S^4$ twisted along a $2$-knot $S^{2}\hookrightarrow S^4$; by using this we obtain a $3$-handle free handlebody description of $\Sigma$ and then show $\Sigma \approx S^4$.