Abstract |
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 whether it is \(S^4\). We show that
\(\Sigma\) is a Gluck twisted \(S^4\) twisted along a 2-knot \(S^{2}\hookrightarrow S^4\); then by this we obtain a 3-handle free handlebody description of \(\Sigma\), and then show \(\Sigma \approx S^4\).
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