Cork twists and automorphisms of $3$-manifolds

TL;DR

This paper investigates smooth contractible 4-manifolds with boundary mapping class groups, identifies a Stein cork with an automorphism, and proves that a certain homotopy 4-sphere constructed from these manifolds is diffeomorphic to the standard 4-sphere.

Contribution

It introduces a Stein cork with a non-trivial automorphism and demonstrates that a homotopy 4-sphere formed by gluing along this cork is actually the standard 4-sphere.

Findings

The Stein cork has a non-trivial automorphism affecting boundary diffeomorphisms.

The constructed homotopy 4-sphere is diffeomorphic to S^4.

The paper provides a handlebody description of the homotopy sphere.

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 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 $\Sigma \approx S^4$.