Homotopy $4$-spheres associated to an infinite order loose cork

TL;DR

This paper proves that a family of homotopy 4-spheres constructed via infinite order loose corks are actually standard 4-spheres, using Gluck twists and handle cancellation techniques.

Contribution

It demonstrates that certain infinite order loose corks produce homotopy spheres diffeomorphic to the standard 4-sphere, expanding understanding of corks and smooth structures.

Findings

Homotopy spheres $ abla_{n}$ are $S^4$.

$ abla_{n}$ are obtained by Gluck twisting $S^4$.

Handle cancellations show $ abla_{n}$ are standard.

Abstract

We show the homotopy spheres $\Sigma_{n} = -W\smile_{f^{n}}W$, formed by doubling the infinite order loose-cork $(W,f)$ by iterates of the cork diffeomorphism $f: \partial W \to \partial W$ is $S^4$. To do this we first show that $\Sigma_{n} $ are obtained by Gluck twistings of $S^4$; then from this we show how to cancel $3$-handles of $\Sigma_{n}$ and identify it by $S^{4}$.