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

Abstract

We show the homotopy spheres ${\Sigma }_n=-W{⌣}_{f^n}W$, 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 $S^4$.

17 pages, 26 figures