Gluck twists of $S^4$ are diffeomorphic to $S^4$

Mark C. Hughes, Seungwon Kim, Maggie Miller

Published 2018-04-24Version 1

We prove that every Gluck twist on an embedded $2$-sphere in $S^4$ is diffeomorphic to $S^4$. We accomplish this by showing that a genus-$g$ surface in $\mathbb{C}P^2$ intersecting the standard $\mathbb{C}P^1$ in a single positive intersection point is isotopic to $\mathbb{C}P^1$ trivially stabilized $g$ times. These results provide answers to questions of Kirby (4.24) and Gabai, as well as a partial answer to a question of Melvin.