Notes on Gompf’s Infinite Order Corks

Abstract

For any positive integer n, we give a ${\mathbb{Z}}^{\mathit{n}}$-cork with a ${\mathbb{Z}}^{\mathit{n}}$-effective embedding in a 4-manifold being homeomorphic to $\mathit{E}(\mathit{n})$. This means that a cork gives a subset ${\mathbb{Z}}^{\mathit{n}}$ in the differential structures on $\mathit{E}(\mathit{n})$. Further, we describe handle decompositions of the twisted doubles (homotopy ${\mathit{S}}^4$) of Gompf’s infinite order cork and show that they are log transforms of ${\mathit{S}}^4$.