Notes on Gompf's infinite order corks

For any positive integer $n$ we give a ${\mathbb{Z}}^n$-cork with a ${\mathbb Z}^n$-effectiveembeddingina4-manifoldbeinghomeomorphicto$E(n)$. This means that a cork gives a subset ${\mathbb{Z}}^n$ in the differential structures on $E(n)$. Further, we describe handle decompositions of the twisted doubles (homotopy $S^4$) of Gompf’s infinite order corks and show that they are Gluck twists and log transforms of $S^4$.