Notes on Gompf's infinite order corks

TL;DR

This paper constructs infinite order corks with ${f Z}^n$-symmetry embedded in elliptic surfaces and analyzes their handle decompositions, revealing they are related to Gluck twists and log transforms of $S^4$.

Contribution

It introduces ${f Z}^n$-corks with effective embeddings in $E(n)$ and describes their handle decompositions as Gluck twists and log transforms.

Findings

Constructed ${f Z}^n$-corks in $E(n)$

Described handle decompositions as Gluck twists

Showed twisted doubles are homotopy $S^4$

Abstract

For any positive integer $n$ we give a ${\mathbb Z}^n$-cork with a ${\mathbb Z}^n$-effective embedding in a 4-manifold being homeomorphic to $E(n)$. 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 $S^4$.