Infinite order corks via handle diagrams

TL;DR

This paper provides handle diagrams for infinite order corks in 4-manifolds, translating previous abstract proofs into visual representations and analyzing conditions under which such twists alter manifold diffeomorphism types.

Contribution

It introduces the first handle diagrams for infinite order corks and relates torus twists to $\delta$-moves, advancing understanding of cork phenomena in 4-manifold topology.

Findings

Handle diagrams for infinite order corks are constructed.

Conditions are given for when torus twists do not change the manifold.

$\delta$-moves are shown to be equivalent to torus twists.

Abstract

The author recently proved the existence of an infinite order cork: a compact, contractible submanifold $C$ of a 4-manifold and an infinite order diffeomorphism $f$ of $\partial C$ such that cutting out $C$ and regluing it by distinct powers of $f$ yields pairwise nondiffeomorphic manifolds. The present paper exhibits the first handle diagrams of this phenomenon, by translating the earlier proof into this language (for each of the infinitely many corks arising in the first paper). The cork twists in these papers are twists on incompressible tori. We give conditions guaranteeing that such twists do not change the diffeomorphism type of a 4-manifold, partially answering a question from the original paper. We also show that the "$\delta$-moves" recently introduced by Akbulut are essentially equivalent to torus twists.