# Higher order corks

## TL;DR

This paper demonstrates that any finite collection of smooth closed simply-connected 4-manifolds homeomorphic to a given manifold can be generated by a single cork operation, and uses this to distinguish different corks within a fixed 4-manifold.

## Contribution

It introduces a method to generate finite families of 4-manifolds via a single cork and distinguishes embedded corks within a fixed 4-manifold.

## Key findings

- Any finite list of such 4-manifolds can be obtained by a single cork operation.
- The paper provides a way to 'separate' finite families of corks embedded in a fixed 4-manifold.
- It advances understanding of corks and their role in 4-manifold topology.

## Abstract

It is shown that any finite list of smooth closed simply-connected 4-manifolds homeomorphic to a given one X can be obtained by removing a single compact contractible submanifold (or cork) from X, and then regluing it by powers of a boundary diffeomorphism. We then use this result to "separate" finite families of corks embedded in a fixed 4-manifold.