Unknotting 3-balls in the 5-ball

TL;DR

This paper proves that any 3-ball embedded in the 4-sphere can be isotoped into the 5-ball relative to its boundary, confirming a conjecture about embeddings in higher dimensions.

Contribution

It provides an affirmative answer to a question about isotopy of 3-balls in the 4-sphere when embedded in the 5-ball, advancing understanding of high-dimensional embeddings.

Findings

Every 3-ball in the 4-sphere becomes isotopic in the 5-ball

Supports conjectures about embeddings and isotopies in higher dimensions

Clarifies the relationship between embeddings in 4- and 5-dimensional spaces

Abstract

The purpose of this note is to answer affirmatively a question posed by both Gay, and Hughes, Kim and Miller as to whether every $3$--ball smoothly embedded in the $4$--sphere becomes isotopic relative to the bounding 2-sphere when pushed into the 5-ball.