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Spanning 3-discs in the 4-sphere pushed into the 5-disc

Mark Powell

TL;DR

The paper proves that any two smooth spanning 3-disc collections for the trivial 2-link in $S^4$, once pushed into $D^5$, become smoothly isotopic rel boundary. The author employs surgery theory on the pushed-in exteriors, constructing degree-one normal maps to a model space and analyzing the obstruction via $L$-theory for the free group $F_m$, showing the relative structure set is trivial. This yields an $s$-cobordism between exteriors and, after gluing back the ambient pieces, a diffeomorphism of the pushed-in complements that induces a 1-parameter isotopy between the original spanning discs in $D^5$. The result extends the known $m=1$ case to arbitrary $m$, aligns with Brunnian-type phenomena and Budney–Gabai-type examples, and demonstrates a smooth isotopy in high dimensions even with multiple components.

Abstract

I prove that any two smooth collections of spanning 3-discs for the trivial 2-link in $S^4$ become smoothly isotopic rel. boundary after pushing them into $D^5$.

Spanning 3-discs in the 4-sphere pushed into the 5-disc

TL;DR

The paper proves that any two smooth spanning 3-disc collections for the trivial 2-link in , once pushed into , become smoothly isotopic rel boundary. The author employs surgery theory on the pushed-in exteriors, constructing degree-one normal maps to a model space and analyzing the obstruction via -theory for the free group , showing the relative structure set is trivial. This yields an -cobordism between exteriors and, after gluing back the ambient pieces, a diffeomorphism of the pushed-in complements that induces a 1-parameter isotopy between the original spanning discs in . The result extends the known case to arbitrary , aligns with Brunnian-type phenomena and Budney–Gabai-type examples, and demonstrates a smooth isotopy in high dimensions even with multiple components.

Abstract

I prove that any two smooth collections of spanning 3-discs for the trivial 2-link in become smoothly isotopic rel. boundary after pushing them into .

Paper Structure

This paper contains 3 sections, 5 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Using surgery to compare pushed-in 3-disc exteriors
  3. Proof of Theorem \ref{['thm']}

Key Result

Theorem A

Let $D_m^0$ and $D_m^1$ be spanning 3-disc collections for the trivial 2-link $U_m$ in $S^4$. Then including $S^4 \subseteq D^5$, $D^0_m$ and $D^1_m$ become smoothly isotopic in $D^5$, relative to $U_m$.

Theorems & Definitions (13)

  • Theorem A
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • ...and 3 more