Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Brunnian spanning 3-disks for the 2-unlink in the 4-sphere

Weizhe Niu

Abstract

We show that the $2$-component unlink in $S^4$ admits infinitely many isotopy classes of spanning $3$-disks that are Brunnian.

Brunnian spanning 3-disks for the 2-unlink in the 4-sphere

Abstract

We show that the -component unlink in admits infinitely many isotopy classes of spanning -disks that are Brunnian.
Paper Structure (5 sections, 10 theorems, 55 equations, 7 figures)

This paper contains 5 sections, 10 theorems, 55 equations, 7 figures.

Table of Contents

  1. Introduction
  2. From spanning disks to embedded $3$-balls in $X$
  3. The Dax invariant and barbell generators
  4. Proof of Theorem A
  5. Proof of Theorem \ref{['cor:B']}

Key Result

Theorem 1

For $k\ge 1$, the barbell diffeomorphisms $\Phi_{\mathcal{B}(t\nu_B\nu_R tu^kt^{-1})}\in \pi_0\mathrm{Diff}(\#_2S^1\times D^3,\partial)$ are non-trivial and pairwise non-isotopic. Moreover, they are detected by the induced invariants $W_3'^{\Delta_i}$ (equivalently, by $W_3^{\Delta_i}$ via the const

Figures (7)

  • Figure 1: The barbell $\mathcal{B}(t\nu_B\nu_R tut^{-1})$ embedded in $\natural_2 S^1\times D^3\subset \#_2 S^1\times D^3$.
  • Figure 2: The relationship between $S^1\times D^3 \natural S^1\times D^3$ and $S^1\times D^3 \# S^1\times D^3$.
  • Figure 3: Representation of the connected-sum sphere $S$ by a loop of embedded intervals.
  • Figure 4: The element $t^3 u^3 t^2$ in $\pi_1\mathrm{Emb}(I,X)$.
  • Figure 5: The embedded barbell induced by $t^3 u^3 t^2$ in $\pi_1\mathrm{Emb}(I,X)$.
  • ...and 2 more figures

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Remark
  • Theorem 3.1
  • Definition 3.2: Admissible pairs
  • Proposition 3.3
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 9 more