Whitehead doubling, rank estimate and nonembeddability of contractible open manifolds

TL;DR

This paper proves a linear lower bound for the rank of the n-th iterated Whitehead double knot group, showing r(WD^n(K)) ≥ n+1 for any nontrivial K, by analyzing the JSJ decomposition and the hyperbolic pieces (via Weidmann’s result). Leveraging this bound, the authors construct contractible open manifolds W(K,m) whose end-structure and knot-theoretic data prevent embedding into any compact locally connected and locally 1-connected metric space, a phenomenon persisting in higher dimensions. They establish a sharp homeomorphism classification for W(K,m), proving W(K,m) ≅ W(K',m') only when m=m' and K isotopic to K'. The work extends Gu21 to produce infinitely many non-homeomorphic contractible open manifolds in every dimension n≥3 that do not embed in the specified compact spaces, using end-structure and Alexander-polynomial arguments to distinguish twisting and knot data.

Abstract

Let $K$ be a nontrivial knot. For each $n\in \mathbb{N}$, we prove that the rank of its $n$th iterated Whitehead doubled knot group $π_1(S^3 \setminus \operatorname{WD}^n(K))$ is bounded below by $n+1$. As an application, we show that there exist infinitely many non-homeomorphic contractible open $n$-manifolds ($n\geq 3$) which cannot embed in a compact, locally connected and locally 1-connected $n$-dimensional metric space.

Figures (5)

• Figure 1: A knot $K_W$ with a trefoil knot as companion.
• Figure 2: The Whitehead link complement in $S^3$ is homeomorphic to the complement of a twist knot with $m$ half-twists in a solid torus, where $m \in 2\mathbb{Z}$.
• Figure 3: $L_l = T_l \setminus \operatorname{Int} T_l' = T_{l}^{\ast}\setminus \operatorname{Int} T_{l-1}^{\ast}$. The "inner" boundary component of $L_l$ is $\partial T_l'$. The "outer" boundary component of $L_l$ is $\partial T_l$. The box $m$ represents $m$ half-twists.
• Figure 4: This picture illustrates the embedding of $T_{l}^{\ast}$ relative to $T_{l+1}^{\ast}$. The $\epsilon$-neighborhoods of $T_{l}^{\ast}$ and $\mathbf{B}_l$ are not shown.
• Figure 5: $L^\ast = T \setminus T'$. The "inner" boundary component of $L^\ast$ is $\partial T'$. The "outer" boundary component of $L^\ast$ is $\partial T$.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Theorem \ref{['Thm: homeomorphism of manifolds']}
• Definition 2.1
• Definition 2.2
• Remark 1
• Lemma 2.1
• proof
• Lemma 2.2