Nonembeddable contractible open manifolds arising from Whitehead doubling

Abstract

We study a family of genus-one contractible open manifolds constructed by iterated Whitehead doubling from a nontrivial knot $K$ and an even half-twist $m$. For each such pair, we prove that the resulting contractible open $3$-manifold $W(K,m)$ does not embed as an open subset of any compact, locally connected and locally $1$-connected metric $3$-space. We also classify this family: $W(K,m)$ is homeomorphic to $W(K^{\prime },m^{\prime })$ if and only if $m=m^{\prime }$ and $K$ is isotopic to $K^{\prime }$. Thus the knot type and the half-twist parameter form a complete invariant for these nonembeddable contractible open manifolds. The proof combines the topology of ends with JSJ decompositions, hyperbolic pieces, and a rank estimate for iterated Whitehead doubled knot groups. The same method gives infinitely many pairwise non-homeomorphic higher-dimensional examples which embed in no compact, locally connected and locally $1$-connected metric space of the same dimension.

24 pages, 5 figures. Changed the title and abstract, and improved the exposition of the introduction to emphasize a classification of embeddable contractible open manifolds. Gave a direct proof of Theorem 3.12 (Theorem 3.9 in v1)