Contractible open manifolds which embed in no compact, locally connected and locally 1-connected metric space

TL;DR

This paper investigates the embedding properties of a famous contractible open 3-manifold proposed by Bing, demonstrating it cannot embed in certain compact, locally connected, and locally 1-connected metric spaces, and extends these results to higher dimensions.

Contribution

It proves that Bing's contractible open 3-manifold cannot embed in a broad class of compact metric spaces and generalizes this non-embedding result to all higher dimensions.

Findings

Bing's manifold cannot embed in compact, locally connected, locally 1-connected metric 3-spaces.

The non-embedding result extends to all contractible open n-manifolds (n≥4).

Provides counterexamples for embedding in specified metric spaces across dimensions.

Abstract

This paper pays a visit to a famous contractible open 3-manifold $W^3$ proposed by R. H. Bing in 1950's. By the finiteness theorem \cite{Hak68}, Haken proved that $W^3$ can embed in no compact 3-manifold. However, until now, the question about whether $W^3$ can embed in a more general compact space such as a compact, locally connected and locally 1-connected metric 3-space was not known. Using the techniques developed in Sternfeld's 1977 PhD thesis \cite{Ste77}, we answer the above question in negative. Furthermore, it is shown that $W^3$ can be utilized to produce counterexamples for every contractible open $n$-manifold ($n\geq 4$) embeds in a compact, locally connected and locally 1-connected metric $n$-space.