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

Abstract

We revisit a famous contractible open 3–manifold $W^3$ proposed by R H Bing in the 1950s. By the finiteness theorem, Haken (1968) proved that $W^3$ does not embed in any compact $3$–manifold. However, until now, the question of 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 unknown. Using the techniques developed in Sternfeld’s 1977 PhD thesis, we answer this question in the negative. Furthermore, it is shown that $W^3$ can be utilized to produce counterexamples to the proposition that every contractible open $n$–manifold ($n\ge 4$) embeds in a compact, locally connected and locally 1–connected metric $n$–space.