#### Volume 21, issue 3 (2021)

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

### Shijie Gu

Algebraic & Geometric Topology 21 (2021) 1327–1350
##### 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.

##### Keywords
contractible manifold, covering space, trefoil knot, Whitehead double, Whitehead manifold
##### Mathematical Subject Classification 2010
Primary: 54E45, 54F65, 57M10
Secondary: 57M25, 57N10, 57N15
##### Publication
Received: 15 February 2019
Revised: 4 May 2020
Accepted: 1 June 2020
Published: 11 August 2021
##### Authors
 Shijie Gu Department of Mathematical Sciences Central Connecticut State University New Britain, CT United States