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

We revisit a famous contractible open 3–manifold W3 proposed by R H Bing in the 1950s. By the finiteness theorem, Haken (1968) proved that W3 does not embed in any compact 3–manifold. However, until now, the question of whether W3 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 W3 can be utilized to produce counterexamples to the proposition that every contractible open n–manifold (n 4) embeds in a compact, locally connected and locally 1–connected metric n–space.

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