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

Gu, Shijie

doi:10.2140/agt.2021.21.1327

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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

DOI: 10.2140/agt.2021.21.1327

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 ( $n \geq 4$ ) embeds in a compact, locally connected and locally 1–connected metric $n$ –space.

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Keywords

contractible manifold, covering space, trefoil knot, Whitehead double, Whitehead manifold

Mathematical Subject Classification 2010

Primary: 54E45, 54F65, 57M10

Secondary: 57M25, 57N10, 57N15

References

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

Volume 21, issue 3 (2021)