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Engulfing and finitely generated groups. (English) Zbl 0607.57008
The examples of contractible, open 3-manifolds, M, not homeomorphic to \({\mathbb{R}}^ 3\) are distinguished by the existence of a piecewise-linear simple loop, \(K\subset M\), which cannot be engulfed in M (does not lie in the interior of a piecewise-linear homotopy 3-ball in M). The author observes that in these examples \(\pi_ 1(M-K)\) is not finitely generated and proves that this is necessarily the case: If M is a simply connected 3-manifold and K is a piecewise-linear simple loop in Int(M), then K can be engulfed in M if and only if \(\pi_ 1(M-K)\) is finitely generated.
Reviewer: J.Hempel

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M05 Fundamental group, presentations, free differential calculus
57M40 Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
Full Text: DOI
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