×

zbMATH — the first resource for mathematics

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

MSC:
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)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] R. H. Bing, Necessary and sufficient conditions that a 3-manifold be \?³, Ann. of Math. (2) 68 (1958), 17 – 37. · Zbl 0081.39202 · doi:10.2307/1970041 · doi.org
[2] Ralph H. Fox and Emil Artin, Some wild cells and spheres in three-dimensional space, Ann. of Math. (2) 49 (1948), 979 – 990. · Zbl 0033.13602 · doi:10.2307/1969408 · doi.org
[3] G. P. Scott, Compact submanifolds of 3-manifolds, J. London Math. Soc. (2) 7 (1973), 246 – 250. · Zbl 0266.57001 · doi:10.1112/jlms/s2-7.2.246 · doi.org
[4] Arnold Shapiro and J. H. C. Whitehead, A proof and extension of Dehn’s lemma, Bull. Amer. Math. Soc. 64 (1958), 174 – 178. · Zbl 0084.19104
[5] Richard Skora, Cantor sets in \?³ with simply connected complements, Topology Appl. 24 (1986), no. 1-3, 181 – 188. Special volume in honor of R. H. Bing (1914 – 1986). · Zbl 0606.57006 · doi:10.1016/0166-8641(86)90060-X · doi.org
[6] J. H. C. Whitehead, A certain open manifold whose group is unity, Quart. J. Math. (Oxford) 6 (1935), 268-279. · Zbl 0013.08103
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.