Characterization of knot complements in the n-sphere

Vo-Thanh Liem; Gerard Venema

Fundamenta Mathematicae (1995)

  • Volume: 147, Issue: 2, page 189-196
  • ISSN: 0016-2736

Abstract

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Knot complements in the n-sphere are characterized. A connected open subset W of S n is homeomorphic with the complement of a locally flat (n-2)-sphere in S n , n ≥ 4, if and only if the first homology group of W is infinite cyclic, W has one end, and the homotopy groups of the end of W are isomorphic to those of S 1 in dimensions less than n/2. This result generalizes earlier theorems of Daverman, Liem, and Liem and Venema.

How to cite

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Liem, Vo-Thanh, and Venema, Gerard. "Characterization of knot complements in the n-sphere." Fundamenta Mathematicae 147.2 (1995): 189-196. <http://eudml.org/doc/212083>.

@article{Liem1995,
abstract = {Knot complements in the n-sphere are characterized. A connected open subset W of $S^n$ is homeomorphic with the complement of a locally flat (n-2)-sphere in $S^n$, n ≥ 4, if and only if the first homology group of W is infinite cyclic, W has one end, and the homotopy groups of the end of W are isomorphic to those of $S^1$ in dimensions less than n/2. This result generalizes earlier theorems of Daverman, Liem, and Liem and Venema.},
author = {Liem, Vo-Thanh, Venema, Gerard},
journal = {Fundamenta Mathematicae},
keywords = {knot; n-sphere; complement; homotopy groups of end; knot complements; -sphere; homotopy groups of the end},
language = {eng},
number = {2},
pages = {189-196},
title = {Characterization of knot complements in the n-sphere},
url = {http://eudml.org/doc/212083},
volume = {147},
year = {1995},
}

TY - JOUR
AU - Liem, Vo-Thanh
AU - Venema, Gerard
TI - Characterization of knot complements in the n-sphere
JO - Fundamenta Mathematicae
PY - 1995
VL - 147
IS - 2
SP - 189
EP - 196
AB - Knot complements in the n-sphere are characterized. A connected open subset W of $S^n$ is homeomorphic with the complement of a locally flat (n-2)-sphere in $S^n$, n ≥ 4, if and only if the first homology group of W is infinite cyclic, W has one end, and the homotopy groups of the end of W are isomorphic to those of $S^1$ in dimensions less than n/2. This result generalizes earlier theorems of Daverman, Liem, and Liem and Venema.
LA - eng
KW - knot; n-sphere; complement; homotopy groups of end; knot complements; -sphere; homotopy groups of the end
UR - http://eudml.org/doc/212083
ER -

References

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  12. [12] P. F. Smith, A note on idempotent ideals in group rings, Arch. Math. (Basel) 27 (1976), 22-27. Zbl0318.16003
  13. [13] G. A. Venema, Duality on noncompact manifolds and complements of topological knots, Proc. Amer. Math. Soc., to appear. Zbl0855.57008
  14. [14] G. A. Venema, Local homotopy properties of topological embeddings in codimension two, in: Proc. 1993 Georgia Internat. Topology Conf., to appear. Zbl0889.57012
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