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On 3-dimensional Poincaré duality complexes and 2-knot groups

Published online by Cambridge University Press:  24 October 2008

Jonathan A. Hillman
Jonathan A. Hillman
Affiliation:
School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, Australia
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Abstract

We show that a PD3-complex P such that π = π1(P) is infinite and has a non-trivial finite normal subgroup must be homotopy equivalent to RP2 × S1. Hence if A is an abelian normal subgroup of a 2-knot group πK which is not contained in the commutator subgroup πK′ and πK′ is infinite then A is torsion free.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 2 , September 1993 , pp. 215 - 218
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Chiswell, I. M.. Exact sequences associated with a graph of groups. J. Pure Appl. Algebra 8 (1976), 63–74.CrossRefGoogle Scholar
[2]Dicks, W. and Dunwoody, M. J.. Groups Acting on Graphs. Cambridge Studies in Advanced Mathematics 17 (Cambridge University Press 1989).Google Scholar
[3]Hillman, J. A.. 2-Knots and their Groups. Australian Mathematical Society Lecture Series 5 (Cambridge University Press, 1989).Google Scholar
[4]Hullman, J. A.. 3-Dimensional Poincaré duality groups which are extensions. Math. Z. 195 (1987), 89–92.CrossRefGoogle Scholar
[5]Specker, E.. Die erste Cohomologiegruppe von Überlagerungen und Homotopie-Eigenschaften dreidimensionaler Mannigfaltigkeiten. Comment. Math. Helv. 23 (1949), 303–333.CrossRefGoogle Scholar
[6]Turaev, V. G.. Three-dimensional Poincaré complexes: homotopy classification and splitting. Math. USSR Sbornik 67 (1990), 261–282.CrossRefGoogle Scholar
[7]Wall, C. T. C.. Poincaré complexes. I. Ann. Math. 86 (1967), 213–245.CrossRefGoogle Scholar
[8]Yoshikawa, K.. Certain abelian subgroups of two-knot groups. In Knots 90 (edited by Kawauchi, A.) (W. de Gruyter, 1992), 231–240.Google Scholar