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Which Abelian Groups Can be Fundamental Groups of Regions in Euclidean Spaces?

  • Bai Ching Chang (a1)

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It is known that there are a lot of properties of the group of a knot in S 3 which fail to generalize to the group of a knotted sphere in S 4; among them are included Dehn's lemma, Hopf's conjecture, and the aspherity of knots. In this paper, we shall investigate the properties of the fundamental groups of regions in S 3 and in S 4, with examples to show that they are not quite the same. Some special consideration will be given to regions that are the complements in S 3 or in S 4 of a finite number of tamely imbedded manifolds of co-dimension 2, and, more generally, to regions that are the complements of subcomplexes in S 3 or in S 4.

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References

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1. Chang, Bai Ching, Which abelian groups can be fundamental groups of regions in Euclidean spaces? Ph.D. Thesis, Princeton University, 1971.
2. Conner, P. E., On the action of a finite group on Sn × Sn , Ann. of Math. 66 (1957), 586588.
3. Eilerberg, S. and Steenrod, N., Foundations of algebraic topology (Princeton University Press, Princeton, 1952).
4. Evan, B. and L. Moser, Solvable fundamental groups of compact 3-manifolds, Trans Amer. Math. Soc. 168 (1972), 189210.
5. Fox, R. H., On the imbedding of polyhedra in 3-space, Ann. of Math. 49 (1948), 462470.
6. Fox, R. H., A quick trip through knot theory, Topology of 3-manifold and related topics (Prentice Hall, New York, 1961).
7. Gutierrez, M., Boundary links and an unlinking theorem (to appear in Trans. Amer. Math. Soc).
8. Papakyriakopoulos, C. D., On Dehn's Lemma and the asphericity of knots, Ann. of Math. 66 (1957), 126.
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Which Abelian Groups Can be Fundamental Groups of Regions in Euclidean Spaces?

  • Bai Ching Chang (a1)

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