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A group with zero homology

Published online by Cambridge University Press:  24 October 2008

D. B. A Epstein
D. B. A Epstein
Affiliation:
University of Warwick
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Extract

In this paper we describe a group G such that for any simple coefficients A and for any i > 0, Hi(G; A) and Hi(G; A) are zero. Other groups with this property have been found by Baumslag and Gruenberg (1). The group G in this paper has cohomological dimension 2 (that is Hi(G; A) = 0 for any i > 2 and any G-module A). G is the fundamental group of an open aspherical 3-dimensional manifold L, and is not finitely generated. The only non-trivial part of this paper is to prove that the fundamental group of the 3-manifold L, which we shall construct, is not the identity group.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 64 , Issue 3 , July 1968 , pp. 599 - 602
Copyright
Copyright © Cambridge Philosophical Society 1968

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References

REFERENCES

(1)Baumslag, G. and Gruenberg, K. W.Some reflections on cohomological dimension and freeness. Journal of Algebra, 6 (1967), 394–409.CrossRefGoogle Scholar
(2)Epstein, D. B. A.Free products with amalgamation and 3-manifolds. Proc. Amer. Math. Soc. 12 (1961), 669–670.Google Scholar
(3)Papakyriakopoulos, C. D.On Dehn's Lemma and the asphericity of knots. Ann. of Math. 66 (1957), 1–26.CrossRefGoogle Scholar