Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-28T07:31:25.850Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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