Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-25T14:33:47.141Z Has data issue: false hasContentIssue false
  • English
  • Français

On groups and 3-manifolds with weak dimension ≤ 1

Published online by Cambridge University Press:  24 October 2008

David E. Galewski
Get access Rights & Permissions [Opens in a new window]

Extract

0. Introduction. A group π has weak dimension (wd) ≤ n (see Cartan and Ellen-berg (2)) if Hk(π, A) = 0 for all right Z(π)-modules A and all k > n. We say that the weak dimension of a manifold M is ≤ n if wd (πl(M))≤ n. In section 1 we show that open, orientable, irreducible 3-manifolds have wd ≤ 1 if and only if they are the monotone on of 1-handle bodies. In his celebrated theorem (10), Stallings proves that finitely presented groups of cohomological dimensions ≤ 1 are free. In section 2 we prove that if π is a finitely presented group which is the fundamental group of any orientable 3-manifold with wd ≤ 1 then π is free. We also give an example to show that the finite generation of π is necessary. (Swan (11) removes the finitely presented hypothesis from Stalling's theorem.) Finally, in section 3 we generalize a theorem of McMillan (5) and prove that if M is an open, orientable, irreducible 3-manifold with finitely generated fundamental group, then M is stably (taking the product with n ≥ 1 copies of ℝ) a connected sum along the boundary of trivial (n+2)-disc Sl bundles.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 75 , Issue 1 , January 1974 , pp. 33 - 35
Copyright
Copyright © Cambridge Philosophical Society 1974

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)Alexander, J. W.An example of a simply connected surface bounding a region which is not simply connected. Proc. Nat. A. Sci. 10 (1924), 8–10.CrossRefGoogle Scholar
(2)Cartan, H. and Elenberg, S.Homological algebra (Princeton University Press, 1956).Google Scholar
(3)Tatsuo, Homma. On the embedding of polyhedra in manifolds. Yokohama Math. J. 10 (1962), 5–10.Google Scholar
(4)Hudson, J. F. P.Piecewise linear topology (New York, Benjamin, 1969).Google Scholar
(5)McMillan, D. R.Cartesian products of contractible open manifolds. Bull. Amer. Math. Soc. 67 (1961), 510–514.CrossRefGoogle Scholar
(6)McMillan, D. R. Canonical neighborhoods in three-manifolds, Topology Seminar Wisconsin, 1965 (Princeton University Press), 1966.Google Scholar
(7)Milnor, J.A unique decomposition theorem for 3-manifolds. Amer. J. Math. 84 (1962), 1–7.CrossRefGoogle Scholar
(8)Siebenmann, L. C. The obstruction to finding a boundary for an open manifold of dimension greater than five. Thesis, Princeton University (1965).Google Scholar
(9)Stallings, J. R.The piecewise-linear structure of Euclidean space. Proc. Cambridge Philos. Soc. 58 (1962), 481–488.CrossRefGoogle Scholar
(10)Stallings, J. R.On torsion free groups with infinitely many ends. Ann. of Math. 88 (1968), 312–334.CrossRefGoogle Scholar
(11)Swan, R. G.Groups of cohomological dimension one. J. Algebra 12 (1969), 585–601.CrossRefGoogle Scholar
(12)Wall, C. T. C.Finiteness conditions for CW-complexes. Ann. of Math. (2) 81 (1965), 56–69.CrossRefGoogle Scholar