Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-28T16:27:31.305Z Has data issue: false hasContentIssue false
  • English
  • Français

Parallelizable manifolds and the fundamental group

Part of: Low-dimensional topology

Published online by Cambridge University Press:  26 February 2010

F. E. A. Johnson  and
J. P. Walton
F. E. A. Johnson
Affiliation:
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT.
J. P. Walton
Affiliation:
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT.
Get access

Extract

§0. Introduction. Low-dimensional topology is dominated by the fundamental group. However, since every finitely presented group is the fundamental group of some closed 4-manifold, it is often stated that the effective influence of π1 ends in dimension three. This is not quite true, however, and there are some interesting border disputes. In this paper, we show that, by imposing the extra condition of parallelizability on the tangent bundle, the dominion of π1 is extended by an extra dimension.

MSC classification

Secondary: 57M50: Geometric structures on low-dimensional manifolds
Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 165 - 172
Copyright
Copyright © University College London 2000

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

1.Epstein, D. B. A.. Finite presentations of groups and 3-manifolds. Quart. J. Math., 12 (1961), 205–212.CrossRefGoogle Scholar
2.Hausmann, J. C. and Weinberger, S.. Caractéristiques d'Euler et groupes fondamentaux des variétés de dimension 4. Comment. Math. Helv., 60 (1985), 139–144.CrossRefGoogle Scholar
3.Higman, G.. Subgroups of finitely presented groups. Proc. Row Soc. (Ser. A), 262 (1961), 455–475.Google Scholar
4.Johnson, F. E. A. and Kotschick, D.. On the signature and Euler characteristic of certain fourmanifolds. Math. Proc. Camb. Phil. Soc., 114 (1993), 431–438.CrossRefGoogle Scholar
5.Kahzdan, D. A.. On the connection between the dual space of a group and the structure of its closed subgroups. Function. Anal. Appl., 1 (1967), 63–65.CrossRefGoogle Scholar
6.Katok, S.. Fuchsian Groups (University of Chicago Press, Chicago, 1992).Google Scholar
7.Kervaire, M. A.. Relative characteristic classes. Amer. J. Math., 79 (1957), 517–558.CrossRefGoogle Scholar
8.Kervaire, M. A.. Some non-stable homotopy groups of Lie groups. Illinois J. Math., 4 (1960), 161–169.CrossRefGoogle Scholar
9.Kervaire, M. A. and Milnor, J. W.. Groups of homotopy spheres I. Ann. Math., 77 (1963), 504–537.CrossRefGoogle Scholar
10.Kotshick, D.. Four-manifold invariants of finitely presented groups. Preprint (University of Basle, 1993).Google Scholar
11.Lott, J.. Deficiencies of lattice subgroups of Lie groups. Preprint (University of Michigan, 1997).Google Scholar
12.Lubotzky, A.. Group presentation, p-adic analytic groups and lattices in SL2(C). Ann. Math., 118 (1983), 115–130.CrossRefGoogle Scholar
13.Matsushima, Y.. On the first Betti number of compact quotient spaces of higher dimensional symmetric spaces. Ann. Math., 75 (1962), 312–330.CrossRefGoogle Scholar
14.Paechter, G. F.. The groups πr(Vn, m). Quart. J. Math. (Ser. 2), 7 (1956), 249–268.CrossRefGoogle Scholar
15.Wang, S. P.. The dual space of semisimple Lie groups. Amer. J. Math., 91 (1969), 921–937.CrossRefGoogle Scholar
16.Winkelnkemper, H. E., Un teorema sobre variedades de dimension 4. Acta Mexicana Ci. Tecn., 2 (1968), 88–89.Google Scholar