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A torus reduction theorem for regular coverings of 3-manifolds by homology 3-spheres

Published online by Cambridge University Press:  24 October 2008

E. Luft  and
D. Sjerve
E. Luft
Affiliation:
Department of Mathematics, University of British Columbia, 121–1984 Mathematics Road, Vancouver, B.C., Canada, V6T 1Y4
D. Sjerve
Affiliation:
Department of Mathematics, University of British Columbia, 121–1984 Mathematics Road, Vancouver, B.C., Canada, V6T 1Y4
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Abstract

For any regular covering p:M→M of 3-dimensional manifolds M, M with M a homology 3-sphere we construct a regular covering p′: M′ → M′ of 3-manifolds M′, M′ with the same group of covering transformations and a degree 1 map f:M → M′ so that M′ is a homology 3-sphere, M′ (and hence M′) is irreducible and does not contain incompressible tori, and the regular covering p:M→M is induced from the regular covering p′: M′ → M′ by the map f. Assuming Thurston's geometrization conjecture it follows that M′ (and hence M′) is either hyperbolic or Seifert fibred.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 109 , Issue 1 , January 1991 , pp. 117 - 124
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Davis, J. and Milgram, R.. A Survey of the Spherical Space Form Problem. Math. Reports, vol. 2 (part 2) (Harwood Academic Publ., 1985).Google Scholar
[2]Dunwoody, M.. An equivariant sphere theorem. Bull. London Math. Soc. 17 (1985), 437–448.CrossRefGoogle Scholar
[3]Ewing, J.. Automorphisms of surfaces and class numbers: an illustration of the G-index theorem. In Topological Topics, London Math. Soc. Lecture Note Series vol. 86 (Cambridge University Press, 1983), pp. 120–127.CrossRefGoogle Scholar
[4]Hempel, J.. 3-manifolds. Ann. of Math. Studies no. 86 (Princeton University Press, 1976).Google Scholar
[5]Hilton, P. and Stammbach, U.. A Course in Homological Algebra. Graduate Texts in Math., vol. 4 (Springer-Verlag, 1970).Google Scholar
[6]Lee, R.. Semicharacteristic classes. Topology 12 (1973), 183–199.CrossRefGoogle Scholar
[7]Luft, E. and Sjerve, D.. Regular coverings of homology 3-spheres by homology 3-spheres. Trans. Amer. Math. Soc. 311 (1989), 467–481.Google Scholar
[8]Luft, E. and Sjerve, D.. Degree 1 maps into lens spaces and free cyclic group actions on homology 3-spheres. Topology Appl. (To appear).Google Scholar
[9]Luft, E. and Sjerve, D.. On regular coverings of 3-manifolds by homology 3-spheres. (To appear.)Google Scholar
[10]Milnor, J.. Groups which act on Sn without fixed points. Amer. J. Math. 79 (1957), 623–631.CrossRefGoogle Scholar
[11]Morgan, J.. On Thurston's uniformization theorem for three-dimensional manifolds. In The Smith Conjecture (editors Morgan, J. and Bass, H.), (Academic Press, 1984) pp. 37–125.Google Scholar
[12]Rong, Y.. Degree 1 maps between geometric 3-manifolds. (To appear.)Google Scholar
[13]Scott, P.. The geometries of 3-manifolds. Bull. London Math. Soc. 15 (1983), 401–487.CrossRefGoogle Scholar
[14]Seifert, H. and Threlfall, W.. Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes. Math. Ann. 104 (1930–1931), 1–70.Google Scholar
[15]Symonds, P.. The cohomology representation of an action of Cp on a surface. Trans. Amer. Math. Soc. 308 (1988), 398–400.Google Scholar
[16]Thomas, C. B.. A reduction theorem for free actions by the group Q(8n, k, l) on S 3. Bull. London Math. Soc. 20 (1988), 65–67.CrossRefGoogle Scholar
[17]Thurston, W.. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. 6 (1982), 357–381.CrossRefGoogle Scholar