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Finite group actions on homologically peripheral 3-manifolds

Published online by Cambridge University Press:  10 June 2011

TORU IKEDA
TORU IKEDA*
Affiliation:
Department of Mathematics, Faculty of Science, Kochi University, 2-5-1 Akebono-cho, Kochi 780-8520, Japan. e-mail: [email protected]
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Abstract

We generalize the notion of totally peripheral 3-manifolds to define homologically peripheral 3-manifolds. The homologically peripheral property survives canonical decompositions of 3-manifolds as well as it defines a sufficiently large class of 3-manifolds containing link exteriors. The aim of this paper is to study finite group actions on a homologically peripheral 3-manifold, which agree on the boundary, up to equivalence relative to the boundary. As an application, we generalize Sakuma's theorems on the uniqueness of symmetries of knots to the case of symmetries of links.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 2 , September 2011 , pp. 319 - 337
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]Brin, M., Johannson, K. and Scott, P.. Totally peripheral 3-manifolds. Pacific J. Math. 118 (1985), 37–51.CrossRefGoogle Scholar
[2]Benedetti, R. and Petronio, C.. Lectures on Hyperbolic Geometry (Universitext, Springer-Verlag, Berlin, 1992).CrossRefGoogle Scholar
[3]Davis, M. W. and Morgan, J. W.. Finite group actions on homotopy 3-spheres. in The Smith Conjecture (Morgan, J. W. and Bass, H., eds.) Pure and Applied Mathematics (Academic Press, 1984), pp. 181–225.Google Scholar
[4]Dinkelbach, J. and Leeb, B.. Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3-manifolds. Geom. Topol. 13 (2009), 1129–1173.CrossRefGoogle Scholar
[5]Edmonds, A. L.. A topological proof of the equivariant Dehn lemma. Trans. Amer. Math. Soc. 297 (1986), 605–615.CrossRefGoogle Scholar
[6]McA, C.. Gordon and R. A. Litherland. Incompressible surfaces in branched coverings. in The Smith conjecture. (Morgan, J. W. and Bass, H., eds.) Pure and Applied Mathematics, (Academic Press, 1984), pp. 139–152.Google Scholar
[7]Harris, L. and Scott, P.. Non-compact totally peripheral 3-manifolds. Pacific J. Math. 167 (1995), 119–127.Google Scholar
[8]Hempel, J.. 3-Manifolds. Ann. Math. Stud. 86 (Princeton University. Press, 1976).Google Scholar
[9]Ikeda, T.. PL finite group actions on 3-manifolds which are conjugate by homeomorphisms. Houston J. Math. 28 (2002), 133–138.Google Scholar
[10]Ikeda, T.. Graph manifolds with certain peripheralities. Topology Proc. 37 (2011), 107–117.Google Scholar
[11]Jaco, W.. Lectures on three-manifold topology. Conference Board of Math. 43 (Amer. Math. Soc., 1980).CrossRefGoogle Scholar
[12]Jaco, W. and Shalen, P.. Seifert fibered spaces in 3-manifolds. Mem. Amer. Math. Soc. 220 (Amer. Math. Soc., 1979).Google Scholar
[13]Johannson, K.. Homotopy equivalences of 3-manifolds with boundaries. Lecture Notes in Mathematics 761 (Springer, 1979).CrossRefGoogle Scholar
[14]McCullough, D., Miller, A. and Zimmermann, B.. Group actions on handlebodies. Proc. London Math. Soc. (3) 59 (1989), 373–416.Google Scholar
[15]Meeks, W. H. and Scott, P.. Finite group actions on 3-manifolds. Invent. Math. 86 (1986), 287–346.CrossRefGoogle Scholar
[16]Meeks, W. H. and Yau, S. T.. The equivariant Dehn's lemma and loop theorem. Comment. Math. Helvetici 56 (1981), 225–239.Google Scholar
[17]Morgan, J. W.. On Thurston's uniformization theorem for three-dimensional manifolds. in The Smith conjecture. (Morgan, J. W. and Bass, H., eds.) Pure and Applied Mathematics (Academic Press, 1984), pp. 37–125.Google Scholar
[18]Newman, W. H. A.. A theorem on periodic transformations of spaces. Quart. J. Math. 2 (1931), 1–8.CrossRefGoogle Scholar
[19]Nielsen, J.. Die Struktur periodischer Transformationen von Flächen. Math. -fys. Medd. Danske Vid. Selsk. 15 (1937), 1–77 (English trans: The structure of periodic surface transformations. in: Jakob Nielsen collected papers 2. (Birkhäuser, Boston, 1986), pp. 65–102).Google Scholar
[20]Plotnick, S. P.. Finite group actions and nonseparating 2-spheres. Proc. Amer. Math. Soc. 90 (1984), 430–432.CrossRefGoogle Scholar
[21]Reni, M.. Cyclic actions on compression bodies. Rend. Circ. Mat. Palermo (2) 46 (1997), 71–88.Google Scholar
[22]Sakuma, M.. Uniqueness of symmetries of knots. Math. Z. 192 (1986), 225–242.Google Scholar
[23]Serre, J. P.. Trees (Springer-Verlag, 1980).Google Scholar
[24]Zimmermann, B.. Genus actions of finite groups on 3-manifolds. Michigan Math. J. 43 (1996), 593–610.Google Scholar
[25]Zimmermann, B.. Finite maximal orientation reversing group actions on handlebodies and 3-manifolds. Rend. Circ. Mat. Palermo (2) 48 (1999), 549–562.Google Scholar