Abstract
We study the topological structure of all 3-manifolds obtained by surgery along principal fibers of a closed orientable \(\mathbb{S}\)-manifold. As a consequence, we give alternative proofs of some classical results due to W. Heil and L. Moser. Moreover, we completely specify the Seifert invariants for the considered manifolds. Finally we classify the manifolds obtained by surgery along certain Seifert links and determine geometric presentations of their fundamental groups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Boileau, M. and Zieschang, H.: Heegaard genus of closed orientable Seifert 3-manifolds, Invent. Math. 76 (1984), 455–468.
Burde. G. and Murasugi, K.: Links and Seifert fiber spaces, Duke Math. J. 37 (1970), 89–93.
Burde, G. and Zieschang, H.: Eine Kennzeichnung der Torusknoten, Math. Ann. 169 (1966), 169–176.
Cavicchioli, A.: Neuwirth manifolds and colourings of graphs, Aequationes Math. 44 (1992), 168–187.
Dehn, M.: Über die Topologie des dreidimensionalen Raumes, Math. Ann. 69 (1910), 137–168.
Eisenbud, D. and Neumann, W.: Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Ann. of Math. Studies 110, Princeton Univ. Press, Princeton, New Jersey, 1985.
Heil, W.: 3-manifolds that are sums of solid tori and Seifert fiber spaces, Proc. Amer. Math. Soc. 37 (1973), 609–614.
Heil, W.: Elementary surgery on Seifert fiber spaces, Yokohama Math. J. 22 (1974), 135–139.
Hempel, J.: A simply connected 3-manifold is \(\mathbb{S}\) 3 if it is the sum of a solid torus and the complement of a torus knot, Proc. Amer. Math. Soc. 15 (1964), 154–158.
Hempel, J.: 3-Manifolds, Princeton Univ. Press, Princeton, New Jersey, 1976.
Kirby, R.: A calculus for framed links in \(\mathbb{S}\) 3, Invent. Math. 45 (1978), 35–56.
Moser, L.: Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), 737–745.
Murasugi, K.: On the center of the group of a link, Proc. Amer. Math. Soc. 16 (1965), 1052–1057.
Neuwirth. L.: A note on torus knots and links determined by their groups, Duke Math. J. 28 (1961), 545–551.
Orlik, P.: Seifert manifolds, Lecture Notes in Math. 291, Springer-Verlag, Berlin, Heidelberg, New York, 1972.
Osborne, R. P. and Stevens, R. S.: Group presentations corresponding to spines of 3-manifolds I, Amer. J. Math. 96 (1974), 454–471.
Osborne, R. P. and Stevens, R. S.: Group presentations corresponding to spines of 3-manifolds II, III, Trans. Amer. Math. Soc. 234 (1977), 213–243, 245–251.
Raymond, F.: Classification of the actions of the circle on 3-manifolds, Trans. Amer. Math. Soc. 131 (1968), 51–78.
Rolfsen, D.: Knots and Links, Math. Lecture Ser. 7, Publish or Perish, Berkeley, 1976.
Seifert, H.: Topology of 3-dimensional fibered spaces, Acta Math. 60 (1933), 147–288.
Seifert, H. and Threlfall, W.: A Textbook of Topology, Academic Press, New York, London, 1980.
Author information
Authors and Affiliations
Additional information
Work performed under the auspices of C.N.R. (National Research Council) of Italy and partially supported by Ministero della Ricerca Scientifica e Tecnologica within the projects ‘Geometria Reale e Complessa’ and ‘Topologia’.
Rights and permissions
About this article
Cite this article
Cavicchioli, A., Hegenbarth, F. Surgery on 3-manifolds with \(\mathbb{S}\) 1-actions. Geom Dedicata 61, 285–313 (1996). https://doi.org/10.1007/BF00150029
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00150029