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Degree-one maps of Seifert manifolds into the Poincaré homology sphere


This paper is devoted to the Legrand-Wang-Zieschang problem of minimal (in the sense of degree-one maps) Seifert manifolds. The main result is that the set of all possible map degrees from a Seifert manifold to a manifold with a finite fundamental group whose base is a sphere or a torus depends only on residues of parameters of exceptional fibers of the Seifert manifold. The minimality of some Seifert manifolds is proved by using this theorem.

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  1. 1.

    C. Hayat-Legrand, S. Matveev, and H. Zieschang, “Computer calculation of the degree of maps into the Poincaré homology sphere,” Experiment. Math., 10, No. 4, 497–508 (2001).

    MATH  Google Scholar 

  2. 2.

    C. Hayat-Legrand, S. Wang, and H. Zieschang, “Minimal Seifert manifolds,” Math. Ann., 308, No. 4, 673–700 (1997).

    MATH  Article  Google Scholar 

  3. 3.

    S. Matveev and A. Perfilyev, “Periodicity of the degrees of maps between Seifert manifolds,” Dokl. Ross. Akad. Nauk, 395, No. 4, 449–451 (2004).

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 173–183, 2005.

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Perfilyev, A.A. Degree-one maps of Seifert manifolds into the Poincaré homology sphere. J Math Sci 144, 4484–4491 (2007).

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  • Manifold
  • Fundamental Group
  • Cellular Structure
  • Geometric Representation
  • Common Multiple