## Osaka Journal of Mathematics

### Self-mapping degrees of 3-manifolds

#### Abstract

For each closed oriented $3$-manifold $M$ in Thurston's picture, the set of degrees of self-maps on $M$ is given.

#### Article information

Source
Osaka J. Math., Volume 49, Number 1 (2012), 247-269.

Dates
First available in Project Euclid: 21 March 2012

https://projecteuclid.org/euclid.ojm/1332337246

Mathematical Reviews number (MathSciNet)
MR2903262

Zentralblatt MATH identifier
1241.55002

Subjects
Primary: 55M25: Degree, winding number 57M10: Covering spaces

#### Citation

Sun, Hongbin; Wang, Shicheng; Wu, Jianchun; Zheng, Hao. Self-mapping degrees of 3-manifolds. Osaka J. Math. 49 (2012), no. 1, 247--269. https://projecteuclid.org/euclid.ojm/1332337246

#### References

• X.M. Du: On self-mapping degrees of $S^{3}$-geometry manifolds, Acta Math. Sin. (Engl. Ser.) 25 (2009), 1243–1252.
• \begingroup P. Derbez: Topological rigidity and Gromov simplicial volume, Comment. Math. Helv. 85 (2010), 1–37. \endgroup
• H.B. Duan and S.C. Wang: Non-zero degree maps between $2n$-manifolds, Acta Math. Sin. (Engl. Ser.) 20 (2004), 1–14.
• A. Hatcher: Notes on basic 3-manifold topology, http://www.math.cornell.edu/ hatcher/.
• J. Hempel: $3$-Manifolds, Princeton Univ. Press, Princeton, NJ, 1976.
• H. Hendriks and F. Laudenbach: Scindement d'une équivalence d'homotopie en dimension $3$, Ann. Sci. École Norm. Sup. (4) 7 (1974), 203–217.
• C. Hayat-Legrand, E. Kudryavtseva, S.C. Wang and H. Zieschang: Degrees of self-mappings of Seifert manifolds with finite fundamental groups, Rend. Istit. Mat. Univ. Trieste 32 (2001), 131–147.
• C. Hayat-Legrand, S.C. Wang and H. Zieschang: Degree-one maps onto lens spaces, Pacific J. Math. 176 (1996), 19–32.
• K. Ireland and M. Rosen: A Classical Introduction to Modern Number Theory, second edition, Graduate Texts in Mathematics 84, Springer, New York, 1990.
• D. Kotschick and C. Löh: Fundamental classes not representable by products, J. Lond. Math. Soc. (2) 79 (2009), 545–561.
• J. Kalliongis and D. McCullough: $\pi_{1}$-injective mappings of compact $3$-manifolds, Proc. London Math. Soc. (3) 52 (1986), 173–192.
• S.V. Matveev and A.A. Perfil'ev: Periodicity of degrees of mappings between Seifert manifolds, Dokl. Akad. Nauk 395 (2004), 449–451, (Russian).
• P. Orlik: Seifert Manifolds, Lecture Notes in Mathematics 291, Springer, Berlin, 1972.
• P. Olum: Mappings of manifolds and the notion of degree, Ann. of Math. (2) 58 (1953), 458–480.
• T. Püttmann: Cohomogeneity one manifolds and self-maps of nontrivial degree, Transform. Groups 14 (2009), 225–247.
• G. de Rham: Sur l'analysis situs des variétés á $n$ dimensions, J. Math. 10 (1931), 115–200.
• Y.W. Rong: Degree one maps between geometric $3$-manifolds, Trans. Amer. Math. Soc. 332 (1992), 411–436.
• Y.W. Rong and S.C. Wang: The preimages of submanifolds, Math. Proc. Cambridge Philos. Soc. 112 (1992), 271–279.
• P. Scott and C.T.C. Wall: Topological methods in group theory; in Homological Group Theory (Proc. Sympos., Durham, 1977), London Math. Soc. Lecture Note Ser. 36, Cambridge Univ. Press, Cambridge, 137–203, 1979.
• P. Scott: The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), 401–487.
• T. Soma: The Gromov invariant of links, Invent. Math. 64 (1981), 445–454.
• \begingroup H.B. Sun: Degree $\pm 1$ self-maps and self-homeomorphisms on prime 3-manifolds, Algebr. Geom. Topol. 10 (2010), 867–890. \endgroup
• H. Sun, S. Wang and J. Wu: Self-mapping degrees of torus bundles and torus semi-bundles, Osaka J. Math. 47 (2010), 131–155.
• W.P. Thurston: Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 357–381.
• S.C. Wang: The $\pi_{1}$-injectivity of self-maps of nonzero degree on $3$-manifolds, Math. Ann. 297 (1993), 171–189.
• S.C. Wang: Non-zero degree maps between 3-manifolds; in Proceedings of the International Congress of Mathematicians, II, (Beijing, 2002), Higher Ed. Press, Beijing, 457–468, 2002.