Advertisement

Bulletin of the Iranian Mathematical Society

, Volume 45, Issue 6, pp 1855–1869 | Cite as

Self-homeomorphisms and Degree \(\pm \,1\) Self-maps on Lens Spaces

  • Xiaotian Pan
  • Bingzhe HouEmail author
  • Zhongyang Zhang
Original Paper
  • 31 Downloads

Abstract

In this article, we give a sufficient and necessary condition to that a degree \(\pm \,1\) self-map on a \((2n-1)\)-dimensional lens space for \(n \ge 2\) is homotopic to a self-homeomorphism. In fact, we show how the endomorphism induced by a homeomorphism can act on the fundamental group of the lens space. Moreover, we provide a specific description of a class of lens spaces which admit self-homeomorphisms inducing nontrivial automorphisms on the fundamental groups. Furthermore, the topologically conjugate classification for a special class of periodic homeomorphisms on \(S^{2n-1}\) is obtained.

Keywords

Lens spaces Self-homeomorphisms Fundamental groups Endomorphisms Topological conjugacies 

Mathematics Subject Classification

20F34 55M25 57N99 37C15 

Notes

References

  1. 1.
    Brody, E.J.: The topological classification of the lens spaces. Ann. Math. 71(2), 163–184 (1960)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Brouwer, L.E.J.: Über Abbildung von Mannigfaltigkeitin. Math. Ann. 71, 97–115 (1911)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brouwer L.E.J.: Sur la notion de “classe” de transformations d’une multiplicité. Proc. 5th Intern. Congr. Math. Cambridge 2, pp. 9–10 (1912)Google Scholar
  4. 4.
    Browder, W., Petrie, T., Wall, C.T.C.: The classification of free actions of cyclic groups of odd order on homotopy spheres. Bull. Am. Math. Soc. 77(3), 455–459 (1971)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bryden, J., Zvengrowski, P.: The cohomology ring of the orientable Seifert manifolds. II. Topol. Appl. 127(1–2), 213–257 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, H.: Counting homotopy classes of mappings via Dijkgraaf–Witten invariants. Topol. Appl. 161, 316–320 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cohen, M.M.: A course in simple-homotopy theory. Springer, New York, Berlin (1973)CrossRefGoogle Scholar
  8. 8.
    Ding, Y.H., Pan, J.Z.: Computing degree of maps between manifolds. Acta Math. Sin. (Engl. Ser.) 21(6), 1277–1284 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Du, X.: On self-mapping degrees of \(S^3\)-geometry manifolds. Acta Math. SCI (Engl. Ser.) 25(8), 1243–1252 (2009)Google Scholar
  10. 10.
    Duan, H., Wang, S.: The degrees of maps between manifolds. Math. Z. 244(1), 67–89 (2003)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Franz, W.: über die Torsion einer überdeckung. J. Reine Angew. Math. 173, 245–254 (1935)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hayat-Legrand, C., Kudryavtseva, E., Wang, S., Zieschang, H.: Degrees of self-mappings of Seifert manifolds with finite fundamental groups. Rend. Istit. Mat. Univ. Trieste 32(suppl. 1), 131–147 (2001)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Hayat-Legrand, C., Wang, S., Zieschang, H.: Degree-one maps onto lens spaces. Pac. J. Math. 176(1), 19–32 (1996)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jahren, B., Kwasik, S.: Free involutions on \(S^1 \times S^n\). Math. Ann. 351(2), 281–303 (2011)Google Scholar
  15. 15.
    Jahren, B., Kwasik, S.: How different can h-cobordant manifolds be? Bull. Lond. Math. Soc. 47(4), 617–630 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lee, Y.M., Xu, F.: Realization of numbers as the degrees of maps between manifolds. Acta Math. Sin. (Engl. Ser.) 26(8), 1413–1424 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    de Medrano, S. Lopez.: Some Results on Involutions of Homotopy Spheres, Proc. Conference Transformation Groups (New Orleans, La.,1967) Springer-Ver lag, New York; Thesis, Princeton University, Princeton, N. J., pp. 167–174 (1968)Google Scholar
  18. 18.
    Luft, E., Sjerve, D.: Degree-1 maps onto lens spaces and free actions on homology spheres. Topol. Appl. 37, 131–136 (1990)CrossRefGoogle Scholar
  19. 19.
    Milnor, J.: Whitehead torsion. Bull. Am. Math. Soc. 72, 358–426 (1966)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Olum, P.: On mappings into spaces in which certain homotopy groups vanish. Ann. Math. 57, 561–574 (1953)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Olum, P.: Mappings of manifolds and the notion of degree. Ann. Math. 58(2), 458–480 (1953)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Sady, R.: Free involutions on complex projective spaces. Mich. Math. J. 24, 51–64 (1977)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Scott, G.P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15, 401–487 (1983)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Su, Y.: Free involutions on \(S^2 \times S^3\). Geom. Dedic. 159(1), 11–28 (2012)Google Scholar
  25. 25.
    Sun, H.: Degree \(\pm 1\) self-maps and self-homeomorphisms on prime 3-manifolds. Algebraic Geom. Topol. 10(2), 867–890 (2010)Google Scholar
  26. 26.
    Tan, L.: On mapping degrees between 1-connected 5-manifolds. Sci. China Ser. A 49(12), 1855–1863 (2006)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Thurston, W.P.: Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Am. Math. Soc. (N.S) 6, 357–381 (1982)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math. 87, 56–88 (1968)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Wall, C.T.C.: Free piecewise-linear involutions on spheres. Bull. Am. Math. Soc. 74, 554–558 (1968)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Wall, C.T.C.: Surgery on compact manifolds. In: Ranicki, A.A. (ed.) Mathematical Surveys and Monographs, vol. 69, 2nd edn. American Mathematical Society, Providence, RI (1999)Google Scholar

Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.School of MathematicsJilin UniversityChangchunPeople’s Republic of China
  2. 2.Department of Basic ScienceAviation University Air ForceChangchunPeople’s Republic of China

Personalised recommendations