Discrete & Computational Geometry

, Volume 59, Issue 4, pp 843–863 | Cite as

Loops in Reeb Graphs of n-Manifolds

  • Irina GelbukhEmail author


The Reeb graph of a smooth function on a connected smooth closed orientable n-manifold is obtained by contracting the connected components of the level sets to points. The number of loops in the Reeb graph is defined as its first Betti number. We describe the set of possible values of the number of loops in the Reeb graph in terms of the co-rank of the fundamental group of the manifold and show that all such values are realized for Morse functions and, except on surfaces, even for simple Morse functions. For surfaces, we describe the set of Morse functions with the number of loops in the Reeb graph equal to the genus of the surface.


Reeb graph Contour tree Number of loops Morse function Co-rank of the fundamental group 

Mathematics Subject Classification

05C38 05E45 58C05 


Compliance with Ethical Standards

Conflict of interest

The author declares that she has no conflict of interest.


  1. 1.
    Arnoux, P., Levitt, G.: Sur l’unique ergodicité des 1-formes fermées singulières. Invent. Math. 84(1), 141–156 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Biasotti, S., Giorgi, D., Spagnuolo, M., Falcidieno, B.: Reeb graphs for shape analysis and applications. Theoret. Comput. Sci. 392(1–3), 5–22 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cole-McLaughlin, K., Edelsbrunner, H., Harer, J., Natarajan, V., Pascucci, V.: Loops in Reeb graphs of 2-manifolds. Discrete Comput. Geom. 32(2), 231–244 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dimca, A., Papadima, S., Suciu, A.: Quasi-Kähler groups, 3-manifold groups, and formality. Math. Z. 268(1–2), 169–186 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society, Providence (2010)zbMATHGoogle Scholar
  6. 6.
    Farber, M.: Topology of Closed One-Forms. Mathematical Surveys and Monographs, vol. 108. American Mathematical Society, Providence (2004)CrossRefGoogle Scholar
  7. 7.
    Gelbukh, I.: Presence of minimal components in a Morse form foliation. Differ. Geom. Appl. 22(2), 189–198 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gelbukh, I.: Number of minimal components and homologically independent compact leaves for a Morse form foliation. Stud. Sci. Math. Hung. 46(4), 547–557 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Gelbukh, I.: On the structure of a Morse form foliation. Czechoslov. Math. J. 59(134)(1), 207–220 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gelbukh, I.: On collinear closed one-forms. Bull. Aust. Math. Soc. 84(2), 322–336 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gelbukh, I.: Ranks of collinear Morse forms. J. Geom. Phys. 61(2), 425–435 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gelbukh, I.: Structure of a Morse form foliation on a closed surface in terms of genus. Differ. Geom. Appl. 29(4), 473–492 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gelbukh, I.: Close cohomologous Morse forms with compact leaves. Czechoslov. Math. J. 63(138)(2), 515–528 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gelbukh, I.: The co-rank of the fundamental group: the direct product, the first Betti number, and the topology of foliations. Math. Slovaca 67(3), 645–656 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gelbukh, I.: Isotropy index for the connected sum and the direct product of manifolds. Publ. Math. Debr. 90(3–4), 287–310 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gelbukh, I.: Sufficient conditions for the compactifiability of a closed one-form foliation. Turk. J. Math. 41, 1344–1353 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gelbukh, I.: On the topology of the Reeb graph (under review)Google Scholar
  18. 18.
    Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities, Graduate Texts in Mathematics, vol. 14. Springer, New York (1973). X+209 ppGoogle Scholar
  19. 19.
    Hirsch, M.: Smooth regular neighborhoods. Ann. Math. 76(3), 524–530 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jaco, W.: Heegaard splittings and splitting homomorphisms. Trans. Am. Math. Soc. 144, 365–379 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Jaco, W.: Geometric realizations for free quotients. J. Aust. Math. Soc. 14, 411–418 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kaluba, M., Marzantowicz, W., Silva, N.: On representation of the Reeb graph as a sub-complex of manifold. Topol. Methods Nonlinear Anal. 45(1), 287–307 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lambe, L., Priddy, S.: Cohomology of nilmanifolds and torsion-free, nilpotent groups. Trans. Am. Math. Soc. 273(1), 39–55 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Leininger, C., Reid, A.: The co-rank conjecture for 3-manifold groups. Algebr. Geom. Topol. 2, 37–50 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Makanin, G.: Equations in a free group. Math. USSR Izv. 21(3), 483–546 (1983)CrossRefzbMATHGoogle Scholar
  26. 26.
    Martinez-Alfaro, J., Meza-Sarmiento, I.S., Oliveira, R.: Topological classification of simple Morse Bott functions on surfaces. In: Real and Complex Singularities. Contemporary Mathematics, vol. 675, pp. 165–179. American Mathematical Society, Providence (2016)Google Scholar
  27. 27.
    Masumoto, Y., Saeki, O.: Smooth function on a manifold with given Reeb graph. Kyushu J. Math. 65(1), 75–84 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Mel’nikova, I.: A test for non-compactness of the foliation of a Morse form. Russ. Math. Surv. 50(2), 444–445 (1995)CrossRefzbMATHGoogle Scholar
  29. 29.
    Mel’nikova, I.: Maximal isotropic subspaces of a skew-symmetric bilinear mapping. Mosc. Univ. Math. Bull. 54(4), 1–3 (1999)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Nicolaescu, L.: An Invitation to Morse Theory. Universitext. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  31. 31.
    Pedersen, E.: Regular neighborhoods in topological manifolds. Mich. J. Math. 24(2), 177–183 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Razborov, A.: On systems of equations in a free group. Math. USSR Izv. 25, 115–162 (1985)CrossRefzbMATHGoogle Scholar
  33. 33.
    Sharko, V.: About Kronrod–Reeb graph of a function on a manifold. Methods Funct. Anal. Topol. 12(4), 389–396 (2006)MathSciNetzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.CIC, IPNMexico CityMexico

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