Discrete & Computational Geometry

, Volume 59, Issue 4, pp 843–863

# Loops in Reeb Graphs of n-Manifolds

Article

## Abstract

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.

## Keywords

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

## Mathematics Subject Classification

05C38 05E45 58C05

## Notes

### Conflict of interest

The author declares that she has no conflict of interest.

## References

1. 1.
Arnoux, P., Levitt, G.: Sur l’unique ergodicité des 1-formes fermées singulières. Invent. Math. 84(1), 141–156 (1986)
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)
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)
4. 4.
Dimca, A., Papadima, S., Suciu, A.: Quasi-Kähler groups, 3-manifold groups, and formality. Math. Z. 268(1–2), 169–186 (2011)
5. 5.
Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society, Providence (2010)
6. 6.
Farber, M.: Topology of Closed One-Forms. Mathematical Surveys and Monographs, vol. 108. American Mathematical Society, Providence (2004)
7. 7.
Gelbukh, I.: Presence of minimal components in a Morse form foliation. Differ. Geom. Appl. 22(2), 189–198 (2005)
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)
9. 9.
Gelbukh, I.: On the structure of a Morse form foliation. Czechoslov. Math. J. 59(134)(1), 207–220 (2009)
10. 10.
Gelbukh, I.: On collinear closed one-forms. Bull. Aust. Math. Soc. 84(2), 322–336 (2011)
11. 11.
Gelbukh, I.: Ranks of collinear Morse forms. J. Geom. Phys. 61(2), 425–435 (2011)
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)
13. 13.
Gelbukh, I.: Close cohomologous Morse forms with compact leaves. Czechoslov. Math. J. 63(138)(2), 515–528 (2013)
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)
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)
16. 16.
Gelbukh, I.: Sufficient conditions for the compactifiability of a closed one-form foliation. Turk. J. Math. 41, 1344–1353 (2017)
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)
20. 20.
Jaco, W.: Heegaard splittings and splitting homomorphisms. Trans. Am. Math. Soc. 144, 365–379 (1969)
21. 21.
Jaco, W.: Geometric realizations for free quotients. J. Aust. Math. Soc. 14, 411–418 (1972)
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)
23. 23.
Lambe, L., Priddy, S.: Cohomology of nilmanifolds and torsion-free, nilpotent groups. Trans. Am. Math. Soc. 273(1), 39–55 (1982)
24. 24.
Leininger, C., Reid, A.: The co-rank conjecture for 3-manifold groups. Algebr. Geom. Topol. 2, 37–50 (2002)
25. 25.
Makanin, G.: Equations in a free group. Math. USSR Izv. 21(3), 483–546 (1983)
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)
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)
29. 29.
Mel’nikova, I.: Maximal isotropic subspaces of a skew-symmetric bilinear mapping. Mosc. Univ. Math. Bull. 54(4), 1–3 (1999)
30. 30.
Nicolaescu, L.: An Invitation to Morse Theory. Universitext. Springer, New York (2011)
31. 31.
Pedersen, E.: Regular neighborhoods in topological manifolds. Mich. J. Math. 24(2), 177–183 (1977)
32. 32.
Razborov, A.: On systems of equations in a free group. Math. USSR Izv. 25, 115–162 (1985)
33. 33.
Sharko, V.: About Kronrod–Reeb graph of a function on a manifold. Methods Funct. Anal. Topol. 12(4), 389–396 (2006)