Realization of a graph as the Reeb graph of a height function on an embedded surface
DOI:
https://doi.org/10.12775/TMNA.2021.058Keywords
{Reeb graph, height function, Morse-Bott function, orientable surface, embeddingAbstract
We show that for a given finite graph $G$ without loop edges and isolated vertices, there exists an embedding of a closed orientable surface in $\mathbb{R}^3$ such that the Reeb graph of the associated height function has the structure of $G$. In particular, this gives a positive answer to the corresponding question posed by Masumoto and Saeki in 2011. We also give a criterion for a given surface to admit such a realization of a given graph, and study the problem in the class of Morse functions and in the class of round Morse-Bott functions. In the case of realization up to homeomorphism, the height function can be chosen Morse-Bott; we estimate from below the number of its critical circles and the number of its isolated critical points in terms of the graph structure.References
A. Bolsinov and A. Fomenko, Integrable Hamiltonian Systems: Geometry, Topology, Classification, CRC Press, USA, 2004.
H. de Fraysseix, P.O. de Mendez and P. Rosenstiehl, Bipolar orientations revisited, Discrete Appl. Math. 56 (1995), 157–179.
I. Gelbukh, Number of minimal components and homologically independent compact leaves for a Morse form foliation, Stud. Sci. Math. Hung. 46 (2009), 547–557.
I. Gelbukh, Loops in Reeb graphs of n-manifolds, Discrete Comput. Geom. 59 (2018), 843–863.
I. Gelbukh, A finite graph is homeomorphic to the Reeb graph of a Morse–Bott function, Math. Slovaca 71 (2021), 757–772.
I. Gelbukh, Morse–Bott functions with two critical values on a surface, Czech. Math. J. 71 (2021), 865–880.
I. Gelbukh, Criterion for a graph to admit a good orientation in terms of leaf blocks, Monatsh. Math. 198 (2022), no. 1, 61–77.
W. Jaco, Geometric realizations for free quotients, J. Aust. Math. Soc. 14 (1972), 411–418.
G. Khimshiashvili and D. Siersma, Remarks on minimal round functions, Banach Center Publ. 62 (2004), 159–172.
E.A. Kudryavtseva, Realization of smooth functions on surfaces as height functions, Sb. Math. 190 (1999), 349–405.
J. Martı́nez-Alfaro, I.S. Meza-Sarmiento and R. Oliveira, Topological classification of simple Morse–Bott functions on surfaces, Real and Complex Singularities 675 (2016), 165–179.
W. Marzantowicz and L.P. Michalak, Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups (2020), preprint, 20 pp., arXiv:2002.02388 [math.GT], 2002.02388.
Y. Masumoto and O. Saeki, Smooth function on a manifold with given Reeb graph, Kyushu J. Math. 65 (2011), 75–84.
L.P. Michalak, Realization of a graph as the Reeb graph of a Morse function on a manifold, Topol. Methods Nonlinear Anal. 52 (2018), 749–762.
L.P. Michalak, Combinatorial modifications of Reeb graphs and the realization problem, Discrete Comput. Geom. 65 (2021), 1038–1060.
G. Reeb, Sur les points singuliers dune forme de Pfaff completement integrable ou dune fonction numerique, C.R.A.S. Paris 222 (1946), 847–849.
O. Saeki, Reeb Spaces of Smooth Functions on Manifolds, Int. Math. Res. Not. (2021), DOI: 10.1093/imrn/rnaa301.
V.V. Sharko, About Kronrod–Reeb graph of a function on a manifold, Methods Funct. Anal. Topol. 12 (2006), 389–396.
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