Constructing Morse functions with given Reeb graphs and level sets
DOI:
https://doi.org/10.12775/TMNA.2025.066Keywords
Smooth functions, Morse(-Bott) functions, Reeb (di)graphs, differential topology, handlebodiesAbstract
We present a new result on realization of graphs as the {\it Reeb graphs} of Morse functions with given level sets. The {\it Reeb graph} of a Morse function on a closed manifold is the quotient space obtained by contracting connected components of its level sets to single points. Reeb graphs are graphs whose vertex sets consist of all points representing components with critical points. They are not only fundamental tools in geometry using differentiable maps, but also in applications of mathematics such as visualizations. Constructing explicit functions with prescribed {\it regular} level sets, was considered first by the author essentially and previously some cases where manifolds of the domains are at most $3$-dimensional have been studied. We generalize these cases by generalizing regular level sets to closed manifolds each component of which is the boundary of a nice compact manifold and construct Morse functions.References
R. Bott Nondegenerate critical manifolds, Ann. of Math. 60 (1954), 248–261.
S.A. Izar, Funções de Morse e Topologia das Superfı́cies I: O grafo de Reeb de f : M → R, PhD thesis, Estudo e Pesquisas em Matemática, Brazil, IBILCE, 1988, https://www.ibilce.unesp.br/Home/Departamentos/Matematica/metrica-31.pdf.
I. Gelbukh, Approximation of metric spaces by Reeb graphs: Cycle Rank of a Reeb Graph, the co-rank of the fundamental group, and large components of level sets on riemannian manifolds, Filomat 59 (2018). no. 4, 843–863.
I. Gelbukh, Criterion for a graph to admit a good orientation in terms of leaf blocks, Monatsh. Math. 198 (2022), 61–77.
I. Gelbukh, Realization of a digraph as the Reeb graph of a Morse–Bott function on a given surface, Topology and its Applications, 2024.
I. Gelbukh, Reeb graphs of Morse–Bott functions on a given surface, Bull. Iranian Math. Soc. 50 (2024), 1–17.
M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Graduate Texts in Mathematics, vol. 14, Springer–Verlag, 1974.
J. Hempel, 3-Manifolds, AMS Chelsea Publishing, 2004.
N. Kitazawa, On round fold maps, RIMS Kokyuroku Bessatsu B38 (2013), 45–59 (in Japanese).
N. Kitazawa, Fold maps with singular value sets of concentric spheres, Hokkaido Math. J. 43 (2014), no. 3, 327–359.
N. Kitazawa, On manifolds admitting fold maps with singular value sets of concentric spheres, PhD thesis, Tokyo Institute of Technology, 2014. https://t2r2.star.titech.ac.jp/rrws/file/CTT100666807/ATD100000413/ (only a short abstract in Japanese).
N. Kitazawa, On Reeb graphs induced from smooth functions on 3-dimensional closed manifolds with finitely many singular values, Topol. Methods Nonlinear Anal. 57 (2022), no. 2B, 897–912.
N. Kitazawa, On Reeb graphs induced from smooth functions on closed or open manifolds, Methods Funct. Anal. Topology 28 (2022), no. 2, 127–143.
N. Kitazawa, On Reeb graphs induced from smooth functions on 3-dimensional closed manifolds which may not be orientable, Methods Funct. Anal. Topology 29 (2023), no. 1, 57–72.
W. Marzantowicz and L.P. Michalak, Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups, Fund. Math. 265 (2024), no. 2, 97–140.
Y. Masumoto and O. Saeki, A smooth function on a manifold with given Reeb graph, Kyushu J. Math. 65 (2011), 75–84.
J. Martinez-Alfaro, I.S. Meza-Sarmiento and R. Oliveira, Topological classification of simple Morse–Bott functions on surfaces, Contemp. Math. 675 (2016), no. 4, 165–179.
L.P. Michalak, Realization of a graph as the Reeb graph of a Morse function on a manifold, Topol. Methods Nonlinear Anal. 52 (2018), no. 2, 749–762.
L.P. Michalak, Combinatorial modifications of Reeb graphs and the realization problem, Discrete Comput. Geom. 65 (2021), no. 2, 1038–1060.
L.P. Michalak, Reeb graph invariants of Morse functions and 3-manifold groups (2024), arXiv: 2403.02291.
J. Milnor, Lecture on the h-Cobordism Theorem, Math. Notes, Princeton Univ. Press, Princeton, N.Y., 1965.
A. Ranicki, Algebraic and Geometric Surgery, Oxford Mathematical Monographs, Oxford Academic, 2007.
G. Reeb, Sur les points singuliers d´une forme de Pfaff complétement intègrable ou d´une fonction numérique, Comptes Rendus Hebdomadaires des Séances de I´Académie des Sciences 22 (1946), 847–849.
O. Saeki, Notes on the topology of folds, J. Math. Soc. Japan 44 (1992), no. 3, 551–566.
O. Saeki, Topology of Singular Fibers of Differntiable Maps, Lecture Notes in Math., vol. 1854, Springer, 2004.
O. Saeki, Reeb spaces of smooth functions on manifolds, International Mathematics Research Notices 11 (2022), 8740–8768.
O. Saeki and T. Yamamoto, Singular fibers of stable maps of 3-manifolds with boundary into surfaces and their applications, Algebr. Geom. Topol. 16 (2016), no. 3, 1379–1402.
V. Sharko, About Kronrod–Reeb graph of a function on a manifold, Methods Funct. Anal. Topology 12 (2006), 389–396.
C.T.C. Wall, Classification problems in differential topology I: Classification of handlebodies, Topology 2 (1963), 253–261.
C.T.C. Wall, Classification problems in differential topology II: Diffeomorphisms of handlebodies, Topology 2 (1963), 263–272.
C.T.C. Wall, Classification problems in differential topology Q: Quadratic forms on finite groups and related topics, Topology 2 (1963), 281–298.
C.T.C. Wall, Classification problems in differential topology III: Applications to special cases, Topology 5 (1965), 291–304.
C.T.C. Wall, Classification problems in differential topology IV: Thickenings, Topology 2 (1966), 73–94.
C.T.C . Wall, Classification problems in differential topology VI: Classification of (s−1)-connected (2s + 1)-manifolds, Topology 6 (1967), no. 3, 273–296.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
