On Reeb graphs induced from smooth functions on 3-dimensional closed manifolds with finitely many singular values
DOI:
https://doi.org/10.12775/TMNA.2021.044Keywords
Singularities of differentiable maps, generic maps, differential topology, Reeb spaces (graphs)Abstract
The {\it Reeb graph} of a smooth function on a smooth manifold is the graph obtained as the space of all connected components of preimages (level sets) such that the set of all vertices coincides with the set of all the connected components of preimages containing some singular points. Reeb graphs are fundamental and important tools in algebraic topological and differential topological theory of Morse functions and their variants. In the present paper, as a related fundamental and important study, for given graphs, we construct certain smooth functions inducing the graphs as the Reeb graphs. Such results have been demonstrated by Masumoto, Michalak, Saeki, Sharko, among others, and also by the author since 2000s. We construct good smooth functions on suitable $3$-dimensional connected, closed and orientable manifolds.References
R. Bott, Nondegenerate critical manifolds, Ann. of Math. 60 (1954), 248–261.
C. Ehresmann, Les connexions infinitesimales dans un espace fibre differentiable, Colloque de Topologie, Bruxelles (1950), 29–55.
I. Gelbukh, Loops in Reeb graphs of n-manifolds, Discrete Comput. Geom. 59 (2018), no. 4, 843–863.
I. Gelbukh, Approximation of metricsSpaces by Reeb graphs: cycle rank of a Reeb graph, the co-rank of the fundamental group, and large components of level sets on Riemannia manifolds, Filomat, arxiv:1903.00777. (in press)
M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Graduate Texts in Mathematics, vol. 14, Springer–Verlag, 1974.
N. Kitazawa, Smooth maps compatible with simplicial structures and preimages, arxiv: 1802.06381v6.
N. Kitazawa, Structures of cobordism-like modules induced from generic maps of codimension −2, arxiv:1901.04994v1.
J. Martinez-Alfaro, I.S. Meza-Sarmiento and R. Oliveira, Topological classification of simple Morse Bott functions on surfaces, Contemp. Math. 675 (2016), 165–179.
Y. Masumoto and O. Saeki, A 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), no. 2, 749–762.
L.P. Michalak, Combinatorial modifications of Reeb graphs and the realization problem, Discrete Comput. Geometry 65 (2021), 1038–1060.
J. Milnor, Lectures on the h-Cobordism Theorem, Math. Notes, Princeton Univ. Press, Princeton, N.J., 1965.
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 222 (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–Verlag, 2004.
O. Saeki, Morse functions with sphere fibers, Hiroshima Math. J. 36 (2006), no. 1, 141–170.
O. Saeki, Reeb spaces of smooth functions on manifolds, Intermational Mathematics Research Notices, maa301, DOI: 10.1093/imrn/maa301, arxiv:2006.01689.
O. Saeki and M. Takase, Desingularizing special generic maps, J. Gökova Geom. Topol. GGT 7 (2013), 1–24.
V. Sharko, About Kronrod–Reeb graph of a function on a manifold, Methods Funct. Anal. Topology 12 (2006), 389–396.
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Naoki Kitazawa
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
