Geodesic
Loops in generalized Reeb graphs associated to stable circle-valued functions
Journal of Singularities, Tome 22 (2020), pp. 104-113

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Let N be a smooth compact, connected and orientable 2-manifold with or without boundary. Given a stable circle-valued function \gamma: N -> S^1, we introduced a topological invariant associated to \gamma, called generalized Reeb graph. It is a generalized version of the classical and well known Reeb graph. The purpose of this paper is to investigate the number of loops in generalized Reeb graphs associated to stable circle-valued functions \gamma: N -> S^1. We show that the number of loops depends on the genus of N, the number of boundary components of N, and the number of open saddles of \gamma. In particular, we show a class of functions whose generalized Reeb graphs have the maximal number of loops.
DOI : 10.5427/jsing.2020.22g
Classification : 58K15, 58K40, 58K65 
@article{10_5427_jsing_2020_22g,
     author = {Erica Boizan Batista and Jo\~ao Carlos Ferreira Costa, and Juan J. Nu\~no-Ballesteros},
     title = {Loops in generalized {Reeb} graphs associated to stable circle-valued functions},
     journal = {Journal of Singularities},
     pages = {104--113},
     publisher = {mathdoc},
     volume = {22},
     year = {2020},
     doi = {10.5427/jsing.2020.22g},
     url = {http://geodesic.mathdoc.fr/articles/10.5427/jsing.2020.22g/}
}
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Erica Boizan Batista; João Carlos Ferreira Costa,; Juan J. Nuño-Ballesteros. Loops in generalized Reeb graphs associated to stable circle-valued functions. Journal of Singularities, Tome 22 (2020), pp. 104-113. doi: 10.5427/jsing.2020.22g

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