Loops in generalized Reeb graphs associated to stable circle-valued functions

Erica Boizan Batista, João Carlos Ferreira Costa, and Juan J. Nuño-Ballesteros

Journal of Singularities
volume 22 (2020), 104-113

Received: 25 February 2019. Received in revised form: 25 May 2020.

DOI: 10.5427/jsing.2020.22g

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Abstract:

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.

2000 Mathematical Subject Classification:

58K15, 58K40, 58K65

Key words and phrases:

generalized Reeb graphs, stable maps, loops

Author(s) information:

Erica Boizan Batista
Centro de Ciências e Tecnologia
Universidade Federal do Cariri - UFCA
Campus de Juazeiro do Norte-CE, Brazil
email: erica.batista@ufca.edu.br

João Carlos Ferreira Costa
Universidade Estadual Paulista (Unesp)
Instituto de Biociências, Letras e Ciências Exatas
Campus de São José do Rio Preto, Brazil
email: joao.costa@unesp.br

Juan J. Nuño-Ballesteros
Departament de Matemàtiques
Universitat de València
Campus de Burjassot 46100 Spain
email: Juan.Nuno@uv.es