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

Abstract

Let N be a smooth compact, connected and orientable 2-manifold with or without boundary. Given a stable circle-valued function γ: N → S1, we introduced a topological invariant associated to γ, 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 γ: N → S1. 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 γ. In particular, we show a class of functions whose generalized Reeb graphs have the maximal number of loops.