On Reeb graphs induced from smooth functions on 3-dimensional closed orientable manifolds with finite singular values

Naoki Kitazawa

Published 2019-02-23Version 1

The Reeb graph of a function on a smooth manifold is the graph obtained as the space of all connected components of inverse images such that the set of all vertices coincides with the set of all connected components of inverse images including singular points. Reeb graphs are fundamental and important in the algebraic and differential topological theory of Morse functions and their generalizations, or in other words, the theory of global singulartiy. In this paper, as a related fundamental and important study, for given graphs, we construct certain smooth functions inducing the graphs as the Reeb graphs. Such works have been demonstrated by Masumoto, Michalak, Saeki, Sharko etc. and also by the author since 2000s. We demonstrate new works and present new smooth functions on $3$-dimensional closed orientable manifolds.