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

Abstract

The Reeb graph of a function on a smooth manifold is the graph obtained as the space of all connected components of level sets such that the set of all vertices coincides with the set of all connected components of level sets including singular points. Reeb graphs are fundamental and important in the algebraic and differential topological theory of Morse functions and their generalizations. 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 present new smooth functions on suitable $3$-dimensional closed orientable manifolds through explicit constructive methods.

13 pages, 6 figures, this version has been accepted for publication by Topological Methods in Nonlinear Analysis (https://apcz.umk.pl/czasopisma/index.php/TMNA/index)

References in corpus (2)

• Reeb spaces of smooth functions on manifolds
• Structures of cobordism-like modules induced from generic maps of codimension -2

Cited by in corpus (6)

• Reeb spaces of smooth functions on manifolds
• On Reeb graphs induced from smooth functions on closed or open manifolds
• Realization problems of graphs as Reeb graphs of Morse functions with prescribed preimages
• Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups
• Global topologies of Reeb spaces of stable fold maps with non-trivial top homology groups
• Branched surfaces homeomorphic to Reeb spaces of simple fold maps