Reeb spaces of smooth functions on manifolds

Abstract

The Reeb space of a continuous function is the space of connected components of the level sets. In this paper we first prove that the Reeb space of a smooth function on a closed manifold with finitely many critical values has the structure of a finite graph without loops. We also show that an arbitrary finite graph without loops can be realized as the Reeb space of a certain smooth function on a closed manifold with finitely many critical values, where the corresponding level sets can also be preassigned. Finally, we show that a continuous map of a smooth closed connected manifold to a finite connected graph without loops that induces an epimorphism between the fundamental groups is identified with the natural quotient map to the Reeb space of a certain smooth function with finitely many critical values, up to homotopy.

arXiv comment21 pages, 10 figures

References in corpus (3)

• On Reeb graphs induced from smooth functions on 3-dimensional closed orientable manifolds with finitely many singular values
• Co-rank and Betti number of a group
• On Reeb graphs induced from smooth functions on closed or open manifolds

Cited by in corpus (6)

• On Reeb graphs induced from smooth functions on 3-dimensional closed orientable manifolds with finitely many singular values
• 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
• Global topologies of Reeb spaces of stable fold maps with non-trivial top homology groups
• Decomposition space theory
• Branched surfaces homeomorphic to Reeb spaces of simple fold maps