Loops in Reeb graphs of n-manifolds

Author: Irina Gelbukh

Discrete & Computational Geometry, 59(4):843–863, 2018.

Abstract: The Reeb graph of a smooth function on a connected smooth closed orientable n-manifold is obtained by contracting the connected components of the level sets to points. The number of loops in the Reeb graph is defined as its first Betti number. We describe the set of possible values of the number of loops in the Reeb graph in terms of the co-rank of the fundamental group of the manifold and show that all such values are realized for Morse functions and, except on surfaces, even for simple Morse functions. For surfaces, we describe the set of Morse functions with the number of loops in the Reeb graph equal to the genus of the surface.

Keywords: Reeb graph; contour tree; number of loops; Morse function; co-rank of the fundamental group

PDF: Loops in Reeb graphs of n-manifolds (preprint version)

Final text on the journal's site: DOI 10.1007/s00454-017-9957-9.