ON REPRESENTATION OF THE REEB GRAPH AS A SUB-COMPLEX OF MANIFOLD

Author(s)

Kaluba, Marek ^[1] ; Marzantowicz, Waclaw ^[1] ; Silva, Nelson ^[1]

Affiliation

^[1] Adam Mickiewicz Univ, Fac Math & Comp Sci, PL-61614 Poznan - Poland

• Document type: Journal article
• Source: TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS; v. 45, n. 1, p. 287-307, MAR 2015.
• Web of Science Citations: 3

Abstract

The Reeb graph R(f) is one of the fundamental invariants of a smooth function f: M -> R with isolated critical points. It is defined as the quotient space M/similar to of the closed manifold M by a relation that depends on f. Here we construct a 1-dimensional complex Gamma(f) embedded into M which is homotopy equivalent to R(f). As a consequence we show that for every function f on a manifold with finite fundamental group, the Reeb graph of f is a tree. If pi(1)(M) is an abelian group, or more general, a discrete amenable group, then R(f) contains at most one loop. Finally we prove that the number of loops in the Reeb graph of every function on a surface M-g is estimated from above by g, the genus of M-g. (AU)

• FAPESP's process: 12/15659-5 - Topological invariants of minimax problems with symmetry
• Grantee: Nelson Antonio Silva
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate