On the topology of the Reeb graph
Irina Gelbukh · Publicationes Mathematicae Debrecen · 2024
The Reeb quotient space $R_f$ of a function $f\from X\to\mathbb{R}$, known as the Reeb graph, can have various properties depending on $X$ and $f$. In the classical case of a smooth function on a closed manifold with a finite number of critical points, $R_f$ has the structure of a finite graph. Recently, Saeki showed that the same is true if $f$ is a smooth function with a finite number of critical values. Expanding his result, we prove that for an arbitrary smooth function on a closed connected manifold, the Reeb space $R_f$ still has a ``good'' structure; namely, $R_f$ is a $1$-dimensional Peano continuum homotopy equivalent to a finite graph.