Non-CW Reeb spaces of continuous real-valued functions on nice Hausdorff spaces can be represented via simplified graphs, with concrete examples.
Realization of a graph as the Reeb graph of a Morse function on a manifold
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abstract
We investigate the problem of the realization of a given graph as the Reeb graph $\mathcal{R}(f)$ of a smooth function $f\colon M\rightarrow \mathbb{R}$ with finitely many critical points, where $M$ is a closed manifold. We show that for any $n\geq2$ and any graph $\Gamma$ admitting the so called good orientation there exist an $n$-manifold $M$ and a Morse function $f\colon M\rightarrow \mathbb{R} $ such that its Reeb graph $\mathcal{R}(f)$ is isomorphic to $\Gamma$, extending previous results of Sharko and Masumoto-Saeki. We prove that Reeb graphs of simple Morse functions maximize the number of cycles. Furthermore, we provide a complete characterization of graphs which can arise as Reeb graphs of surfaces.
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Certain 3-manifolds are characterized by Morse functions with regular levels consisting only of spheres, tori, or Klein bottles.
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Representations of Reeb spaces via simplified graphs and examples
Non-CW Reeb spaces of continuous real-valued functions on nice Hausdorff spaces can be represented via simplified graphs, with concrete examples.
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Morse functions with regular level sets consisting of $2$-dimensional spheres, $2$-dimensional tori, or Klein Bottles
Certain 3-manifolds are characterized by Morse functions with regular levels consisting only of spheres, tori, or Klein bottles.