# A Finite Graph is Homeomorphic to The Reeb Graph of a Morse–Bott Function

**Author:** Irina Gelbukh

Mathematica Slovaca,
2020.

**Abstract:**
We prove that a finite graph (allowing loops and multiple edges) is homeomorphic
(isomorphic up to vertices of degree two) to the Reeb graph of a Morse–Bott function on a
smooth closed n-manifold, for any dimension n ≥ 2. The manifold can be chosen orientable or
non-orientable; we estimate the co-rank of its fundamental group (or the genus in the case of
surfaces) from below in terms of the cycle rank of the graph. The function can be chosen with
any number k ≥ 3 of critical values, and in a few special cases with k < 3.
In the case of surfaces, the function can be chosen, except for a few special cases,
as the height function associated with an immersion in ℝ^{3}.

**Keywords:** Reeb graph, Morse–Bott function, immersion, height function, surface

**PDF:** A Finite Graph is Homeomorphic to The Reeb Graph of a Morse–Bott Function

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