A finite graph is homeomorphic to the Reeb graph of a Morse-Bott function. (English) Zbl 1478.58005
Summary: 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 \(n\geq 2\). 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 \(k\geq 3\) 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 \(\mathbb{R}^3\).
MathOverflow Questions:
Immersion in \(\mathbb R^3\) of a Klein bottle with Morse-Bott height function without centersMSC:
| 58C05 | Real-valued functions on manifolds |
| 58K65 | Topological invariants on manifolds |
| 68U05 | Computer graphics; computational geometry (digital and algorithmic aspects) |
| 05C60 | Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) |
Software:
MathOverflowReferences:
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