# Loops in Reeb Graphs of n-Manifolds

@article{Gelbukh2018LoopsIR, title={Loops in Reeb Graphs of n-Manifolds}, author={Irina Gelbukh}, journal={Discrete \& Computational Geometry}, year={2018}, volume={59}, pages={843-863} }

The Reeb graph of a smooth function on a connected smooth closed orientable n-manifold is obtained by contracting the connected components of the level sets to points. The number of loops in the Reeb graph is defined as its first Betti number. We describe the set of possible values of the number of loops in the Reeb graph in terms of the co-rank of the fundamental group of the manifold and show that all such values are realized for Morse functions and, except on surfaces, even for simple Morse…

## 11 Citations

### ON REEB GRAPHS INDUCED FROM SMOOTH FUNCTIONS ON 3-DIMENSIONAL CLOSED MANIFOLDS WITH FINITELY MANY SINGULAR VALUES

- Mathematics
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The Reeb graph of a smooth function on a smooth manifold is the graph obtained as the space of all connected components of preimages (level sets) such that the set of all vertices coincides with the…

### A finite graph is homeomorphic to the Reeb graph of a Morse–Bott function

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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…

### On Reeb graphs induced from smooth functions on 3-dimensional closed orientable manifolds with finitely many singular values

- Mathematics
- 2019

The Reeb graph of a function on a smooth manifold is the graph obtained as the space of all connected components of inverse images such that the set of all vertices coincides with the set of all…

### Combinatorial Modifications of Reeb Graphs and the Realization Problem

- MathematicsDiscrete & Computational Geometry
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We prove that, up to homeomorphism, any graph subject to natural necessary conditions on orientation and the cycle rank can be realized as the Reeb graph of a Morse function on a given closed…

### Reeb Spaces of Smooth Functions on Manifolds

- Mathematics
- 2020

The Reeb space of a continuous function is the space of connected components of the level sets. In this paper we first prove that the Reeb space of a smooth function on a closed manifold with…

### Approximation of metric spaces by Reeb graphs: Cycle rank of a Reeb graph, the co-rank of the fundamental group, and large components of level sets on Riemannian manifolds

- MathematicsFilomat
- 2019

For a connected locally path-connected topological space X and a continuous
function f on it such that its Reeb graph Rf is a finite topological graph,
we show that the cycle rank of Rf, i.e., the…

### Realization of a graph as the Reeb graph of a Morse function on a manifold

- MathematicsTopological Methods in Nonlinear Analysis
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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 $M$…

### Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups

- Mathematics
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In this work we present a construction of correspondence between epimorphisms $\varphi \colon \pi_1(M) \to F_r$ from the fundamental group of a compact manifold $M$ onto the free group of rank $r$,…

### Title REEB GRAPHS OF SMOOTH FUNCTIONS ON MANIFOLDS (Singularity theory of differentiable maps

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In this article we announce three theorems on Reeb spaces of smooth functions on closed manifolds with finitely many critical values.

### ON THE MAXIMUM NUMBER OF PERIOD ANNULI FOR SECOND ORDER CONSERVATIVE EQUATIONS

- MathematicsMathematical Modelling and Analysis
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We consider a second order scalar conservative differential equation whose potential function is a Morse function with a finite number of critical points and is unbounded at infinity. We give an…

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