# 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.

@article{Gelbukh2019ApproximationOM, title={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.}, author={Irina Gelbukh}, journal={arXiv: Metric Geometry}, year={2019} }

For a connected locally path-connected topological space $X$ and a continuous function $f$ on it such that its Reeb graph $R_f$ is a finite topological graph, we show that the cycle rank of $R_f$, i.e., the first Betti number $b_1(R_f)$, in computational geometry called \emph{number of loops}, is bounded from above by the co-rank of the fundamental group $\pi_1(X)$, the condition of local path-connectedness being important since generally $b_1(R_f)$ can even exceed $b_1(X)$. We give some… Expand

#### 6 Citations

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

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- 2019

Combinatorial Modifications of Reeb Graphs and the Realization Problem

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- Discret. Comput. Geom.
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#### References

SHOWING 1-10 OF 42 REFERENCES

The co-rank of the fundamental group: The direct product, the first Betti number, and the topology of foliations

- Mathematics
- 2015

Combinatorial Modifications of Reeb Graphs and the Realization Problem

- Mathematics, Computer Science
- Discret. Comput. Geom.
- 2021