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

Abstract

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 b₁(R_f), in computational geometry called number of loops, is bounded from above by the co-rank of the fundamental group π₁(X), the condition of local path-connectedness being important since generally b₁(R_f) can even exceed b₁(X). We give some practical methods for calculating the co-rank of π₁(X) and a closely related value, the isotropy index. We apply our bound to improve upper bounds on the distortion of the Reeb quotient map, and thus on the Gromov-Hausdorff approximation of the space by Reeb graphs, for the distance function on a compact geodesic space and for a simple Morse function on a closed Riemannian manifold. This distortion is bounded from below by what we call the Reeb width b(M) of a metric space M, which guarantees that any real-valued continuous function on M has large enough contour (connected component of a level set). We show that for a Riemannian manifold, b(M) is non-zero and give a lower bound on it in terms of characteristics of the manifold. In particular, we show that any real-valued continuous function on a closed Euclidean unit ball E of dimension at least two has a contour C with diam(C ∩ ∂E) ≥ √3.