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 b 1 (R f ), in computational geometry called number of loops, is bounded from above by the co-rank of the fundamental group π 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 practical methods for calculating the co-rank of π 1 (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.

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

“…where Ab(G) is the abelianization of G. For more information about the corank and its properties we refer to [4,6,8,14,18]. In the case when G = π 1 (X) the corank of G is also called the first non-commutative Betti number of X (cf.…”

“…The Reeb graph R(f ) of a Morse function f : M → R, as an invariant of the pair (M, f ), is a tool of global analysis attracting more attention recently due to its applications to computer graphics as well as its importance to purely mathematical problems (for more details see [2,5,7,8,18,19,23]). It is constructed by contracting the connected components of levels sets of the function f .…”

Morse-Bott functions with two critical values on a surface

“…For a connected locally path-connected topological space X and a continuous function f : X → R whose Reeb graph R f is a finite topological graph, it holds [6], Theorem 3.1…”

Reeb spaces of smooth functions on manifolds

“…About realization of Reeb graphs, there have been a lot of studies, e.g. by Sharko [29], Martínez-Alfaro, Meza-Sarmiento and Oliveira [18,19,20], Masumoto and Saeki [21], Gelbukh [2,3,4,5], Kaluba, Marzantowicz and Silva [12], Michalak [22,23], Michalak and Marzantowicz [17], Kitazawa [13,14,15], etc. Our theorems generalize some of them.…”