[Paper Review] Reeb spaces of smooth functions on manifolds

Osamu Saeki|arXiv (Cornell University)|Jun 2, 2020

Topological and Geometric Data Analysis14 references11 citations

TL;DR

This paper establishes that the Reeb space of a smooth function on a closed manifold with finitely many critical values is a finite graph without loops, and proves that any such graph can be realized as the Reeb space of a smooth function with prescribed level set topologies. It further shows that any continuous map from a closed manifold to a finite graph inducing a surjection on fundamental groups is homotopic to the quotient map to the Reeb space of such a function.

ABSTRACT

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 finitely many critical values has the structure of a finite graph without loops. We also show that an arbitrary finite graph without loops can be realized as the Reeb space of a certain smooth function on a closed manifold with finitely many critical values, where the corresponding level sets can also be preassigned. Finally, we show that a continuous map of a smooth closed connected manifold to a finite connected graph without loops that induces an epimorphism between the fundamental groups is identified with the natural quotient map to the Reeb space of a certain smooth function with finitely many critical values, up to homotopy.

Motivation & Objective

• To characterize the topological structure of the Reeb space of smooth functions on closed manifolds with finitely many critical values.
• To prove that every finite graph without loops arises as the Reeb space of a smooth function on a closed manifold with finitely many critical values.
• To establish a homotopy classification of continuous maps from closed manifolds to finite loopless graphs via Reeb space quotients.
• To provide a realization theorem for decorated graphs, where level set diffeomorphism types are preassigned to vertices and edges.
• To show that the fundamental group epimorphism condition is both necessary and sufficient for a map to be homotopic to a Reeb space quotient.

Proposed method

• Prove that the Reeb space of a smooth function with finitely many critical values is a finite graph without loops using topological techniques, particularly Lemma 3.8 on finite components at isolated critical values.
• Construct smooth functions on manifolds by gluing non-singular functions on handles and constant functions on vertices, using consistency conditions on boundary diffeomorphisms.
• Use surgery techniques to embed (m−1)-dimensional submanifolds in the source manifold corresponding to edges of the target graph.
• Apply isotopy and tubular neighborhood constructions to ensure disjointness and smooth gluing of components during function assembly.
• Leverage the contractibility of the graph minus its vertices to establish homotopy equivalence between the quotient map and a given continuous map.
• Utilize the notion of m-decorated graphs to encode preassigned diffeomorphism types of level set components.

Experimental results

Research questions

• RQ1 What topological structure does the Reeb space of a smooth function on a closed manifold with finitely many critical values admit?
• RQ2 Can any finite graph without loops be realized as the Reeb space of a smooth function on a closed manifold with finitely many critical values and prescribed level set topologies?
• RQ3 Under what conditions is a continuous map from a closed manifold to a finite loopless graph homotopic to the quotient map to the Reeb space of a smooth function with finitely many critical values?
• RQ4 What is the relationship between the fundamental group of a manifold and the realizability of a given graph as a Reeb space?
• RQ5 How can one construct a smooth function with finitely many critical values whose Reeb space matches a given decorated graph?

Key findings

• The Reeb space of a smooth function on a closed manifold with finitely many critical values is always a finite graph without loops.
• Every finite graph without loops can be realized as the Reeb space of a smooth function on a closed manifold with finitely many critical values, with preassigned diffeomorphism types for level set components.
• A continuous map from a closed connected manifold of dimension $m \geq 2$ to a finite connected graph without loops is homotopic to the Reeb space quotient map of a smooth function with finitely many critical values if and only if it induces an epimorphism on fundamental groups.
• The condition on the fundamental group is necessary and sufficient for such a homotopy realization.
• The first Betti number of the graph is bounded above by the co-rank of the fundamental group of the manifold, providing a topological obstruction to realization.
• The path Reeb space does not necessarily have a graph structure, even for smooth functions with finitely many critical values, but coincides with the standard Reeb space when the function has finitely many critical points.