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Reeb spaces of smooth functions on manifolds
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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.
Forward citations
Cited by 2 Pith papers
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Compactifying real analytic functions and resulting Reeb spaces
Explicit real-analytic projections on manifolds in Euclidean space realize prescribed normal graph diagrams for NF of their Reeb digraph compactifications.
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Reconstruction of real algebraic functions into curves with prescribed Reeb graphs
For any finite graph satisfying genericity conditions and any dimension at least 2, the paper constructs real algebraic maps to curves whose Reeb graph is isomorphic to the graph.
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