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arXiv · 2006.01689

Reeb spaces of smooth functions on manifolds

Osamu Saeki
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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.

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BibTeXRIS

Osamu Saeki. 2020-06-02. Reeb spaces of smooth functions on manifolds. https://arxiv.org/abs/2006.01689

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