TY - JOUR

T1 - Reeb Spaces of Smooth Functions on Manifolds

AU - Saeki, Osamu

N1 - Publisher Copyright:
© 2021 The Author(s). Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com.

PY - 2022/6/1

Y1 - 2022/6/1

N2 - 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. Dedicated to Professor Toshizumi Fukui on the occasion of his 60th birthday.

AB - 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. Dedicated to Professor Toshizumi Fukui on the occasion of his 60th birthday.

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U2 - 10.1093/imrn/rnaa301

DO - 10.1093/imrn/rnaa301

M3 - Article

AN - SCOPUS:85136155015

VL - 2022

SP - 8740

EP - 8768

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 11

ER -