Reeb Spaces of Smooth Functions on Manifolds

Abstract The Reeb space of a continuous function is the connected components level sets. In this paper, we first prove that smooth on closed manifold with finitely many critical values has structure finite graph without loops. We also show an arbitrary loops can be realized as certain values, where corresponding sets preassigned. Finally, map to induces epimorphism between fundamental groups identified natural quotient up homotopy. Dedicated Professor Toshizumi Fukui occasion his 60th birthday.