Maps on manifolds onto graphs locally regarded as the quotient maps onto Reeb spaces of some differentiable maps and a new construction problem

The Reeb space of a function or a map on a manifold is defined as the space of all connected components of preimages and represents the manifold compactly. In fact, Reeb spaces are fundamental and useful tools in geometric theory of so-called Morse functions and more general maps which are sufficiently tame. Can we construct an explicit good function inducing a given graph as the Reeb space (Reeb graph)? These problems were launched by Sharko in 2000s and have been explicitly solved by several researchers. As related pioneering studies, the author also found and solved problems adding constraints on singularities and preimages for example. The present paper concerns new problems on these works. We define the classes of maps onto graphs locally regarded as ones onto the Reeb spaces induced from smooth functions of suitable classes and consider and challenge the problems for the classes.