On the Reeb spaces of definable maps

We prove that the Reeb space of a proper definable map in an arbitrary o-minimal expansion of the reals is realizable as a proper definable quotient. We also show that the Betti numbers of the Reeb space of a map f can be arbitrarily large compared to those of X, unlike in the special case of Reeb graphs of manifolds. Nevertheless, in the special case when f : X → Y is a semi-algebraic map and X is closed and bounded, we prove a singly exponential upper bound on the Betti numbers of the Reeb space of f in terms of the number and degrees of the polynomials defining X,Y and f .