Before revision:  Nevertheless, in the special case when f:X \rightarrow 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.
After revision:  Nevertheless, in the special case when f:X \rightarrow 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.