On the Reeb spaces of definable maps
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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-algebraicmap 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.
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