On the Reeb spaces of definable maps

Saugata Basu; Nathanael Cox; Sarah Percival

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arXiv · 1804.00605

On the Reeb spaces of definable maps

Saugata Basu·Nathanael Cox·Sarah Percival

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Abstract

We prove that the Reeb space of a proper definable map $f:X \rightarrow Y$ in an arbitrary o-minimal expansion of a real closed field is realizable as a proper definable quotient. This result can be seen as an o-minimal analog of Stein factorization of proper morphisms in algebraic geometry. We also show that the Betti numbers of the Reeb space of $f$ 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 $f:X \rightarrow Y$ 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 $f$.

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Saugata Basu, Nathanael Cox, Sarah Percival. 2018-04-02. On the Reeb spaces of definable maps. https://arxiv.org/abs/1804.00605

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