On the Reeb spaces of definable maps

TL;DR

This paper establishes that Reeb spaces of proper definable maps in o-minimal structures are proper definable quotients and explores their topological complexity, providing bounds on Betti numbers in specific cases.

Contribution

It proves the Reeb space can be realized as a proper definable quotient and offers bounds on Betti numbers for semi-algebraic maps with bounded domains.

Findings

Reeb space is a proper definable quotient.

Betti numbers of Reeb spaces can be arbitrarily large.

Singly exponential bounds on Betti numbers for semi-algebraic maps.

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 $f$.