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On the number of homotopy types of fibres of a definable map. (English) Zbl 1131.14060
The main result of the paper is as follows: given a definable subset of \({\mathbb R}^{m+n}\) over an o-minimal structure over the reals (for example, a semi-algebraic or a restricted sub-Pfaffian set), the number of homotopy types of its projection to \({\mathbb R}^n\) admits a single exponential in \(mn\) upper bound. On the other hand, the only known upper bound for the number of topological types of such a projection is double exponential in \(mn\).
For the proof, the authors define a Whitney stratification of the given set by sign-invariant subsets of a family of real polynomials, then bound the number of homotopy types of fibres by the number of connected components of the complement in \({\mathbb R}^n\) of the projections of strata of dimension \(< n\) united with the critical values of the projection over the strata of dimension \(\geq n\), and finally use the upper bounds to the Betti numbers of projections of semi-algebraic sets established by A. Gabrielov, N. Vorobjov and T. Zell [J. Lond. Math. Soc. 69, 27–43 (2004; Zbl 1087.14038)]. Among applications of the main result are single exponential upper bounds to the number of homotopy types of semi-algebraic sets defined by fewnomials and for the radii of balls guaranteeing the local contractibility for semi-algebraic sets defined by integral polynomials.

14P10 Semialgebraic sets and related spaces
14P25 Topology of real algebraic varieties
32B20 Semi-analytic sets, subanalytic sets, and generalizations
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