Approximation of definable sets by compact families, and upper bounds on homotopy and homology

Reference:

Gabrielov, A. and Vorobjov, N., 2009. Approximation of definable sets by compact families, and upper bounds on homotopy and homology. Journal of the London Mathematical Society, 80 (1), pp. 35-54.

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Official URL:

http://dx.doi.org/10.1112/jlms/jdp006

Abstract

We prove new upper bounds on homotopy and homology groups of o-minimal sets in terms of their approximations by compact o-minimal sets. In particular, we improve the known upper bounds on Betti numbers of semialgebraic sets defined by quantifier-free formulae and obtain, for the first time, a singly exponential bound on Betti numbers of sub-Pfaffian sets.

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