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Journal of the London Mathematical Society Advance Access published online on May 10, 2009
Journal of the London Mathematical Society, doi:10.1112/jlms/jdp006
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© 2009 London Mathematical Society

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

Andrei Gabrielov

Department of Mathematics
Purdue University
West Lafayette
IN 47907
USA
agabriel@math.purdue.edu

Nicolai Vorobjov

Department of Computer Science
University of Bath
Bath
BA2 7AY
United Kingdom

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.

2000 Mathematics Subject Classification 14P25 (primary), 14P15, 03C64 (secondary).

The first author was supported in part by NSF grant DMS-0801050.

Received October 18, 2007; revised October 15, 2008;
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