KTH/SU Mathematics Colloquium

10 December 2008

Nikolai Vorobiev, University of Bath, England

Compact approximations and topological complexity of definable sets

ABSTRACT Around 1950 Petrovskii and Oleinik, and then in 1960s Milnor and Thom, proved explicit upper bounds on total Betti numbers of real algebraic sets in terms of degrees and numbers of variables of the defining polynomials. The principal difficulty with expanding these results to formulae more general than conjunctions of polynomial equations arises when sets are not locally closed. We describe a construction for approximating arbitrary definable sets by compact ones with dominating topological complexity. This allows, in particular, to improve the known upper bounds on Betti numbers of semialgebraic sets and to obtain a singly exponential bound on Betti numbers of sub-Pfaffian sets.