Homotopy, Complexity and O-Minimality

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Andrei Gabrielov proposes to investigate homotopy types of definable sets in o-minimal expansions of real closed fields, and the algorithmic complexity of operations with such sets and their families. The fundamental question is to estimate the topological complexity of a definable set in terms of the structural complexity of its defining formula. Recent developments in algorithmic real algebraic geometry and topological o-minimality suggest that an answer to that question can be obtained for a wide class of sets definable in o-minimal structures with the Bezout-type finiteness property. Notably, a combinatorial-geometric construction suggested by A. Gabrielov and N. Vorobjov allows one to approximate arbitrary definable sets by homotopy equivalent definably compact sets, simplifying considerably the study of their topology. Furthermore, homotopy colimit construction allows one to approximate a set defined by a formula with existential quantifiers by a homotopy equivalent simplicial object defined by a quantifier-free formula, and to employ the descent spectral sequence to compute topological invariants of the original set. The proposed research will advance our understanding of the topological properties of the sets definable in o-minimal structures, and of the algorithmic complexity of operations with such sets and their families. It will provide new tools for the o-minimal algebraic topology.The goal of the proposed research is to develop new upper bounds on the topological complexity of semialgebraic sets (defined by formulas with equations and inequalities between polynomials in several real variables)and their generalizations known as definable sets in o-minimal structures. Given an appropriate measure of complexity, such as Bezout theorem bounding the number of zeros of a polynomial, the topological complexity of a definable set depends on the structural complexity of its defining formula. Recently A. Gabrielov and N. Vorobjov suggested a construction replacing a general definable set with a homotopy equivalent compact set, applying a simple combinatorial procedure to the defining formula of the original set. Thus the problem of the topological complexity of the general definable sets can be reduced to the more tractable problem for the compact sets. The proposed research will establish new connections between o-minimal theory, topology, combinatorics, and real algebraic geometry. It will provide new combinatorial and topological tools for development of faster computational algorithms in real algebraic and analytic geometry and its applications in control theory, visualization, and computer-aided design.