We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in @math , each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in @math . More precisely, we prove the following. Let @math be a real closed field and let [ P = P_1,...,P_m [Y_1,...,Y_ ,X_1,...,X_k], ] with @math . Let @math be a semi-algebraic set, defined by a Boolean formula without negations, whose atoms are of the form, @math . Let @math be the projection on the last k co-ordinates. Then, the number of stable homotopy types amongst the fibers @math is bounded by [ (2^m k d)^ O(mk) . ]
In another direction Agrachev @cite_5 studied the topology of semi-algebraic sets defined by quadratic inequalities, and he defined a certain spectral sequence converging to the homology groups of such sets. A parametrized version of Agrachev's construction is in fact a starting point of our proof of the main theorem in this paper.
@cite_5
1972610374
In this paper we study sets of real solutions of systems of quadratic equations and inequalities. The results are used for the local study of more general systems of smooth equations and inequalities.