Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities.

We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in Rℓ, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in ℓ. More precisely, we prove the following. Let R be a real closed field and let ℘ = {P1, … , Pm} ⊂ R[Y1, … ,Yℓ,X1, … ,Xk], with degY(Pi) ≤ 2, degX(Pi) ≤ d, 1 ≤ i ≤ m. Let S ⊂ Rℓ be a semi-algebraic set, defined by a Boolean formula without negations, with atoms of the form P ≥ 0, P ≤ 0, P ∈ ℘. Let π : Rℓ → Rk be the projection on the last k coordinates. Then the number of stable homotopy types amongst the fibers Sx = π−1(x) ∩ S is bounded by (2mℓkd)O(mk). [ABSTRACT FROM AUTHOR]