Let ${ \mathcal S}(\R)$ be an o-minimal structure over $\R$, $T \subset \R^{ k_1+k_2+\ell}$ a closed definable set, and $$ \displaylines{ \pi_1: \R^{ k_1+k_2+\ell}\to \R^{ k_1 + k_2}, \pi_2: \R^{ k_1+k_2+\ell}\to \R^{ \ell}, \ \pi_3: \R^{ k_1 + k_2} \to \R^{ k_2}} $$ the projection maps. For any collection ${ \mathcal A} = \{ A_1,...,A_n\}$ of subsets of $\R^{ k_1+k_2}$, and $\z \in \R^{ k_2}$, let $\A_\z$ denote the collection of subsets of $\R^{ k_1}$, $\{ A_{ 1,\z},..., A_{ n,\z}\}$, where $A_{ i,\z} = A_i \cap \pi_3^{ -1}(\z), 1 \leq i \leq n$. We prove that there exists a constant $C = C(T) > 0,$ such that for any family ${ \mathcal A} = \{ A_1,...,A_n\}$ of definable sets, where each $A_i = \pi_1(T \cap \pi_2^{ -1}(\y_i))$, for some $\y_i \in \R^{ \ell}$, the number of distinct stable homotopy types of $\A_\z, \z \in \R^{ k_2}$, is bounded by $ \displaystyle{ C \cdot n^{ (k_1+1)k_2},} $ while the number of distinct homotopy types is bounded by $ \displaystyle{ C \cdot n^{ (k_1+3)k_2}.} $ This generalizes to the general o-minimal setting, bounds of the same type proved in \cite{ BV} for semi-algebraic and semi-Pfaffian families. One main technical tool used in the proof of the above results, is a topological comparison theorem which might be of independent interest in the study of arrangements.
Categories: math.CO math.GT