On the number of topological types occurring in a parametrized family of arrangements

Let $\mathcal{S}(\mathbb{R})$ be an o-minimal structure over $\mathbb{R}$, $T \subset \R^{k_1+k_2+\ell}$acloseddefinableset,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 $\mathcal{A}=\{A_1,\mathrm{.}\mathrm{.}\mathrm{.},A_n\}$ of subsets of ${\mathbb{R}}^{k_1+k_2}$, and $\text{z}\in {\mathbb{R}}^{k_2}$, let ${\text{A}}_{\text{z}}$ denote the collection of subsets of ${\mathbb{R}}^{k_1}$, $\{A_{1,\text{z}},\mathrm{.}\mathrm{.}\mathrm{.},A_{n,\text{z}}\}$, where $A_{i,\z} = A_i \cap \pi_3^{-1}(\z), 1 \leq i \leq n$\mathrm{.}Weprovethatthereexistsaconstant$C = C(T) > 0,$suchthatforanyfamily${\mathcal A} = {A_1,…,A_n}$ of definable sets, where each $A_i={\pi }_1(T\cap {\pi }_2^{-1}({\text{y}}_i))$, for some ${\text{y}}_i\in {\mathbb{R}}^{\mathrm{\ell }}$, the number of distinct stable homotopy types of $\A_\z, \z \in \R^{k_2}$,isboundedby$ \displaystyle{C \cdot n^{(k_1+1)k_2},} $whilethenumberofdistincthomotopytypesisboundedby$ \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.