On the number of topological types occurring in a parametrized family of arrangements
arXiv:0704.0295
Abstract
Let be an o-minimal structure over , a closed definable set, and $$ \displaylines{π_1: \R^{k_1+k_2+\ell}\to \R^{k_1 + k_2}, π_2: \R^{k_1+k_2+\ell}\to \R^{\ell}, \ π_3: \R^{k_1 + k_2} \to \R^{k_2}} $$ the projection maps. For any collection of subsets of , and $\z \in \R^{k_2}$, let $\A_\z$ denote the collection of subsets of , $\{A_{1,\z},..., A_{n,\z}\}$, where $A_{i,\z} = A_i \cap π_3^{-1}(\z), 1 \leq i \leq n$. We prove that there exists a constant such that for any family of definable sets, where each $A_i = π_1(T \cap π_2^{-1}(\y_i))$, 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 while the number of distinct homotopy types is bounded by 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.
20 pages, 2 figures. Revised version with updated bibliography and corrected references