关于参数化构形族中出现的拓扑型的个数
On the number of topological types occurring in a parametrized family of arrangements
摘要：
设${\mathcal S}(\R)$是$\R$上的o-极小结构，$T \subset \R^{k_1+k_2+\ell}$是闭可定义集，$$ \displaylines{π_1: \R^{k_1+k_2+\ell}\to \R^{k_1 + k_2}, π_2: \R^{k_1+k_2+\ell}\to \R^{\ell}, \ π_3: \R^{k_1 + k_2} \to \R^{k_2}} $$是投影映射.对于$\R^{k_1+k_2}$和$\z \in \R^{k_2}$的子集的任何集合${\mathcal A} = \{A_1,...,A_n\}$，令$\A_\z$表示$\R^{k_1}$，$\{A_{1,\z},..., A_{n,\z}\}$的子集的集合，其中$A_{i,\z} = A_i \cap π_3^{-1}(\z), 1 \leq i \leq n$。本文证明了存在一个常数$C = C(T) > 0,$，使得对于任何可定义集族${\mathcal A} = \{A_1,...,A_n\}$，其中每个$A_i = π_1(T \cap π_2^{-1}(\y_i))$，对于某些$\y_i \in \R^{\ell}$，$\A_\z, \z \in \R^{k_2}$的不同稳定同伦类型的数目，由$ \displaystyle{C \cdot n^{(k_1+1)k_2},} $有界，而不同同伦类型的数目由$ \displaystyle{C \cdot n^{(k_1+3)k_2}.} $有界。在\cite{BV}中证明了半代数和半Pfidian族的相同类型的界限。一个主要的技术工具，用于证明上述结果，是一个拓扑比较定理，这可能是独立的利益，在研究安排。
英文摘要：
Let ${\mathcal S}(\R)$ be an o-minimal structure over $\R$, $T \subset \R^{k_1+k_2+\ell}$ a closed definable set, and $$ \displaylines{π_1: \R^{k_1+k_2+\ell}\to \R^{k_1 + k_2}, π_2: \R^{k_1+k_2+\ell}\to \R^{\ell}, \ π_3: \R^{k_1 + k_2} \to \R^{k_2}} $$ the projection maps. For any collection ${\mathcal A} = \{A_1,...,A_n\}$ 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 $A_{i,\z} = A_i \cap π_3^{-1}(\z), 1 \leq i \leq n$. 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 $A_i = π_1(T \cap π_2^{-1}(\y_i))$, 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 $ \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.
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