Abstract
Let \({\mathcal{S}}({\mathbb{R}})\) be an o-minimal structure over ℝ, \(T\subset {\mathbb{R}}^{k_{1}+k_{2}+\ell}\) a closed definable set, and
the projection maps as depicted below:
For any collection \({\mathcal{A}}=\{A_{1},\ldots,A_{n}\}\) of subsets of \({\mathbb{R}}^{k_{1}+k_{2}}\) , and \(\mathbf{z}\in {\mathbb{R}}^{k_{2}}\) , let \(\mathcal {A}_{\mathbf{z}}\) denote the collection of subsets of \({\mathbb{R}}^{k_{1}}\)
where \(A_{i,\mathbf{z}}=A_{i}\cap\pi_{3}^{-1}(\mathbf{z}),\;1\leq i\leq n\) . We prove that there exists a constant C=C(T)>0 such that for any family \({\mathcal{A}}=\{A_{1},\ldots,A_{n}\}\) of definable sets, where each A i =π 1(T∩π −12 (y i )), for some y i ∈ℝℓ, the number of distinct stable homotopy types amongst the arrangements \(\mathcal {A}_{\mathbf{z}},\mathbf{z}\in {\mathbb{R}}^{k_{2}}\) is bounded by \(C\cdot n^{(k_{1}+1)k_{2}}\) while the number of distinct homotopy types is bounded by \(C\cdot n^{(k_{1}+3)k_{2}}.\) This generalizes to the o-minimal setting, bounds of the same type proved in Basu and Vorobjov (J. Lond. Math. Soc. (2) 76(3):757–776, 2007) for semi-algebraic and semi-Pfaffian families. One technical tool used in the proof of the above results is a pair of topological comparison theorems reminiscent of Helly’s theorem in convexity theory. These theorems might be of independent interest in the quantitative study of arrangements.
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The author was supported in part by NSF grant CCF-0634907.
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Basu, S. On the Number of Topological Types Occurring in a Parameterized Family of Arrangements. Discrete Comput Geom 40, 481–503 (2008). https://doi.org/10.1007/s00454-008-9079-5
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DOI: https://doi.org/10.1007/s00454-008-9079-5