On the Number of Topological Types Occurring in a Parameterized Family of Arrangements
Let S(R) be an o-minimal structure over R, T subset of Rk1+k2+l a closed definable set, and pi(1) : Rk1+k2+l -> Rk1+k2, pi(2) : Rk1+k2+l -> R-l, pi(3) : Rk1+k2 -> R-k2 the projection maps as depicted below: [GRAPHICS] For any collection A = {A(1),...,A(n)} of subsets of Rk1+k2, and z epsilon R-k2, let A(z) denote the collection of subsets of R-k1 {A(1), z,..., A(n,z)}, where Ai,z = Ai boolean AND pi(-1)(3) (z), 1   0 such that for any family A = {A(1),...,A(n)} of definable sets, where each A(i) = pi(1)(T boolean AND pi(-1)(2) (y(i))), for some y(i) epsilon R-l, the number of distinct stable homotopy types amongst the arrangements A(z), z epsilon R-k2 is bounded by C center dot n((k1+ 1)k2) while the number of distinct homotopy types is bounded by C center dot n((k1+3)k2). 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.
arrangements o-minimal structures homotopy types combinatorial complexity