Suspensions of affine arrangements (Q1374729)

An affine arrangement is a finite set of affine subspaces of \(\mathbb{R}^N\). If one takes the complement of such an arrangement and suspends it ``often enough'', one obtains a space whose homotopy type depends only on the poset of intersections together with the dimension function. In this paper it is shown that from Spanier-Whitehead duality one gets explicit upper bounds for the number of suspensions needed, and that these bounds may be strengthened considerably. The method is based on an analysis of a construction of a homotopy equivalence between the compactification of the arrangement and its combinatorial model, given by \textit{G. M. Ziegler} and \textit{R. T. Živaljević} [Math. Ann. 295, No. 3, 527-548 (1993; Zbl 0792.55002)], and on the application of the Spanier-Whitehead duality theorem to non-stable equivalences. The author describes special classes of arrangements for which the results may be applied. In particular, it follows that in the case of complex arrangements (or, more generally, \((\geq 2)\)-arrangements) one single suspension suffices.