A homotopy theorem on oriented matroids. (English) Zbl 0781.52008
Summary: Consider a finite family of hyperplanes \({\mathcal H}=\{H_ 1,\ldots,H_ n\}\) in the finite-dimensional vector space \(\mathbb{R}^ d\). We call chambers (determined by \({\mathcal H})\) the connected components of \(\mathbb{R}^ d\backslash\bigcup^ n_{i=1}H_ i\). Galleries are finite families of chambers \((C_ 0,C_ 1,\ldots,C_ m)\), where exactly one hyperplane separates \(C_{i+1}\) from \(C_ i\), for \(0\leq i\leq m\), and exactly \(m\) hyperplanes separate \(C_ 0\) from \(C_ m\). Using oriented matroid theory, we prove that any two galleries with the same extremities can be derived from each other by a finite number of deformations of the same kind (elementary deformations). When the chambers are simplicial cones, this is a result of P. Deligne [Invent. Math. 17, 273-302 (1972; Zbl 0238.20034)]. Our theorem generalizes also a result of M. Salvetti [Invent. Math. 88, 603-618 (1987; Zbl 0594.57009)].
MSC:
| 52B40 | Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) |
| 05B35 | Combinatorial aspects of matroids and geometric lattices |
References:
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