For a real oriented hyperplane arrangement, we show that the corresponding Salvetti complex is homotopy equivalent to the complement of the complexified arrangement. This result was originally proved by M. Salvetti. Our proof follows the framework of a proof given by L. Paris and relies heavily on the notation of oriented matroids. We also show that homotopy equivalence is preserved when we quotient by the action of the corresponding reflection group. In particular, the Salvetti complex of the braid arrangement in $\ell$ dimensions modulo the action of the symmetric group is a cell complex which is homotopy equivalent to the space of unlabelled configurations of $\ell$ distinct points. Lastly, we describe a construction of the orbit complex from the dual complex for all finite reflection arrangements in dimension 2. This description yields an easy derivation of the so-called "braid relations" in the case of braid arrangement.
我们对库拉托斯基图平面性准则的陈述和罗伯逊-西摩-托马斯内在联系准则的陈述的6/7提供以下解释。我们称一个胞腔复形为“二分的”，如果每一个胞腔都对应一个唯一的具有顶点补集的胞腔。则每个二分胞腔复形都PL同胚于一个球面;恰好存在两个3维二分胞腔复形，它们的1-η是K_5和K_{3,3} ;恰好存在六个4维二分胞腔复形，它们的1-η是Petersen族中除一个图外的所有图。在n>2的高维中，我们观察到，为了用大量的“禁止子式”来刻画嵌入到S^{2n}中的紧致n-多面体，只要建立所有（n-1）-连通n维有限胞腔复形的列表的有限性就足够了，这些有限细胞复形不嵌入到S^{2n}中，但它们的所有真子复形和真象细胞的组合子都嵌入到那里。我们的主要结果是，该列表包含（2n+1）维二分胞腔复形的n-ε。5维二分胞腔复形形的2-η（除了（2 i +2）-单纯形的i-η的三个连接之外）至少包括十个非单纯复形。
Quantifying the homology of periodic cell complexes
School of Mathematical Sciences, Queen Mary University of London（伦敦大学玛丽女王学院数学科学学院）；Research School of Physics, The Australian National University（澳大利亚国立大学物理研究学院）
