论文标题
定向矩形和Poset准核能的模块化平面
Modular flats of oriented matroids and poset quasi-fibrations
论文作者
论文摘要
我们研究了定向矩阵的模块化平面的组合及其对其Salvetti复合物的拓扑后果。我们表明，我们所谓的poset quasi-bibration在核心模块化平面的局部萨尔维蒂建筑群上的自然地图 - 这是源自质量$ k $ - 理论的Quillen的基本定理B的概念。直接的结果，几何晶格的定向矩阵的salvetti复合物是$ k（π，1）$ - 空间 - 由于Falk，Randell和TeraO而导致的超溶剂超平面布置的经典结果的概括。此外，可超隔离的矩阵的Salvetti复合体的基本组是有限生成的自由组的迭代半领产品 - 类似于可实现的情况。 我们的主要工具是离散的摩尔斯理论，即定向的矩阵的共振复合物的某些子复合物的可撒性，定位图的POSET纤维的不错的组合分解以及与模块化元件相关的共证posets的同构。 我们提供了简单的可超过方向的矩阵的结构。这给出了许多不可交易的超偏置的基生态学，并且通过我们的主要结果，非球面CW-复合物。
We study the combinatorics of modular flats of oriented matroids and the topological consequences for their Salvetti complexes. We show that the natural map to the localized Salvetti complex at a modular flat of corank one is what we call a poset quasi-fibration -- a notion derived from Quillen's fundamental Theorem B from algebraic $K$-theory. As a direct consequence, the Salvetti complex of an oriented matroid whose geometric lattice is supersolvable is a $K(π,1)$-space -- a generalization of the classical result for supersolvable hyperplane arrangements due to Falk, Randell and Terao. Furthermore, the fundamental group of the Salvetti complex of a supersolvable oriented matroid is an iterated semidirect product of finitely generated free groups -- analogous to the realizable case. Our main tools are discrete Morse theory, the shellability of certain subcomplexes of the covector complex of an oriented matroid, a nice combinatorial decomposition of poset fibers of the localization map, and an isomorphism of covector posets associated to modular elements. We provide a simple construction of supersolvable oriented matroids. This gives many non-realizable supersolvable oriented matroids and by our main result aspherical CW-complexes.
