Mathematical Sciences: RUI: The Homotopy Theory of Arrangements

研究者将继续研究复n空间超平面并补的拓扑不变量对相关排列的组合结构的依赖。研究的不变量包括全称阿贝尔盖的同调、高同伦群的同调、Milnor纤维的同调以及上同环。这些将用下面的拟阵、有向的拟阵及其对非实排列的推广和有界复合体来描述。所得结果将应用于未解决的格拉斯曼矩阵分层、组合不变量的超平面补的同调分类、自由排列的组合结构和orliko - solomon代数的因数分解性质等问题。这项研究的最初重点将放在三维的复杂排列上。主要的拓扑工具是研究者构建的通用覆盖的2-复模型T。T的结构以一种简单的方式由排列实部有界面的复合体决定。这一方面方便了第二同伦群的几何分析。同时，这个泛复盖给出了多项式环上模的阿贝尔复盖同调的代数描述。自由安排的研究将集中在调查员和他的同事引入的各种形式的概念上。n空间中超平面排列的补就是n空间被若干（n-1）个空间切割后的块的集合。这个看似无害的结构从代数和拓扑学的各个部分唤起了惊人数量的重型机械，并导致了一些令人满意的有吸引力的分析和一般组合原理。虽然这项工作是理论性的，并导致严格的证明，但在中间阶段，实验和测试示例是有用的。在这一点上，一些机器计算是必不可少的，特别是使用像Cayley这样的符号操作器，研究者可以使用Sun SPARCstation。

The investigator will continue to study the dependence of topological invariants of the complement of a union of hyperplanes in complex n-space on the combinatorial structure of the associated arrangement. Invariants to be studied include the homology of the universal abelian cover, the higher homotopy groups, the homology of the Milnor fiber, and the cohomology ring. These will be described in terms of the underlying matroid, the oriented matroid and its generalization to non-real arrangements, and the bounded complex. The results will be applied to unsolved problems concerning the matroid stratification of the Grassmannian, the homology classification of hyperplane complements in terms of combinatorial invariants, the combinatorial structure of free arrangements, and the factorization properties of the Orlik-Solomon algebra. The initial focus of this research will be on complexified arrangements in dimension three. The main topological tool is the 2-complex model T of the universal cover constructed by the investigator. The structure of T is determined in a simple way by the complex of bounded faces of the real part of the arrangement. On the one hand, this facilitates a geometric analysis of the second homotopy group. At the same time, this universal cover complex yields an algebraic description of the homology of abelian covers in terms of modules over polynomial rings. The study of free arrangements will center on the various notions of formality introduced by the investigator and his co- workers. The complement of an arrangement of hyperplanes in n-space is just the collection of chunks into which n-space falls when cleaved by a number of (n-1)-spaces. This seemingly innocuous structure has called forth a surprising amount of heavy machinery from various parts of algebra and topology and led to some gratifyingly attractive analysis and general combinatorial principles. Although the work is theoretical and leads to rigorous proofs, at intermediate stages it is useful to experiment and test examples. Some machine computation is indispensable at this point, particularly using such symbolic manipulators as Cayley, for which a Sun SPARCstation is available to the investigator.