Mod 2 cohomology of combinatorial Grassmannians
Mod 2 cohomology of combinatorial Grassmannians
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组合格拉斯曼函数的 Mod 2 上同调
DOI：
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发表时间：
1999
期刊：
影响因子：
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通讯作者：
James F. Davis
中科院分区：
文献类型：
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作者：
L. Anderson;James F. Davis
抽象的。拟阵丛由麦克弗森提出，是实数向量丛的组合类似物。本文建立了拟阵丛的基础。它定义了从实向量丛的同构类到拟阵丛的同构类的自然变换。然后，通过表明拟阵丛的几何实现是球形准纤维，给出了从拟阵丛到球形准纤维的变换。固定秩的有向拟阵的偏序集对拟阵丛进行分类，并且上述变换给出了从拓扑到组合学再到拓扑的分裂。结果是，阶 k 定向拟阵的偏序集（该偏序集对拟阵丛进行分类）的 mod 2 上同调包含前 k 个 Stiefel-Whitney 类上的自由多项式环。
Abstract. Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles. It defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles. It then gives a transformation from matroid bundles to spherical quasifibrations, by showing that the geometric realization of a matroid bundle is a spherical quasifibration. The poset of oriented matroids of a fixed rank classifies matroid bundles, and the above transformations give a splitting from topology to combinatorics back to topology. A consequence is that the mod 2 cohomology of the poset of rank k oriented matroids (this poset classifies matroid bundles) contains the free polynomial ring on the first k Stiefel-Whitney classes.