Chow–Witt rings and topology of flag varieties

IF 0.8 2区 数学 Q2 MATHEMATICS Journal of Topology Pub Date : 2024-11-17 DOI:10.1112/topo.70004
Thomas Hudson, Ákos K. Matszangosz, Matthias Wendt
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Abstract

The paper computes the Witt-sheaf cohomology rings of partial flag varieties in type A in terms of the Pontryagin classes of the subquotient bundles. The proof is based on a Leray–Hirsch-type theorem for Witt-sheaf cohomology for the maximal rank cases, and a detailed study of cohomology ring presentations and annihilators of characteristic classes for the general case. The computations have consequences for the topology of real flag manifolds: we show that all torsion in the integral cohomology is 2-torsion, which was not known in full generality previously. This allows for example to compute the Poincaré polynomials of complete flag varieties for cohomology with twisted integer coefficients. The computations also allow to describe the Chow–Witt rings of flag varieties, and we sketch an enumerative application to counting flags satisfying multiple incidence conditions to given hypersurfaces.

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周维特环和旗变拓扑学
这篇论文根据子曲束的庞特里亚金类计算了 A 型偏旗变体的维特-舍夫同调环。证明基于最大秩情况下维特-舍夫同调的勒雷-赫希类型定理,以及对一般情况下同调环呈现和特征类湮没器的详细研究。这些计算对实旗流形的拓扑学有影响:我们证明了积分同调中的所有扭转都是2扭转,而这在以前是不为人所知的。举例来说,这使得我们可以计算具有扭曲整数系数的同调的完整旗流形的波恩卡列多项式。计算还可以描述旗状变体的 Chow-Witt 环,我们还勾画了一个枚举应用,用于计算满足给定超曲面多重入射条件的旗状变体。
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来源期刊
Journal of Topology
Journal of Topology 数学-数学
CiteScore
2.00
自引率
9.10%
发文量
62
审稿时长
>12 weeks
期刊介绍: The Journal of Topology publishes papers of high quality and significance in topology, geometry and adjacent areas of mathematics. Interesting, important and often unexpected links connect topology and geometry with many other parts of mathematics, and the editors welcome submissions on exciting new advances concerning such links, as well as those in the core subject areas of the journal. The Journal of Topology was founded in 2008. It is published quarterly with articles published individually online prior to appearing in a printed issue.
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