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Cohomology of Artin groups

Artin 群的上同调

基本信息

批准号：
16J00125
负责人：
劉曄
金额：
$ 0.83万
依托单位：
Hokkaido University
依托单位国家：
日本
项目类别：
Grant-in-Aid for JSPS Fellows
财政年份：
2016
资助国家：
日本
起止时间：
2016-04-22 至 2018-03-31
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16J00125/
关键词：
Artin group group cohomology hyperplane arrangement Tutte polynomial Artin groups Coxeter groups Group homology

项目摘要

(A)Joint work with T. Akita “Second mod 2 homology of Artin groups” has been published in the journal “Algebraic & Geometric Topology”. The paper uses group-theoretic methods to compute the second mod 2 homology of arbitrary Artin groups, without assuming the K(\pi,1) conjecture holds. The result is an example of the application of the Hopf formula on the second homology of groups.(B)A project joint with T. N. Tran and M. Yoshinaga has been started. The motivation of this project is to explore the topology of complement to an arrangement and relation to the combinatorics of the arrangement.We introduced a “G-Tutte polynomial”, associated to a list of elements of a finitely generated abelian group, where the variable G is an auxiliary abelian group. Our G-Tutte polynomial has the property that by choosing special G, we recover the ordinary and arithmetic polynomials. The G-Tutte polynomials govern the topological and enumerative information of the so-call G-plexification, which is a common generalization of (real) hyperplane arrangements and their complexifications, c-plexification, toric arrangements and mod q reduction arrangements. Our main results show that the characteristic and Poincare polynomials of the G-plexifications are specializations of the G-Tutte polynomials. As consequences, many known results about those polynomials of complexifications, c-plexifications and toric arrangements are uniformly recovered, as well as new results are obtained. For example, we obtained a formula for the characteristic quasi-polynomial of the mod q reduction arrangements.

(A)与T. Akita合著的《Second mod 2 homology of Artin groups》发表在《algeaic & Geometric Topology》杂志上。在不假设K（\pi,1）猜想成立的情况下，利用群论方法计算了任意Artin群的第二模2同调。结果是Hopf公式在群的二次同调上应用的一个例子。(B)与t.n. Tran和吉永先生合作的一个项目已经开始。这个项目的动机是探索一个排列的互补拓扑和排列的组合关系。我们引入了一个“G- tutte多项式”，它与有限生成的阿贝尔群的元素列表相关联，其中变量G是辅助阿贝尔群。我们的G- tutte多项式具有这样的性质：通过选择特殊的G，我们可以恢复普通多项式和算术多项式。G-Tutte多项式控制着所谓的g -丛化的拓扑和枚举信息，g -丛化是（实）超平面排列及其复化、c-丛化、环形排列和模q约简排列的一般推广。我们的主要结果表明g -丛化的特征多项式和庞加莱多项式是g -图特多项式的特化。结果统一地恢复了复数化多项式、c-丛化多项式和环构多项式的许多已知结果，并得到了新的结果。例如，我们得到了模q约简排列的特征拟多项式的一个公式。