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 的联合工作“Artin 群的第二模 2 同源性”已发表在《代数与几何拓扑》杂志上，论文使用群论方法计算任意 Artin 群的第二模 2 同源性，不假设 K(\pi,1) 猜想成立。结果是 Hopf 公式在群的第二同调上的应用示例。(B) 与 T. N. Tran 联合的项目M. Yoshinaga 已经开始了这个项目的动机是探索排列补集的拓扑以及与排列组合的关系。我们引入了一个“G-Tutte 多项式”，与有限的元素列表相关联。生成的阿贝尔群，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。多项式控制所谓的 G 复杂化的拓扑和枚举信息，这是（真实）超平面排列及其复化、c 复杂化、环面排列和 mod q 还原排列的常见概括。 G-复合化的庞加莱多项式是 G-Tutte 多项式的特化，因此，关于复化、c-复合化和的多项式有许多已知的结果。环面排列得到了一致的恢复，并且得到了新的结果，例如，我们得到了mod q归约排列的特征拟多项式的公式。