Cohomology of Artin groups

(A)与T.Akita合作的“Artin群的第二mod 2同调”已发表在“代数与几何拓扑学”期刊上。本文用群论方法计算了任意Artin群的二次mod2同调，而不假定K(\pi，1)猜想成立。这一结果是Hopf公式在群的二次同调中应用的一个例子。(B)与T.N.Tran和M.Yoshinaga联合的项目已经开始。这个项目的动机是探索一个排列的补集的拓扑及其与该排列的组合学的关系。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有关，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-丛化的拓扑和计数信息，G-丛化是(实)超平面排列及其复化、c-丛化、环面排列和mod-q约化排列的常见推广。我们的主要结果表明，G-丛的特征和Poincare多项式是G-Tutte多项式的特化。作为推论，我们得到了许多关于复化、c-复化和环面排列多项式的已知结果，并得到了一些新的结果。例如，我们得到了mod-q约简排列的特征拟多项式的一个公式。

(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.