Cohomology of Artin groups

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(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.