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2017 Fiscal Year Annual Research Report

Cohomology of Artin groups

Research Project

Project/Area Number 16J00125
Research InstitutionHokkaido University

Principal Investigator

劉 曄  北海道大学, 大学院理学研究院, 特別研究員(PD)

Project Period (FY) 2016-04-22 – 2018-03-31
KeywordsArtin group / group cohomology / hyperplane arrangement / Tutte polynomial
Outline of Annual Research Achievements

(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.

Research Progress Status

29年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

29年度が最終年度であるため、記入しない。

  • Research Products

    (9 results)

All 2018 2017 Other

All Journal Article (2 results) (of which Peer Reviewed: 1 results) Presentation (6 results) (of which Int'l Joint Research: 3 results,  Invited: 1 results) Remarks (1 results)

  • [Journal Article] Second mod 2 homology of Artin groups2018

    • Author(s)
      Toshiyuki Akita, Ye Liu
    • Journal Title

      Algebraic & Geometric Topology

      Volume: 18 Pages: 547-568

    • DOI

      10.2140/agt.2018.18.547

    • Peer Reviewed
  • [Journal Article] Homology of Artin groups: A combinatorial group theoretic approach2018

    • Author(s)
      Ye Liu
    • Journal Title

      Oberwolfach Report

      Volume: 印刷中 Pages: 印刷中

    • DOI

      10.4171/OWR/2018/2

  • [Presentation] Homology of Artin groups: A combinatorial group theoretic approach2018

    • Author(s)
      Ye Liu
    • Organizer
      Topology of Arrangements and Representation Stability
    • Int'l Joint Research / Invited
  • [Presentation] G-Tutte polynomials2018

    • Author(s)
      Ye Liu
    • Organizer
      The 13th East Asian School of Knots and Related Topics
    • Int'l Joint Research
  • [Presentation] On chromatic functors of graphs2018

    • Author(s)
      Ye Liu
    • Organizer
      第23回代数若手研究会
  • [Presentation] Garside groups and Artin groups2018

    • Author(s)
      Ye Liu
    • Organizer
      Problems around hyperplane arrangements
  • [Presentation] Homology of tropical moduli spaces2017

    • Author(s)
      Ye Liu
    • Organizer
      International Workshop on Topology and Combinatorics
    • Int'l Joint Research
  • [Presentation] Chromatic quasisymmetric functions vs Hessenberg varieties2017

    • Author(s)
      Ye Liu
    • Organizer
      離散幾何構造セミナー
  • [Remarks] Ye Liu's homepage

    • URL

      https://sites.google.com/view/yeliumath

URL: 

Published: 2018-12-17  

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