• About
  • Policies
  • What is openaccess
  • Library
  • Contact
Advanced search
      View Item 
      •   BUIR Home
      • Scholarly Publications
      • Faculty of Science
      • Department of Mathematics
      • View Item
      •   BUIR Home
      • Scholarly Publications
      • Faculty of Science
      • Department of Mathematics
      • View Item
      JavaScript is disabled for your browser. Some features of this site may not work without it.

      The Euler class of a subset complex

      Thumbnail
      View / Download
      365.0 Kb
      Author
      Güçlükan, A.
      Yalçın, E.
      Date
      2010
      Source Title
      Quarterly Journal of Mathematics
      Print ISSN
      0033-5606
      Electronic ISSN
      1464-3847
      Publisher
      Oxford University Press
      Volume
      61
      Issue
      1
      Pages
      43 - 68
      Language
      English
      Type
      Article
      Item Usage Stats
      115
      views
      106
      downloads
      Abstract
      The subset complex Δ(G) of a finite group G is defined as the simplicial complex whose simplices are non-empty subsets of G. The oriented chain complex of Δ(G) gives a G-module extension of by , where is a copy of integers on which G acts via the sign representation of the regular representation. The extension class ζG ∈ ExtGG-1 (, ) of this extension is called the Ext class or the Euler class of the subset complex Δ (G). This class was first introduced by Reiner and Webb [The combinatorics of the bar resolution in group cohomology, J. Pure Appl. Algebra 190 (2004), 291-327] who also raised the following question: What are the finite groups for which ζG is non-zero?In this paper, we answer this question completely. We show that ζG is non-zero if and only if G is an elementary abelian p-group or G is isomorphic to /9, /4 × /4 or (/2)n × /4 for some integer n ≥ 0. We obtain this result by first showing that ζG is zero when G is a non-abelian group, then by calculating ζG for specific abelian groups. The key ingredient in the proof is an observation by Mandell which says that the Ext class of the subset complex Δ (G) is equal to the (twisted) Euler class of the augmentation module of the regular representation of G.We also give some applications of our results to group cohomology, to filtrations of modules and to the existence of Borsuk-Ulam type theorems. © 2008. Published by Oxford University Press. All rights reserved.
      Permalink
      http://hdl.handle.net/11693/22398
      Published Version (Please cite this version)
      http://dx.doi.org/10.1093/qmath/han025
      Collections
      • Department of Mathematics 614
      Show full item record

      Browse

      All of BUIRCommunities & CollectionsTitlesAuthorsAdvisorsBy Issue DateKeywordsTypeDepartmentsThis CollectionTitlesAuthorsAdvisorsBy Issue DateKeywordsTypeDepartments

      My Account

      Login

      Statistics

      View Usage StatisticsView Google Analytics Statistics

      Bilkent University

      If you have trouble accessing this page and need to request an alternate format, contact the site administrator. Phone: (312) 290 1771
      Copyright © Bilkent University - Library IT

      Contact Us | Send Feedback | Off-Campus Access | Admin | Privacy