International Mathematics Research Notices

International Mathematics Research Notices (2008) Vol. 2008 : article ID rnm119, 36 pages, doi:10.1093/imrn/rnm119 published on January 15, 2008

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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Alexander Polynomials: Essential Variables and Multiplicities

Alexandru Dimca¹, Stefan Papadima² and Alexander I. Suciu^3,*

¹ Laboratoire J.A. Dieudonné, UMR du CNRS 6621, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice, France
² Institute of Mathematics Simion Stoilow, P.O. Box 1-764, RO-014700 Bucharest, Romania
³ Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston, MA 02115, USA

Correspondence: ^* Correspondence to be sent to: a.suciu{at}neu.ed

We explore the codimension-one strata in the degree-one cohomology jumping loci of a finitely generated group, through the prism of the multivariable Alexander polynomial. As an application, we give new criteria that must be satisfied by fundamental groups of smooth, quasi-projective complex varieties. These criteria establish precisely which fundamental groups of boundary manifolds of complex line arrangements are quasi-projective. We also give sharp upper bounds for the twisted Betti ranks of a group, in terms of multiplicities constructed from the Alexander polynomial. For Seifert links in homology 3-spheres, these bounds become equalities, and our formula shows explicitly how the Alexander polynomial determines all the characteristic varieties.

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This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Request Permissions How to cite this article Google Scholar Articles by Dimca, A. Articles by Suciu, A. I. Search for Related Content Social Bookmarking          What's this?