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arXiv · 0706.2499

Alexander polynomials: Essential variables and multiplicities

Alexandru Dimca·Stefan Papadima·Alexander I. Suciu
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Abstract

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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Alexandru Dimca, Stefan Papadima, Alexander I. Suciu. 2007-06-17. Alexander polynomials: Essential variables and multiplicities. https://doi.org/10.1093/imrn%2Frnm119

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