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The boundary manifold of a complex line arrangement

Daniel C Cohen and Alexander I Suciu

Geometry & Topology Monographs 13 (2008) 105–146

DOI: 10.2140/gtm.2008.13.105

arXiv: math.GT/0607274

Abstract

We study the topology of the boundary manifold of a line arrangement in CP2, with emphasis on the fundamental group G and associated invariants. We determine the Alexander polynomial Δ(G), and more generally, the twisted Alexander polynomial associated to the abelianization of G and an arbitrary complex representation. We give an explicit description of the unit ball in the Alexander norm, and use it to analyze certain Bieri–Neumann–Strebel invariants of G. From the Alexander polynomial, we also obtain a complete description of the first characteristic variety of G. Comparing this with the corresponding resonance variety of the cohomology ring of G enables us to characterize those arrangements for which the boundary manifold is formal.

For Fred Cohen on the occasion of his sixtieth birthday

Keywords

line arrangement, graph manifold, fundamental group, twisted Alexander polynomial, BNS invariant, cohomology ring, holonomy Lie algebra, characteristic variety, resonance variety, tangent cone, formality

Mathematical Subject Classification

Primary: 32S22

Secondary: 57M27

References
Publication

Received: 30 May 2006
Revised: 29 May 2007
Accepted: 30 May 2007
Published: 22 February 2008

Authors
Daniel C Cohen
Department of Mathematics
Louisiana State University
Baton Rouge LA 70803
USA
http://www.math.lsu.edu/~cohen/
Alexander I Suciu
Department of Mathematics
Northeastern University
Boston MA 02115
USA
http://www.math.neu.edu/~suciu/