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Zbl 1137.32013
Cohen, Daniel C.; Suciu, Alexander I.
The boundary manifold of a complex line arrangement. (English)
[A] Iwase, Norio (ed.) et al., Proceedings of the conference on groups, homotopy and configuration spaces, University of Tokyo, Japan, July 5--11, 2005 in honor of the 60th birthday of Fred Cohen. Coventry: Geometry \& Topology Publications. Geometry and Topology Monographs 13, 105-146 (2008).

Summary: We study the topology of the boundary manifold of a line arrangement in $\Bbb C P^{2}$, with emphasis on the fundamental group $G$ and associated invariants. We determine the Alexander polynomial $\Delta (G)$, 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.

MSC 2000:: *32S22 Relations with arrangements of hyperplanes
57M27 Invariants of knots and 3-manifolds
52C35 Arrangements of points, flats, hyperplanes

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

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