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The boundary manifold of a complex line arrangement. (English) Zbl 1137.32013

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 \(\mathbb C P^{2}\), 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.
For the entire collection see [Zbl 1133.55001].

MSC:

32S22 Relations with arrangements of hyperplanes
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)

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