Fundamental groups, Alexander invariants, and cohomology jumping loci

Details Fundamental groups, Alexander invariants, and cohomology jumping loci Fundamental groups, Alexander invariants, and cohomology jumping loci

2009-10-08

arxiv.org/abs/0910.1559v3

Contemporary Math., Amer. Math. Soc., vol. 538 (2011), pp. 179-223

Mathematics

Geometric Topology

45 pages; accepted for publication in Contemporary Mathematics

Scientific paper

We survey the cohomology jumping loci and the Alexander-type invariants associated to a space, or to its fundamental group. Though most of the material is expository, we provide new examples and applications, which in turn raise several questions and conjectures. The jump loci of a space X come in two basic flavors: the characteristic varieties, or, the support loci for homology with coefficients in rank 1 local systems, and the resonance varieties, or, the support loci for the homology of the cochain complexes arising from multiplication by degree 1 classes in the cohomology ring of X. The geometry of these varieties is intimately related to the formality, (quasi-) projectivity, and homological finiteness properties of \pi_1(X). We illustrate this approach with various applications to the study of hyperplane arrangements, Milnor fibrations, 3-manifolds, and right-angled Artin groups.

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