Fundamental groups, Alexander invariants, and cohomology jumping loci

Abstract

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 π_1(X). We illustrate this approach with various applications to the study of hyperplane arrangements, Milnor fibrations, 3-manifolds, and right-angled Artin groups.

45 pages; accepted for publication in Contemporary Mathematics

References in corpus (14)

• Topology and geometry of cohomology jump loci
• Completely reducible hypersurfaces in a pencil
• Toric complexes and Artin kernels
• Geometric and algebraic aspects of 1-formality
• Which 3-manifold groups are Kähler groups?
• Boundary manifolds of projective hypersurfaces
• Quasi-Kähler groups, 3-manifold groups, and formality
• Characteristic varieties of nilpotent groups and applications
• The boundary manifold of a complex line arrangement
• Non-formality of Milnor fibers of line arrangements
• Resonance varieties and Dwyer-Fried invariants
• Characterization of Line Arrangement for which the Fundamental Group of the Complement is a Direct Product of Free Groups
• On Parallel Lines and Free Group
• Alexander and Thurston norms of graph links

Cited by in corpus (9)

• Line arrangements and configurations of points with an unusual geometric property
• Hyperplane arrangements and Milnor fibrations
• Chen ranks and resonance
• Characteristic varieties and Betti numbers of free abelian covers
• Around the tangent cone theorem
• Resonance varieties and Dwyer-Fried invariants
• Chen ranks and resonance varieties of the upper McCool groups
• Geometric and homological finiteness in free abelian covers
• Milnor fibrations of arrangements with trivial algebraic monodromy