Cohomology jump loci of 3-manifolds
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© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Alexander I. Suciu
Cohomology jump loci of 3-manifolds Received: 17 October 2019 / Accepted: 17 November 2020 Abstract. The cohomology jump loci of a space X are of two basic types: the characteristic varieties, defined in terms of homology with coefficients in rank one local systems, and the resonance varieties, constructed from information encoded in either the cohomology ring, or an algebraic model for X . We explore here the geometry of these varieties and the delicate interplay between them in the context of closed, orientable 3-dimensional manifolds and link complements. The classical multivariable Alexander polynomial plays an important role in this analysis. As an application, we derive some consequences regarding the formality and the existence of finite-dimensional models for such 3-manifolds.
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Resonance varieties . . . . . . . . . . . . . . . . . . . . . . . . 3. Characteristic varieties and the Alexander polynomial . . . . . . 4. Algebraic models and the tangent cone theorem . . . . . . . . . 5. Resonance varieties of 3-manifolds . . . . . . . . . . . . . . . . 6. Alexander polynomials and characteristic varieties of 3-manifolds 7. A tangent cone theorem for 3-manifolds . . . . . . . . . . . . . . 8. Connected sums . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Graph manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 10.Links in the 3-sphere . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction 1.1. Cohomology jump loci Let X be a finite, connected CW-complex and let π = π1 (X ) be its fundamental group. The characteristic varieties Vki (X ) are the Zariski closed subsets of the This work was supported by Simons Foundation Collaboration Grants for Mathematicians #354156 and #693825. Alexander I. Suciu (B): Department of Mathematics, Northeastern University, Boston, MA 02115, USA. e-mail: [email protected] URL: http://web.northeastern.edu/suciu/ Mathematics Subject Classification: Primary 55N25 · 57M27 · Secondary 16E45 · 55P62 · 57M05 · 57M25 · 57N10
https://doi.org/10.1007/s00229-020-01264-5
A. I. Suciu
algebraic group Hom(π, C∗ ) consisting of those characters ρ : π → C∗ for which the i-th homology group of X with coefficients in the rank 1 local system defined by ρ has dimension at least k; in particular, the trivial character 1 belongs to Vki (X ) precisely when the i-th Betti number bi (X ) is at least k. Now let H · = H ·(X, C) be the cohomology algebra of X . For each a ∈ H 1 , we may form a cochain complex, (H, a), with differentials δa : H i → H i+1 given by left-multiplication by a. The resonance varieties Rki (X ), then, are the subvarieties of the affine
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