Poincaré duality and resonance varieties

Suciu, Alexander I.

doi:10.1017/prm.2019.55

Cited by 6 publications

(14 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…with differentials δ i a puq " a ¨u for u P H i . It is readily checked that the specialization of the cochain complex (68) at a coincides with (74); see for instance [19,65].…”

Section: Resonance Varietiesmentioning

confidence: 97%

“…These sets are homogeneous algebraic subvarieties of the affine space H The following (well-known) lemma shows that the resonance varieties are determinantal varieties of the infinitesimal Alexander module, and thus, Zariski closed subsets of the affine space H 1 . Proofs in various levels of generality have been given, for instance, in [43,52,65]. We give here a quick proof, in a slightly greater generality, along the lines of the proof of Lemma 11.…”

Section: Resonance Varietiesmentioning

confidence: 99%

“…A linear subspace U Ă H 1 is said to be isotropic if the restriction of Y G : H 1^H1 Ñ H 2 to U^U vanishes; that is, ab " 0 for all a, b P U. As noted in [65,Lemma 2.2], the variety R k pGq contains every isotropic subspace of H 1 whose dimension is at most k`1; moreover, R 1 pGq is the union of all isotropic planes in H 1 .…”

Section: Alexander I Suciumentioning

confidence: 99%

See 2 more Smart Citations

Alexander invariants and cohomology jump loci in group extensions

Suciu

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We study the integral, rational, and modular Alexander invariants, as well as the cohomology jump loci of groups arising as extensions with trivial algebraic monodromy. Our focus is on extensions of the form 1→K→G→Q→1, where Q is an abelian group acting trivially on H1(K;ℤ), with suitable modifications in the rational and mod-p settings. We find a tight relationship between the Alexander invariants, the characteristic varieties, and the resonance varieties of the groups K and G. This leads to an inequality between the respective Chen ranks, which becomes an equality in degrees greater than 1 for split extensions.

show abstract

“…with differentials δ i a puq " a ¨u for u P H i . It is readily checked that the specialization of the cochain complex (68) at a coincides with (74); see for instance [19,65].…”

Section: Resonance Varietiesmentioning

confidence: 97%

Section: Resonance Varietiesmentioning

confidence: 99%

See 1 more Smart Citation

Alexander invariants and cohomology jump loci in group extensions

Suciu

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Resonance Varietiesmentioning

confidence: 99%

Alexander invariants and cohomology jump loci in group extensions

Suciu¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We study the integral, rational, and modular Alexander invariants, as well as the cohomology jump loci of groups arising as extensions with trivial algebraic monodromy. Our focus is on extensions of the form 1 Ñ K Ñ G Ñ Q Ñ 1, where Q is an abelian group acting trivially on H 1 pK; Zq, with suitable modifications in the rational and mod-p settings. We find a tight relationship between the Alexander invariants, the characteristic varieties, and the resonance varieties of the groups K and G. This leads to an inequality between the respective Chen ranks, which becomes an equality in degrees greater than 1 for split extensions.

show abstract

“…satisfies Poicaré duality and the differential d vanishes on A m−1 . Building on work from [25,33], we show in Theorem 4.6 that, for such cdgas, the involution a → −a on A 1 restricts to isomorphisms…”

mentioning

confidence: 96%

Cohomology, Bocksteins, and resonance varieties in characteristic 2

Suciu¹

2022

Preprint

Self Cite

View full text Add to dashboard Cite

We use the action of the Bockstein homomorphism on the cohomology ring H . (X, Z 2 ) of a finite-type CW-complex X in order to define the resonance varieties of X in characteristic 2. Much of the theory is done in the more general framework of the Maurer-Cartan sets and the resonance varieties attached to a finite-type commutative differential graded algebra. We illustrate these concepts with examples mainly drawn from closed manifolds, where Poincaré duality over Z 2 has strong implications on the nature of the resonance varieties.

show abstract

Poincaré duality and resonance varieties

Cited by 6 publications

References 35 publications

Alexander invariants and cohomology jump loci in group extensions

Alexander invariants and cohomology jump loci in group extensions

Alexander invariants and cohomology jump loci in group extensions

Cohomology, Bocksteins, and resonance varieties in characteristic 2

Contact Info

Product

Resources

About