Cup and cap products in intersection (co)homology

Friedman, Greg; McClure, James E.

doi:10.1016/j.aim.2013.02.017

Cited by 31 publications

(100 citation statements)

References 27 publications

(69 reference statements)

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“…, is a quasi-isomorphism for any ring R ([10, Theorem B]). This extends the Poincaré duality theorem of Friedman and McClure [16] established for field coefficients.…”

supporting

confidence: 81%

Blown-up intersection cochains and Deligne’s sheaves

Chataur¹,

Saralegi-Aranguren²,

Tanré³

2019

Geom Dedicata

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In a series of papers the authors introduced the so-called blown-up intersection cochains. These cochains are suitable to study products and cohomology operations of intersection cohomology of stratified spaces. The aim of this paper is to prove that the sheaf versions of the functors of blown-up intersection cochains are realizations of Deligne's sheaves. This proves that Deligne's sheaves can be incarnated at the level of complexes of sheaves by soft sheaves of perverse differential graded algebras. We also study Poincaré and Verdier dualities of blown-up intersections sheaves with the use of Borel-Moore chains of intersection.

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“…, is a quasi-isomorphism for any ring R ([10, Theorem B]). This extends the Poincaré duality theorem of Friedman and McClure [16] established for field coefficients.…”

supporting

confidence: 81%

Blown-up intersection cochains and Deligne’s sheaves

Chataur¹,

Saralegi-Aranguren²,

Tanré³

2019

Geom Dedicata

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“…This way, we obtain a sequence of δ-compatible morphisms f D * : π D * I p H r → π ′ D * I p H r . With I p C * (X) the singular rational intersection chain complex as in [18], we define intersection cochains by I p C * (X) = Hom(I p C * (X), Q) and intersection cohomology by I p H * (X) = H * (I p C * (X)). Then the universal coefficient theorem…”

Section: Examples Of Precosheavesmentioning

confidence: 99%

Intersection Spaces, Equivariant Moore Approximation and the Signature

Banagl¹,

Chriestenson²

2017

jsing

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We generalize the first author's construction of intersection spaces to the case of stratified pseudomanifolds of stratification depth 1 with twisted link bundles, assuming that each link possesses an equivariant Moore approximation for a suitable choice of structure group. As a by-product, we find new characteristic classes for fiber bundles admitting such approximations. For trivial bundles and flat bundles whose base has finite fundamental group these classes vanish. For oriented closed pseudomanifolds, we prove that the reduced rational cohomology of the intersection spaces satisfies global Poincaré duality across complementary perversities if the characteristic classes vanish. The signature of the intersection spaces agrees with the Novikov signature of the top stratum. As an application, these methods yield new results about the Goresky-MacPherson intersection homology signature of pseudomanifolds. We discuss several nontrivial examples, such as the case of flat bundles and symplectic toric manifolds.

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“…We consider the perversities of [21] defined on each stratum. They are already used in [22], [23], [12], [13], [15].…”

Section: Definition 11mentioning

confidence: 99%

“…Thus the proof is reduced to the study of the right-hand map. First, recall from [15] and [24,Proposition 2.2], that C * p,c (X) = lim − →K C * p (X, X\K) = C * p (X, X\{(0, v)}) and C ∞,p * (X) = lim ← −K C p * (X, X\K) = C * p (X, X\{(0, v)}). Thus, it is sufficient to prove the existence of a quasi-isomorphism,…”

Section: Lemma 25 the Conclusion Of Theorem 22 Is True Ifmentioning

confidence: 99%

“…In intersection homology, this method was investigated by G. Friedman and J.E. McClure in [15] and taken over in [11,Section 8.2]. As cohomology groups, the authors consider the homology of the dual C * p (X; Z) = Hom(C p * (X; Z), Z) of the complex of p-intersection chains and, under the same restriction as in Goresky and Siegel's paper, they prove the existence of an isomorphism induced by a cap product with a fundamental class, H k p,c (X; Z) ∼ = − → H Dp n−k (X; Z), (0.6) for a locally p-torsion free, oriented, paracompact, n-dimensional stratified pseudomanifold X.…”

Section: Introductionmentioning

confidence: 99%

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Variations on Poincaré duality for intersection homology

Saralegi-Aranguren

Tanré

2020

Enseign. Math.

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Intersection homology with coefficients in a field restores Poincaré duality for some spaces with singularities, as stratified pseudomanifolds. But, with coefficients in a ring, the behaviours of manifolds and stratified pseudomanifolds are different. This work is an overview, with proofs and explicit examples, of various possible situations with their properties.We first set up a duality, defined from a cap product, between two intersection cohomologies: the first one arises from a linear dual and the second one from a simplicial blow up. Moreover, from this property, Poincaré duality in intersection homology looks like the Poincaré-Lefschetz duality of a manifold with boundary. Besides that, an investigation of the coincidence of the two previous cohomologies reveals that the only obstruction to the existence of a Poincaré duality is the homology of a well defined complex. This recovers the case of the peripheral sheaf introduced by Goresky and Siegel for compact PL-pseudomanifolds. We also list a series of explicit computations of peripheral intersection cohomology. In particular, we observe that Poincaré duality can exist in the presence of torsion in the "critical degree" of the intersection homology of the links of a stratified pseudomanifold.

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Cup and cap products in intersection (co)homology

Abstract: We construct cup and cap products in intersection (co)homology with field coefficients. The existence of the cap product allows us to give a new proof of Poincaré duality in intersection (co)homology which is similar in spirit to the usual proof for ordinary (co)homology of manifolds.

Cited by 31 publications

References 27 publications

Blown-up intersection cochains and Deligne’s sheaves

Blown-up intersection cochains and Deligne’s sheaves

Intersection Spaces, Equivariant Moore Approximation and the Signature

Variations on Poincaré duality for intersection homology

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