Intersection Spaces, Equivariant Moore Approximation and the Signature

Banagl, Markus; Chriestenson, Bryce

doi:10.5427/jsing.2017.16g

Cited by 10 publications

(85 citation statements)

References 25 publications

(51 reference statements)

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“…In [5] the theory was extended to a larger class of X, allowing the singular set to have arbitrary dimension, but requiring it be smooth and for ∂T to have the structure of a fiber bundle over Σ. The local topological tool developed was equivariant Moore…”

Section: Motivationmentioning

confidence: 99%

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Algebraic intersection spaces

Geske

2019

J. Topol. Anal.

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We define a variant of intersection space theory that applies to many compact complex and real analytic spaces X, including all complex projective varieties; this is a significant extension to a theory which has so far only been shown to apply to a particular subclass of spaces with smooth singular sets. We verify existence of these so-called algebraic intersection spaces and show that they are the (reduced) chain complexes of known topological intersection spaces in the case that both exist. We next analyze "local duality obstructions", which we can choose to vanish, and verify that algebraic intersection spaces satisfy duality in the absence of these obstructions. We conclude by defining an untwisted algebraic intersection space pairing, whose signature is equal to the Novikov signature of the complement in X of a tubular neighborhood of the singular set.

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Section: Motivationmentioning

confidence: 99%

“…As in Example 4.4, this fiberwise truncation constitutes a topological intersection approximation for T . The associated topological intersection space is the cone on the composition:ft <c−1−p(c) ∂T → ∂T → X − T • .This is precisely[5, Definition 9.2], the definition of the Banagl-Chriestenson intersection space.…”

mentioning

confidence: 99%

Algebraic intersection spaces

Geske

2019

J. Topol. Anal.

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“…Before discussing our results (see Sect. 2), which build on work of Klimczak [23] and Banagl-Chriestenson [7], we give in the following an outline of several existing results in the theory of intersection spaces.…”

Section: Introductionmentioning

confidence: 99%

“…This is already evident in the case of arbitrary two strata pseudomanifolds: Surprisingly, even if an intersection spaces can be constructed, the existence of a generalized Poincaré duality isomorphism turns out to be obstructed in general. As discovered by Banagl and Chriestenson [7], the failure of duality is precisely measured by local duality obstructions, which are certain characteristic classes associated to the link bundle over the singular stratum of the pseudomanifold. These obstruction classes are abstractly definable for fiberwise truncatable fiber bundles, and they vanish for product bundles and certain flat bundles, but not for generally twisted bundles.…”

Section: Introductionmentioning

confidence: 99%

A Fundamental Class for Intersection Spaces of Depth One Witt Spaces

Wrazidlo

2020

manuscripta math.

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By a theorem of Banagl-Chriestenson, intersection spaces of depth one pseudomanifolds exhibit generalized Poincaré duality of Betti numbers, provided that certain characteristic classes of the link bundles vanish. In this paper, we show that the middle-perversity intersection space of a depth one Witt space can be completed to a rational Poincaré duality space by means of a single cell attachment, provided that a certain rational Hurewicz homomorphism associated to the link bundles is surjective. Our approach continues previous work of Klimczak covering the case of isolated singularities with simply connected links. For every singular stratum, we show that our condition on the rational Hurewicz homomorphism implies that the Banagl-Chriestenson characteristic classes of the link bundle vanish. Moreover, using Sullivan minimal models, we show that the converse implication holds at least in the case that twice the dimension of the singular stratum is bounded by the dimension of the link. As an application, we compare the signature of our rational Poincaré duality space to the Goresky-MacPherson intersection homology signature of the given Witt space. We discuss our results for a class of Witt spaces having circles as their singular strata.

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“…Note, that Moore approximation is an Eckmann-Hilton dual notion of Postnikov approximation. If the singularities are not isolated, one has to perform the Moore approximation equivariantly, see [BC16]. Having a perversity internal cup product, intersection space cohomology cannot be isomorphic to intersection cohomology.…”

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confidence: 99%

Intersection space cohomology of three-strata pseudomanifolds

Essig

2019

J. Topol. Anal.

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The theory of intersection spaces assigns cell complexes to certain stratified topological pseudomanifolds depending on a perversity function in the sense of intersection homology. The main property of the intersection spaces is Poincaré duality over complementary perversities for the reduced singular (co)homology groups with rational coefficients. This (co)homology theory is not isomorphic to intersection homology, instead they are related by mirror symmetry. Using differential forms, Banagl extended the intersection space cohomology theory to 2-strata pseudomanifolds with a geometrically flat link bundle. In this paper we use differential forms on manifolds with corners to generalize the intersection space cohomology theory to a class of 3-strata spaces with flatness assumptions for the link bundles. We prove Poincaré duality over complementary perversities for the cohomology groups. To do so, we investigate fiber bundles on manifolds with boundary. At the end, we give examples for the application of the theory.

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Intersection Spaces, Equivariant Moore Approximation and the Signature

Cited by 10 publications

References 25 publications

Algebraic intersection spaces

Algebraic intersection spaces

A Fundamental Class for Intersection Spaces of Depth One Witt Spaces

Intersection space cohomology of three-strata pseudomanifolds

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