Intersection homology Poincaré spaces and the characteristic variety theorem

Pardon, William

doi:10.1007/bf02566603

Cited by 35 publications

(26 citation statements)

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“…In particular, taking n = (n, ∅), the ordinary Deligne-sheaf P * X,n,E satisfies P * [26]) requires that InH j−1 (L; E| L ) be torsion-free for each link L 2j−1 while InH j (L; E| L ) = 0 for each link L 2j . The spaces in Cappell-Shaneson [7] have only even codimension strata and/or link intersection homology modules that are all torsion.…”

Section: Torsion-free Coefficientsmentioning

confidence: 99%

Generalizations of Intersection Homology and Perverse Sheaves with Duality over the Integers

Friedman¹

2019

Michigan Math. J.

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We provide a generalization of the Deligne sheaf construction of intersection homology theory, and a corresponding generalization of Poincaré duality on pseudomanifolds, such that the Goresky-MacPherson, Goresky-Siegel, and Cappell-Shaneson duality theorems all arise as special cases. Unlike classical intersection homology theory, our duality theorem holds with ground coefficients in an arbitrary PID and with no local cohomology conditions on the underlying space. Self-duality does require local conditions, but our perspective leads to a new class of spaces more general than the Goresky-Siegel IP spaces on which upper-middle perversity intersection homology is self-dual. We also examine categories of perverse sheaves that contain our "torsionsensitive" Deligne sheaves as intermediate extensions.

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Section: Torsion-free Coefficientsmentioning

confidence: 99%

Generalizations of Intersection Homology and Perverse Sheaves with Duality over the Integers

Friedman¹

2019

Michigan Math. J.

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“…(2) An object is a compact oriented IP-space with a given stratification, and similarly for the bordisms. Pardon [Par90] does not make it clear which definition he is using, but fortunately the two definitions give the same bordism groups by [Fa]. We will use the first definition.…”

Section: Review Of Ip Bordismmentioning

confidence: 99%

The L-homology fundamental class for IP-spaces and the stratified Novikov conjecture

Banagl

Laures

McClure

2019

Sel. Math. New Ser.

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An IP-space is a pseudomanifold whose defining local properties imply that its middle perversity global intersection homology groups satisfy Poincaré duality integrally. We show that the symmetric signature induces a map of Quinn spectra from IP bordism to the symmetric L-spectrum of Z, which is, up to weak equivalence, an E∞ ring map. Using this map, we construct a fundamental L-homology class for IP-spaces, and as a consequence we prove the stratified Novikov conjecture for IP-spaces whose fundamental group satisfies the Novikov conjecture.2010 Mathematics Subject Classification. 55N33, 57R67, 57R20, 57N80, 19G24.

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“…IP spaces. R-IP spaces [27] are defined by the property that if a link L is even-dimensional then ImH dim(L)/2 (L; R) = 0 and if L is odd-dimensional then the R-torsion submodule of ImH dim(L)−1 2 (L; R) is trivial. Let E R-IP be the class of all compact classical pseudomanifolds |Z| of dimension > 0 that satisfy these homological properties (plus the empty set).…”

Section: Examplesmentioning

confidence: 99%

Stratified and unstratified bordism of pseudomanifolds

Friedman¹

2015

Topology and its Applications

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We study bordism groups and bordism homology theories based on pseudomanifolds and stratified pseudomanifolds. The main seam of the paper demonstrates that when we uses classes of spaces determined by local link properties, the stratified and unstratified bordism theories are identical; this includes the known examples of pseudomanifold bordism theories, such as bordism of Witt spaces and IP spaces. Along the way, we relate the stratified and unstratified points of view for describing various classes of pseudomanifolds.

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Intersection homology Poincaré spaces and the characteristic variety theorem

Cited by 35 publications

References 10 publications

Generalizations of Intersection Homology and Perverse Sheaves with Duality over the Integers

Generalizations of Intersection Homology and Perverse Sheaves with Duality over the Integers

The L-homology fundamental class for IP-spaces and the stratified Novikov conjecture

Stratified and unstratified bordism of pseudomanifolds

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