Topological invariance of intersection homology without sheaves

King, Henry C.

doi:10.1016/0166-8641(85)90075-6

Cited by 92 publications

(68 citation statements)

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“…This discourages one from taking a purely sheaf theoretic approach to intersection homology on these spaces. Fortunately, a singular chain version of intersection homology exists, due initially to King [33], and this is the version of intersection homology that Quinn demonstrated was a topological invariant of MHSSs in [38]. In [20], we showed that the singular intersection chains also generate a sheaf complex and that it is quasi-isomorphic to the Deligne sheaf complex on topological pseudomanifolds.…”

Section: Sheaves Vs Singular Chainsmentioning

confidence: 99%

“…For more details, the reader is urged to consult King [33] and the author [20] for singular intersection homology and the original papers of Goresky and MacPherson [21; 22] and the book of Borel [7] for the simplicial and sheaf definitions. Singular chain intersection homology theory was introduced in [33] with finite chains (compact supports) and generalized in [20] to include locally finite but infinite chains (closed supports).…”

Section: Intersection Homologymentioning

confidence: 99%

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Intersection homology and Poincaré duality on homotopically stratified spaces

Friedman

2009

Geom. Topol.

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We show that intersection homology extends Poincaré duality to manifold homotopically stratified spaces (satisfying mild restrictions). These spaces were introduced by Quinn to provide "a setting for the study of purely topological stratified phenomena, particularly group actions on manifolds." The main proof techniques involve blending the global algebraic machinery of sheaf theory with local homotopy computations. In particular, this includes showing that, on such spaces, the sheaf complex of singular intersection chains is quasi-isomorphic to the Deligne sheaf complex.55N33, 57N80, 57P99

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Section: Sheaves Vs Singular Chainsmentioning

confidence: 99%

Section: Intersection Homologymentioning

confidence: 99%

Intersection homology and Poincaré duality on homotopically stratified spaces

Friedman

2009

Geom. Topol.

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“…A useful definition of intersection homology groups IH p,q * (X) for real algebraic varieties X, defined by a pair of loose perversity conditions (p, q) on semialgebraic chains, should have at least the following two properties: (1) If X is nonsingular, then IH p,q * (X) ∼ = H * (X; Z/2) (compact or closed supports), (2) IH p,q * (X) is independent of the good stratification S. Using general position arguments, we can show that the groups IH p,q * (X) satisfy (1) and (2) if the following conditions hold for all i ≥ 0 (cf. [9] Theorem 9, and the proofs of Theorem 2.7 above and Theorem 3.3 below):…”

Section: Corollary 28 Ifmentioning

confidence: 99%

“…For simplicity, in this informal discussion we identify a semialgebraic k-chain with its support, a closed k-dimensional semialgebraic subset of X. Following [9], we define a loose perversity to be a sequence of integers p = (p 0 , p 1 , p 2 , p 3 . .…”

Section: Corollary 28 Ifmentioning

confidence: 99%

Real intersection homology

McCrory

Parusiński

2018

Topology and its Applications

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Abstract. We present a definition of intersection homology for real algebraic varieties that is analogous to Goresky and MacPherson's original definition of intersection homology for complex varieties.Let X be a real algebraic variety. For certain stratifications S of X we define homology groups IH S k (X) with Z/2 coefficients that generalize the standard intersection homology groups [3] if all strata have even codimension. Whether there is a good analog of intersection homology for real algebraic varieties was stated as a problem by Goresky and MacPherson [6] (Problem 7, p. 227). They observed that if such a theory exists then it cannot be purely topological; indeed our groups are not homeomorphism invariants.We consider a class of algebraic stratifications introduced in [16] that have a natural general position property for semialgebraic subsets. After presenting the definition and properties of these stratifications S, we define the real intersection homology groups IH S k (X) and show that they are independent of the stratification. We prove that if X is nonsingular and pure dimensional then IH k (X) = H k (X; Z/2), classical homology with Z/2 coefficients. More generally, we prove that if X is irreducible and X admits a small resolution π : X → X then IH k (X) is canonically isomorphic to H k ( X; Z/2). Thus any two small resolution of X have the same homology.If X is not compact, we have two versions of real intersection homology: IH c k (X) with compact supports and IH cl k (X) with closed supports. We define an intersection pairing IH c k (X) × IH cl n−k (X) → Z/2, where n = dim X. We prove that if X has isolated singularities this pairing is nonsingular, so IH c k (X) ∼ = IH cl n−k (X) for all k ≥ 0. But our intersection pairing is singular for some real algebraic varieties X, so our groups fail to have the key selfduality property suggested by Goresky and MacPherson. We will present a counterexample in a subsequent paper.We benefitted from conversations more than a decade ago with Joost van Hamel, whose approach to real intersection homology [8] was quite different from ours. We dedicate this paper to his memory. Good algebraic stratificationsAn algebraic stratification of a real algebraic variety X is the stratification associated to a filtration of X by algebraic subvarietiessuch that for each j, Sing(X j ) ⊂ X j−1 , and either dim X j = j or X j = X j−1 . This filtration induces a decomposition X = S i , where the S i are the connected components of all the semialgebraic sets X j \ X j−1 . The sets S i , which are nonsingular semialgebraic subsets of X, 1

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“…That is to say, define it as the cohomology of global sections for a complex of fine sheaves for which the local calculation near the boundary is the standard one, and the local calculation near the singular stratum is as in the definition of standard intersection cohomology of perversity p . Define IH [17]). Here we are still considering the generalized definition of intersection cohomology near the singular stratum of X as in equation (11), but for simplicity of notation, we will suppress the B .…”

Section: Intersection Pairingsmentioning

confidence: 99%

Hodge and signature theorems for a family of manifolds with fibre bundle boundary

Hunsicker

2007

Geom. Topol.

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Over the past fifty years, Hodge and signature theorems have been proved for various classes of noncompact and incomplete Riemannian manifolds. Two of these classes are manifolds with incomplete cylindrical ends and manifolds with cone bundle ends, that is, whose ends have the structure of a fibre bundle over a compact oriented manifold, where the fibres are cones on a second fixed compact oriented manifold. In this paper, we prove Hodge and signature theorems for a family of metrics on a manifold M with fibre bundle boundary that interpolates between the incomplete cylindrical metric and the cone bundle metric on M . We show that the Hodge and signature theorems for this family of metrics interpolate naturally between the known Hodge and signature theorems for the extremal metrics. The Hodge theorem involves intersection cohomology groups of varying perversities on the conical pseudomanifold X that completes the cone bundle metric on M . The signature theorem involves the summands i of Dai's invariant [10] that are defined as signatures on the pages of the Leray-Serre spectral sequence of the boundary fibre bundle of M . The two theorems together allow us to interpret the i in terms of perverse signatures, which are signatures defined on the intersection cohomology groups of varying perversities on X .

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Topological invariance of intersection homology without sheaves

Cited by 92 publications

References 4 publications

Intersection homology and Poincaré duality on homotopically stratified spaces

Intersection homology and Poincaré duality on homotopically stratified spaces

Real intersection homology

Hodge and signature theorems for a family of manifolds with fibre bundle boundary

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