Arc-wise analytic stratification, Whitney fibering conjecture and Zariski equisingularity

Parusiński, Adam; Păunescu, Laurenţiu

doi:10.1016/j.aim.2017.01.016

Cited by 26 publications

(82 citation statements)

References 63 publications

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“…Let h : X → S be a regular mapping. Then h −1 (S(s, ǫ)) is AS-homeomorphic to h −1 (S(s, ǫ ′ )) for ǫ, ǫ ′ small enough by Corollary 9.7 in [33], therefore the class of h −1 (lk(s, S)) in K 0 (AS) is well-defined. It just depends on the class [h : X → S] of h : X → S in K 0 (R Var S ) because a regular isomorphism is in particular an AShomeomorphism.…”

Section: 4mentioning

confidence: 87%

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On Grothendieck rings and algebraically constructible functions

Fichou

2017

Math. Ann.

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Section: 4mentioning

confidence: 87%

“…The virtual Poincaré polynomial of the link of an algebraic variety at a point is well-defined by [19]. Even more, the link, as an algebraic set, is well-defined up to AS-homeomorphism and does not depend on the embedding of the algebraic variety in the ambient smooth space (by Proposition 7.4 in [33]).…”

Section: 4mentioning

confidence: 99%

On Grothendieck rings and algebraically constructible functions

Fichou

2017

Math. Ann.

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“…Let S be an algebraic stratification of V , and let ρ : V → R, ρ(v) = |v − v 0 | 2 . By [9], Theorem 9.3, there is a stratification T of R such that the restriction of ρ to the preimage of each stratum T ∈ T is a locally trivial fibration. Thus there exists η > 0 such that for every ǫ with 0 < ǫ < η, there exists δ > 0 and a stratum preserving arc-analytic semialgebraic homeomorphism…”

Section: The Obstructionmentioning

confidence: 99%

“…It follows that L ǫ (V, v 0 ) is independent of ǫ up to stratified arc-analytic semialgebraic homeomorphism (cf. [9], Cor. 9.6).…”

Section: The Obstructionmentioning

confidence: 99%

Real intersection homology II: A local duality obstruction

McCrory

Parusiński

2020

Topology and its Applications

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We prove that the intersection pairing on the real intersection homology of a real algebraic variety is not a dual pairing in general. A classical argument of Thom for manifolds, adapted to real intersection homology, shows that if this intersection pairing is nonsingular and X is the link of a point in a real algebraic variety, with the dimension of X even, then the intersection homology euler characteristic Iχ(X) is even. Using an existence theorem of Akbulut and King [1] we show there is a singular algebraic surface X that is the link of a point in a 3-dimensional real algebraic variety, such that Iχ(X) is odd. Thus our definition of real intersection homology [6] does not have the key self-duality property suggested by Goresky and MacPherson ([3], p.227), though it does enjoy their small resolution property.Date: May 16, 2019. Adam Parusiński partially supported by ANR project LISA (ANR-17-CE40-0023-03).

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“…(For further details see [16], [17]. ) We say the algebraic stratification S of X ⊂ P r is good if it is a Whitney stratification and there exists an algebraic Whitney stratification T of P r such that S = T |X (i.e.…”

Section: Good Algebraic Stratificationsmentioning

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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Arc-wise analytic stratification, Whitney fibering conjecture and Zariski equisingularity

Cited by 26 publications

References 63 publications

On Grothendieck rings and algebraically constructible functions

On Grothendieck rings and algebraically constructible functions

Real intersection homology II: A local duality obstruction

Real intersection homology

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