Lipschitz stratifications and Lipschitz isotopies

Mostowski, Tadeusz

doi:10.4064/bc65-0-13

Cited by 7 publications

(12 citation statements)

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“…In [21] he succeeded in solving a conjecture of Sullivan, showing that a complex analytic variety admits a locally Lipschitz trivial stratification. Following his work, A. Parusiński proved the corresponding results in several real categories ( [26,25,27]).…”

Section: Introductionmentioning

confidence: 99%

The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms

Koike¹,

Păunescu²

2009

Ann. inst. Fourier

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Let A ⊂ R n be a set-germ at 0 ∈ R n such that 0 ∈ A. We say thatWe study the problem of whether the dimension of the common direction set, dim(D(A) ∩ D(B)) is preserved by bi-Lipschitz homeomorphisms. We show that although it is not true in general, it is preserved if the images of A and B are also subanalytic. In particular if two subanalytic set-germs are bi-Lipschitz equivalent their direction sets must have the same dimension.

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Section: Introductionmentioning

confidence: 99%

The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms

Koike¹,

Păunescu²

2009

Ann. inst. Fourier

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“…Finally, we observe that this work lays the ground for possible future work in different directions. For example, it is interesting to explore how strong our invariants are in obstructing Lipschitz equisingularity in higher dimensions (see [19]). Its relation with Zariski equisingularity is also worth exploring because of the work done in [20,23].…”

Section: Mdh Bmentioning

confidence: 99%

Moderately Discontinuous Homology

Bobadilla

Heinze

Pereira³

et al. 2021

Comm Pure Appl Math

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We introduce a new metric homology theory, which we call Moderately Discontinuous Homology, designed to capture Lipschitz properties of metric singular subanalytic germs. The main novelty of our approach is to allow "moderately discontinuous" chains, which are specially advantageous for capturing the subtleties of the outer metric phenomena. Our invariant is a finitely generated graded abelian group MDH b for any b 2 1; C1 and homomorphisms MDH b ! MDH b 0 for any b b 0 . Here b is a "discontinuity rate". The homology groups of a subanalytic germ with the inner or outer metric are proved to be finitely generated and only finitely many homomorphisms MDH b ! MDH b 0 are essential. For b D 1 Moderately Discontinuous Homology recovers the homology of the tangent cone for the outer metric and of the Gromov tangent cone for the inner one. In general, for b D C1 the MD-Homology recovers the homologyof the punctured germ. Hence, our invariant can be seen as an algebraic invariant interpolating from the germ to its tangent cone. Our homology theory is a bi-Lipschitz subanalytic invariant, is invariant by suitable metric homotopies, and satisfies versions of the relative and Mayer-Vietoris long exact sequences. Moreover, fixed a discontinuity rate b we show that it is functorial for a class of discontinuous Lipschitz maps, whose discontinuities are b-moderated; this makes the theory quite flexible. In the complex analytic setting we introduce an enhancement called Framed MD homology, which takes into account information from fundamental classes. As applications we prove that Moderately Discontinuous Homology characterizes smooth germs among all complex analytic germs, and recovers the number of irreducible components of complex analytic germs and the embedded topological type of plane branches. Framed MD homology recovers the topological type of any plane curve singularity and relative multiplicities of complex analytic germs.

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“…We note (cf. [5,Proposition 7.1]) that each Lipschitz stratification satisfies the Kuo-Verdier condition (w) (cf. [9]), and hence any Lipschitz stratification of a subanalytic set is Whitney regular.…”

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confidence: 99%

“…The proof of Theorem 1 is fairly elementary and uses the method introduced in [5]. The author thanks T. Mostowski for interesting discussions concerning the problem.…”

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confidence: 99%

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Bi-Lipschitz trivialization of the distance function to a stratum of a stratification

Parusiński

2005

Ann. Polon. Math.

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Abstract. Given a Lipschitz stratification X that additionally satisfies condition (δ) of Bekka-Trotman (for instance any Lipschitz stratification of a subanalytic set), we show that for every stratum N of X the distance function to N is locally bi-Lipschitz trivial along N . The trivialization is obtained by integration of a Lipschitz vector field.The existence of Lipschitz stratifications of complex analytic or real subanalytic sets ([5], [7], [8]) allows one to trivialize these sets locally along each stratum so that the trivialization is bi-Lipschitz. In this paper we show the following refinement of this result that answers positively a question posed to us by M. Ferrarotti and E. Fortuna. Theorem 1. Let X be a Lipschitz stratification of a locally closed subset X ⊂ R n and let N be a stratum of X . Suppose additionally that X satisfies condition (δ) of Bekka-Trotman along N . Then X can be trivialized locally along N in such a way that the distance to N is preserved by the trivialization and the trivialization is bi-Lipschitz.We explain in Remark 2 below what we precisely mean by bi-Lipschitz trivialization.We always assume that the strata of X are C 2 subvarieties of R n . We say that X satisfies condition (δ) of Bekka-Trotman along N (see [1]) if for any p ∈ N , there are c 0 > 0 and ε > 0 such that for all q ∈ X satisfying dist(p, q) < ε, there is a unit vector v ∈ T q S, where S is the stratum containing q, such that

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Lipschitz stratifications and Lipschitz isotopies

Cited by 7 publications

References 0 publications

The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms

The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms

Moderately Discontinuous Homology

Bi-Lipschitz trivialization of the distance function to a stratum of a stratification

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