Metric homology

Birbrair, Lev; Brasselet, Jean‐Paul

doi:10.1002/1097-0312(200011)53:11<1434::aid-cpa5>3.0.co;2-s

Cited by 11 publications

(11 citation statements)

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“…We found a 1-dimensional nontrivial cycle σ in H 1 (X − {0}), given by σ = ∂η where η = Y ∩ B[0, ǫ]. It means that 1-dimensional characteristic exponent of X at 0 (see [4], [2], [3]) is bigger than or equal to µ(η, 0) = β + 1 > 2. Therefore, by results of [3], X at 0 is not strongly metrically conic, because otherwise the corresponding exponent must be smaller than or equal to 2.…”

Section: It Is Clear Thatmentioning

confidence: 95%

“…In fact, we proved that if the real part of X has empty intersection with the union of the coordinates hyperplane in C 3 − {0}, then X is not strongly metrically conic at 0. In order to show this result, we use the theory of "Characteristic Exponents" developed in [4] and "Metric Homology Theory" developed in [2], [3]. If the intersection of real part of X with the link of X at x 0 presents a nontrivial cycle in 1-dimensional homology of this link, we use the methods developed in [5] to compute a characteristic exponent of this singularity.…”

Section: Introductionmentioning

confidence: 99%

“…We show that the Cheeger's cycle and fundamental cycle are independent. Note that the existence of Cheeger's cycles shows the Filtration Theorem proved in [2] for 1-dimensional Local Metric Homology is not true for 3-dimensional Local Metric Homology. This fact is important for the further development Metric Homology Theory.…”

Section: Introductionmentioning

confidence: 99%

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Inner metric geometry of complex algebraic surfaces with isolated singularities

Birbrair

Fernandes

2008

Comm Pure Appl Math

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Abstract. We produce examples of complex algebraic surfaces with isolated singularities such that these singularities are not metrically conic, i.e. the germs of the surfaces near singular points are not bi-Lipschitz equivalent, with respect to the inner metric, to cones. The technique used to prove the nonexistence of the metric conic structure is related to a development of Metric Homology. The class of the examples is rather large and it includes some surfaces of Brieskorn.

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Section: It Is Clear Thatmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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Inner metric geometry of complex algebraic surfaces with isolated singularities

Birbrair

Fernandes

2008

Comm Pure Appl Math

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“…The notion of Normal Embedding (or in other words Lipschitz Normal Embedding) became rather popular in recent development of Singularity Theory. Is is used in Metric Homology Theory of Birbrair and Brasselet [5], in Vanishing Homology of Valette [6], in Lipschitz Regularity theorem [10]. Several authors are investigating some special algebraic and semialgebraic sets in the spirit of their normal embedding.…”

Section: Introductionmentioning

confidence: 99%

Arc Criterion of Normal Embedding

Birbrair

Mendes

2018

Springer Proceedings in Mathematics &Amp; Statistics

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We present a criterion of local normal embedding of a semialgebraic (or definable in a polynomially bounded o-minimal structure) germ contained in R n in terms of orders of contact of arcs. Namely, we prove that a semialgebraic germ is normally embedded if and only if for any pair of arcs, coming to this point the inner order of contact is equal to the outer order of contact.the anonymous referee for his patience and extremely useful comments and corrections. Normally embedded setsLet X ⊂ R n be a connected semialgebraic set. We define an inner metric on X as follows: Let x, y ∈ X. The inner distance d X (x, y) is defined as the infimum of lengths of rectifiable arcs γ : [0, 1] → X such that γ(0) = x and γ(1) = y. Notice that for connected semialgebraic sets the inner metric is well-defined.1991 Mathematics Subject Classification. 14B05; 32S50 .

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“…The study of the link can provide interesting metric invariants ( [B], [M], [BB2]). Recently L. Birbrair and J. P. Brasselet have introduced a metric homology [BB1] and studied it in the case of a germ of an isolated singularity.…”

Section: Introductionmentioning

confidence: 99%

The link of the germ of a semi-algebraic metric space

Valette

2007

Proc. Amer. Math. Soc.

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Abstract. In this paper we investigate the metric properties of semi-algebraic germs. More precisely we introduce a counterpart to the notion of link for semi-algebraic metric spaces, which is often used to study the topology. We prove that it totally determines the metric type of the germ. We give a nice consequence for semi-algebraically bi-Lipschitz homeomorphic semi-algebraic germs. In [V1] the notion of Lipschitz triangulation is introduced. This tool has been introduced in order to prove a bi-Lipschitz version of Hardt's Theorem. This object gives a counterpart to the notion of triangulations (which usually is devoted to the topology) for the metric type. Here we come back to this notion to describe the metric type of a semi-algebraic germ. IntroductionThe local conic structure is a well known property of semi-algebraic germs (see [BCR1], Theorem 9.3.5). It says that a representative of a semi-algebraic germ is homeomorphic to a cone whose link is the intersection of a "sufficiently small sphere" with our given set. Actually we know that the link is independent of the considered (semi-algebraic) distance function and more generally that it can be defined from the level surface of an arbitrary positive continuous semi-algebraic function satisfying ρ −1 (0) = {0} (see for instance [CK], Proposition 1). As a matter of fact the link completely characterizes the germ up to a semi-algebraic homeomorphism of the germ and two semi-algebraic homeomorphic subsets can be related by a homeomorphism preserving the distance to the origin.In this paper we investigate the metric point of view. It means that we will consider the sets up to semi-algebraic bi-Lipschitz homeomorphisms. By considering the two bi-Lipschitz inequivalent cusps {(x; y) ∈ R 2 |y 2 = x 3 } and {(x; y) ∈ R 2 |y 2 = x 5 }, one sees that there may be two subsets not bi-Lipschitz homeomorphic with links semi-algebraically bi-Lipschitz homeomorphic.

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Metric homology

Cited by 11 publications

References 11 publications

Inner metric geometry of complex algebraic surfaces with isolated singularities

Inner metric geometry of complex algebraic surfaces with isolated singularities

Arc Criterion of Normal Embedding

The link of the germ of a semi-algebraic metric space

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