Rectified Homotopical Depth and Grothendieck Conjectures

Hamm, Helmut A.; Tráng, Dũng Lê

doi:10.1007/978-0-8176-4575-5_7

Cited by 18 publications

(13 citation statements)

References 20 publications

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“…The complex link of the hypersurface f −1 (0) at the origin, denoted lk(f −1 (0), 0), is homotopy equivalent to a wedge of spheres ∨S n−1 . This follows from the fact that f −1 (0) inherits the property (*), see [HL,Th. 3…”

Section: A Geometric Viewpointmentioning

confidence: 96%

The vanishing neighbourhood of non-isolated singularities

Tibăr¹

2007

Isr. J. Math.

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We study the vanishing neighbourhood of non-isolated singularities of functions on singular spaces by associating a general linear function. We use the carrousel monodromy in order to show how to get a better control over the attaching of thimbles. For one-dimensional singularities, we prove obstructions to integer (co)homology groups and to the eigenspaces of the monodromy via monodromies of nearby sections. Our standpoint allows one to find, in certain cases, the structure of the Milnor fibre up to the homotopy type.

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Section: A Geometric Viewpointmentioning

confidence: 96%

The vanishing neighbourhood of non-isolated singularities

Tibăr¹

2007

Isr. J. Math.

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“…Notice that, when X is non-singular, the application of the Lefschetz Hyperplane Section Theorems mentioned in the introduction is valid for L. Indeed, the genericity of the hyperplane required for these theorems can be specified as its transversality to a Whitney stratification of Z (cf. [HL2,Appendix]…”

Section: Generic Pencils and Monodromiesmentioning

confidence: 99%

Homotopical variations and high-dimensional Zariski-van Kampen theorems

Chéniot

Eyral

2005

Trans. Amer. Math. Soc.

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“…For instance, if X is a purely dimensional non-singular or local complete intersection variety, then q(Y, Z) = dim X − 1 (cf. [17]). In the present paper, we are interested in the special class of varieties for which the integer q(Y, Z) is equal to 1.…”

Section: Introductionmentioning

confidence: 99%

Zariski-van Kampen theorems for singular varieties—an approach via the relative monodromy variation

Eyral¹,

Petrov²

2019

Kodai Math. J.

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The classical Zariski-van Kampen theorem gives a presentation of the fundamental group of the complement of a complex algebraic curve in P 2 . The first generalization of this theorem to singular (quasi-projective) varieties was given by the first author. In both cases, the relations are generated by the standard monodromy variation operators associated with the special members of a generic pencil of hyperplane sections. In the present paper, we give a new generalization in which the relations are generated by the relative monodromy variation operators introduced by D. Chéniot and the first author. The advantage of using the relative operators is not only to cover a larger class of varieties but also to unify the Zariski-van Kampen type theorems for the fundamental group and for higher homotopy groups. In the special case of non-singular varieties, the main result of this paper was conjectured by D. Chéniot and the first author.

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Rectified Homotopical Depth and Grothendieck Conjectures

Cited by 18 publications

References 20 publications

The vanishing neighbourhood of non-isolated singularities

The vanishing neighbourhood of non-isolated singularities

Homotopical variations and high-dimensional Zariski-van Kampen theorems

Zariski-van Kampen theorems for singular varieties—an approach via the relative monodromy variation

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