Homotopical variations and high-dimensional Zariski-van Kampen theorems

Abstract

Abstract. We give a new definition of the homotopical variation operators occurring in a recent high-dimensional Zariski-van Kampen theorem, a definition which opens the way to further generalizations of theorems of this kind.

Lectures on topology of complements and fundamental groups

“…Recently, Cheniot and Eyral proposed definition of homotopy variation operator in general showed that the map as in the above theorem is surjective (cf. [10]; see also [78] for another discussion of variation operators).…”

Fundamental groups of singular quasi-projective varieties

“…The link with a conjecture of [7] Corollary 5.3 implies the following result. This result, when d = 2 (which is the only interesting case (see above)), may be connected to the case d = 2 of [7, Conjecture 6.1].…”

“…Proof of Corollary 6.1. By [7,Lemma 4.8], the images of operators VAR i; 1 are contained in the kernel the natural map 1 (X z ; x 0 ) → 1 (X; x 0 ). Conversely, since each element of this kernel can be expressed with the help of operators Var i (cf.…”

“…This paper is thus a natural continuation of original Zariski's and van Kampen's papers [28,26] about the fundamental groups of the complements of plane algebraic curves (which are special cases of our main result); Chà eniot's papers [3,4] containing the ÿrst complete proof of the Zariski-van Kampen theorem, and [6] concerning similar problems for the high-dimensional homology groups of non-singular quasi-projective varieties; Libgober's and Chà eniot-Libgober's papers [19,8] about the high-dimensional homotopy groups of the complements of hypersurfaces with isolated singularities; Shimada's papers [21,22] on the fundamental groups of non-singular irreducible quasi-projective varieties; and the paper of Chà eniot and the author [7] concerning the high-dimensional homotopy groups of the complements of hypersurfaces with isolated singularities (note that [7] also gives a general conjecture concerning extensions of [19,8,7] to high-dimensional homotopy groups of non-singular quasi-projective varieties).…”

“…The main results are stated in Section 5. In Section 6, we discuss the conjecture given in [7] and we make explicit the link between it and some results introduced in this paper. Finally, in Section 7, we give the proof of our main result about singular varieties.…”