Zariski–van Kampen Theorem for Higher-Homotopy Groups

Abstract

This paper gives an extension of the classical Zariski-van Kampen theorem describing the fundamental groups of the complements of plane singular curves by generators and relations. It provides a procedure for computation of the first non-trivial higher homotopy groups of the complements of singular projective hypersurfaces in terms of the homotopy variation operators introduced here.

Homotopical variations and high-dimensional Zariski-van Kampen theorems

“…Knowing that L ∩ X is pathwise connected (cf. [CL,Lemma 2.9]), these facts imply that L ∩X is (n−2)-connected, andχ is then an isomorphism by the Hurewicz isomorphism theorem. Thus, for each i, and for every • ∈ p −1 ( * ), there is a homomorphism…”

Lectures on topology of complements and fundamental groups

“…There was further progress in the study of the complements in higher dimensions on generalizations of Zariski-van Kampen's theorem (cf. [48], [11], [26], [78]). Nevertheless, despite tremendous progress, since the first works by Enriques, Zariski and van Kampen, many problems still remains open and complete understanding of the topology of the complements to curves and hypersurfaces still out of reach.…”

“…The idea of defining homotopy variation operator comes from the fact that the homotopy groups on question are the homology groups of covering spaces. A description of the homotopy groups using variation operators was carried out in [11].…”

Isolated non-normal crossings

“…Notice that the study of the homotopy groups of the complements with module structure when the fundamental group is Z started in [13], [14], [1], [2] and in the context of non isolated singularities in [18].…”