Kulikov, Vik. S. The fundamental group of the complement of a hypersurface in \(\mathbb{C}^ n\). (English. Russian original) Zbl 0802.14007 Math. USSR, Izv. 38, No. 2, 399-418 (1992); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 55, No. 2, 407-428 (1991). Summary: Let \(D\) be a complex algebraic hypersurface in \(\mathbb{C}^ n\) not passing through the point \(o \in \mathbb{C}^ n\). The generators of the fundamental group \(\pi_ 1 (\mathbb{C}^ n \backslash D, o)\) and the relations among them are described in terms of the real cone over \(D\) with apex at \(o\). This description is a generalization to the algebraic case of Wirtinger’s corepresentation of the fundamental group of a knot in \(\mathbb{R}^ 3\). – A new proof of Zariski’s conjecture about commutativity of the fundamental group \(\pi_ 1 (\mathbb{P}^ 2 \backslash C)\) for a projective nodal curve \(C\) is given in the second part of the paper based on the description of the generators and the relations in the group \(\pi_ 1 (\mathbb{C}^ n \backslash D,o)\) obtained in the first part. Cited in 1 ReviewCited in 1 Document MSC: 14F35 Homotopy theory and fundamental groups in algebraic geometry 32C18 Topology of analytic spaces 14J70 Hypersurfaces and algebraic geometry 14H45 Special algebraic curves and curves of low genus 14J40 \(n\)-folds (\(n>4\)) Keywords:fundamental group of the complement of a hypersurface; fundamental group of a knot; Zariski’s conjecture; commutativity of the fundamental group PDFBibTeX XMLCite \textit{Vik. S. Kulikov}, Math. USSR, Izv. 38, No. 2, 399--418 (1991; Zbl 0802.14007); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 55, No. 2, 407--428 (1991) Full Text: DOI