Abstract:
Let $D$ be a complex algebraic hypersurface in $\mathbf C^n$ not passing through the point $o\in\mathbf C^n$. The generators of the fundamental group $\pi_1(\mathbf C^n\setminus D,o)$ 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 $\mathbf R^3$. A new proof of Zariski's conjecture about commutativity of the fundamental group $\pi_1(\mathbf P^2\setminus C)$ 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 $\pi_1(\mathbf C^n\setminus D)$ obtained in the first part.
\Bibitem{Kul91}
\by Vik.~S.~Kulikov
\paper The fundamental group of the scomplement to a hypersurface in $\mathbf C^n$
\jour Math. USSR-Izv.
\yr 1992
\vol 38
\issue 2
\pages 399--418
\mathnet{http://mi.mathnet.ru/eng/im1016}
\crossref{https://doi.org/10.1070/IM1992v038n02ABEH002205}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1133305}
\zmath{https://zbmath.org/?q=an:0802.14007}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?1992IzMat..38..399K}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1992HR86300009}
Linking options:
https://www.mathnet.ru/eng/im1016
https://doi.org/10.1070/IM1992v038n02ABEH002205
https://www.mathnet.ru/eng/im/v55/i2/p407
This publication is cited in the following 8 articles:
Anar Akhmedov, B. Doug Park, “Exotic smooth structures on small 4-manifolds with odd signatures”, Invent math, 2010
Vik. S. Kulikov, “On the fundamental groups of complements of toral curves”, Izv. Math., 61:1 (1997), 89–112
Vik. S. Kulikov, “A geometric realization of $C$-groups”, Russian Acad. Sci. Izv. Math., 45:1 (1995), 197–206
Vik. S. Kulikov, “Alexander polynomials of plane algebraic curves”, Russian Acad. Sci. Izv. Math., 42:1 (1994), 67–89
Vik. S. Kulikov, “Generalized and local Jacobian problems”, Russian Acad. Sci. Izv. Math., 41:2 (1993), 351–365
Vik. S. Kulikov, “On the Lefschetz theorem for the complement of a curve in $\mathbf P^2$”, Russian Acad. Sci. Izv. Math., 41:1 (1993), 169–184
Vik. S. Kulikov, “On the structure of the fundamental group of the complement of algebraic curves in $\mathbf C^2$”, Russian Acad. Sci. Izv. Math., 40:2 (1993), 443–454
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