4 research outputs found
Zariski-van Kampen theorem for higher homotopy groups
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.Comment: 37 pages, LaTeX2e with amsmath, amsthm and amscd packages. To appear
in J. Inst. Math. Jussieu (2003) with the first proof of Theorem 7.1
significantly developped and new references added. Due to copyright
restrictions, this final version will only be available at Cambridge Journals
Online (http://journals.cambridge.org) when published. Thus the content of
the paper here is the same as that of version 1 of 3 March 200
Homotopical variations and high-dimensional Zariski-van Kampen theorems
20 pages, Plain TeXIn 1933, van Kampen described the fundamental groups of the complements of plane complex projective algebraic curves. Recently, Chéniot-Libgober proved an analogue of this result for higher homotopy groups of the complements of complex projective hypersurfaces with isolated singularities. Their description is in terms of some "homotopical variation operators". We generalize here the notion of "homotopical variation" to (singular) quasi-projective varieties. This is a first step for further generalizations of van Kampen's theorem. A conjecture, with a first approach, is stated in the special case of non-singular quasi-projective varieties