Hypersurface complements, Milnor fibers and higher homotopy groups of arrangements. (English) Zbl 1068.32019

This paper concerns the algebraic topology of complements of complex algebraic subvarieties of complex affine or projective space, especially complements of hypersurfaces and hyperplane arrangements. The first set of results concern the gradient map of a nonconstant homogeneous polynomial \(h\) of \(n\) complex variables. It is shown that the degree of the gradient is determined by the number of \(n\)-cells that have to be added to a generic hyperplane section of the domain of the gradient in order to obtain the complement of the projective hypersurface of \(h\). This extends recent work of I. V. Dolgachev [Mich. Math. J. 48, Spec. Vol., 191–202 (2000; Zbl 1080.14511)] and answers one of his conjectures. Another consequence is the existence of a minimal \(CW\)-structure for complements of hyperplane arrangements, a result obtained independently by R. Randell [Proc. Am. Math. Soc. 130, No. 9, 2737–2743 (2002; Zbl 1004.32010)]. The next set of results concerns the higher homotopy groups of complements of hyperplane arrangements. The authors provide a unified treatment encompassing most of what is already known about explicit computations and provide many interesting examples of such complements with non-trivial higher homotopy.


32S22 Relations with arrangements of hyperplanes
32S55 Milnor fibration; relations with knot theory
55Q52 Homotopy groups of special spaces
55R80 Discriminantal varieties and configuration spaces in algebraic topology
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