Hypersurface complements, Milnor fibers and higher homotopy groups of arrangements

Abstract

We describe a new relation between the topology of hypersurface complements, Milnor fibers and degree of gradient mappings. In particular we show that any projective hypersurface has affine parts which are bouquets of spheres. The main tools are the polar curves and the affine Lefschetz theory developped by H. Hamm, D.T. Lê and A. Némethi. In the special case of the hyperplane arrangements, we strengthen some results due to Orlik and Terao (see Math. Ann. 301(1995)) and obtain the minimality of hyperplane arrangements (see Randell math.AT/0011101 for another proof of this result). This is then used to compute some higher homotopy groups of hyperplane arrangements using the ideas from Papadima-Suciu, see math.AT/0002251. The second version contains applications of the above ideas to the polar Cremona transformations and gives a positive answer to Dolgachev's Conjecture (see Michigan Math. J. 48 (2000), volume dedicated to W. Fulton). The third version corrects some errors and provides new applications.