  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. L\^e and A. N\'emethi. 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.
