Lefschetz theorems for singular varieties

Hypersurface complements, Milnor fibers and higher homotopy groups of arrangements

“…In the case of Theorem 1, the homotopy-type claim is a direct consequence of [H,Th. 5], the new part being the relation between the number of n-cells and the degree of the gradient map grad(h).…”

“…On the other hand, in Theorem 3 the main claim is that concerning the homotopy-type and this follows from [H,Prop. 3], by a geometric argument described in Section 3 and involving a key result by Hironaka.…”

“…In view of Hamm's affine Lefschetz theory, see [H,Th. 5], the only thing to prove is the equality between the number k n of n-cells attached and the degree of the gradient.…”

“…For the convenience of the reader we recall, in the special case needed, a basic result of Hamm, see [H,Prop. 3], with our addition concerning the condition (c0) in [DP,Lemma 3 and Ex.…”

“…We are going to derive this easy result from [H,Th. 5], by using the key homological features of arrangement complements.…”