For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ring, H ≤2 (X), to the second nilpotent quotient, G/G 3 . We define invariants of G/G 3 by counting normal subgroups of a fixed prime index p, according to their abelianization. We show how to compute this distribution from the resonance varieties of the Orlik-Solomon algebra mod p. As an application, we establish the cohomology classification of 2-arrangements of n ≤ 6 planes in R 4 . 2.For q = 1, questions (I) and (II) are equivalent, provided H 1 is torsion free. Indeed, H 1 (X) = G/G 2 under that assumption. For q = 2, the two questions are still equivalent, under some additional conditions: If H 2 is also torsion-free, and the cup-product map µ : H 1 ∧ H 1 → H 2 is surjective, then:

We prove that the irreducible components of the characteristic varieties of quasi-projective manifolds are either pull-backs of such components for orbifolds, or torsion points. This gives an interpretation for the so-called \emph{translated} components of the characteristic varieties, and shows that the zero-dimensional components are indeed torsion. The main result is used to derive further obstructions for a group to be the fundamental group of a quasi-projective manifold.Comment: 33 pages, no figures. General rearrangement and proof of Proposition 4.2 is expande

We study the homotopy types of complements of arrangements of n transverse planes in R 4 , obtaining a complete classification for n ≤ 6, and lower bounds for the number of homotopy types in general. Furthermore, we show that the homotopy type of a 2-arrangement in R 4 is not determined by the cohomology ring, thereby answering a question of Ziegler. The invariants that we use are derived from the characteristic varieties of the complement. The nature of these varieties illustrates the difference between real and complex arrangements.1991 Mathematics Subject Classification. Primary 57M05, 57M25, 52B30; Secondary 14M12, 20F36.

We characterize the groups of branched twist spins of classical knots in terms of 3-manifold groups, and also give a purely algebraic, conjectural characterization in terms of P D 3 -groups. We show also that each group is the group of at most finitely many branched twist spins.

We give a necessary and sufficient condition in order for a hyperplane arrangement to be of Torelli type, namely that it is recovered as the set of unstable hyperplanes of its Dolgachev sheaf of logarithmic differentials. Decompositions and semistability of non-Torelli arrangements are investigated.Comment: 2 Figue