Cohomology rings and nilpotent quotients of real and complex arrangements. (English) Zbl 0974.32020
Falk, Michael (ed.) et al., Arrangements - Tokyo 1998. Proceedings of a workshop on mathematics related to arrangements of hyperplanes, Tokyo, Japan, July 13-18, 1998. In honor of the 60th birthyear of Peter Orlik. Tokyo: Kinokuniya Company Ltd. Adv. Stud. Pure Math. 27, 185-215 (2000).
Summary: For an arrangement with complement \(X\) and fundamental group \(G\), we relate the truncated cohomology ring, \(H^{\leq 2}(X)\), 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 \(n\leq 6\) planes in \(\mathbb{R}^4\).
For the entire collection see [Zbl 0952.00034].
For the entire collection see [Zbl 0952.00034].
MSC:
| 32S22 | Relations with arrangements of hyperplanes |
| 57M05 | Fundamental group, presentations, free differential calculus |
| 20J05 | Homological methods in group theory |
| 20F14 | Derived series, central series, and generalizations for groups |
