## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Arrangements – Tokyo 1998, M. Falk and H. Terao, eds. (Tokyo: Mathematical Society of Japan, 2000), 185 - 215

### Cohomology rings and nilpotent quotients of real and complex arrangements

Daniel Matei and Alexander I. Suciu

#### Abstract

For an arrangement with complement $X$ and fundamental group $G$, we relate the truncated cohomology ring, $H^{\le2} (X)$, 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 $n \le 6$ planes in ${\mathbb{R}}^4$.

#### Article information

**Source***Arrangements – Tokyo 1998*, M. Falk and H. Terao, eds. (Tokyo: Mathematical Society of Japan, 2000), 185-215

**Dates**

First available in Project Euclid:
20 August 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1534788972

**Digital Object Identifier**

doi:10.2969/aspm/02710185

**Mathematical Reviews number (MathSciNet)**

MR1796900

**Zentralblatt MATH identifier**

0974.32020

**Subjects**

Primary: 32S22: Relations with arrangements of hyperplanes [See also 52C35] 57M05: Fundamental group, presentations, free differential calculus

Secondary: 20F14: Derived series, central series, and generalizations 20J05: Homological methods in group theory

**Keywords**

complex hyperplane arrangement 2-arrangement cohomology ring resonance variety nilpotent quotient prime index subgroup

#### Citation

Matei, Daniel; Suciu, Alexander I. Cohomology rings and nilpotent quotients of real and complex arrangements. Arrangements – Tokyo 1998, 185--215, Mathematical Society of Japan, Tokyo, Japan, 2000. doi:10.2969/aspm/02710185. https://projecteuclid.org/euclid.aspm/1534788972