A survey of combinatorial aspects in the topology of complex hyperplane arrangements

TL;DR

This survey explores the interplay between topology and combinatorics in complex hyperplane arrangements, covering fundamental groups, Lie algebras, homotopy, cohomology, and Milnor fibers.

Contribution

It provides a comprehensive overview of the combinatorial and topological aspects of complex hyperplane arrangements, highlighting recent developments and open problems.

Findings

Connections between combinatorics and topology are crucial in understanding hyperplane arrangements.

The structure of fundamental groups and cohomology rings reveals deep combinatorial properties.

Milnor fibers exhibit complex topological features influenced by combinatorial data.

Abstract

We survey interactions between the topology and the combinatorics of complex hyperplane arrangements. Without claiming to be exhaustive, we examine in this setting combinatorial aspects of fundamental groups, associated graded Lie algebras, higher homotopy groups, cohomology rings, twisted homology with rank 1 complex coefficients, and Milnor fibers.