Semi-algebraic partition and basis of Borel-Moore homology of hyperplane arrangements

TL;DR

This paper introduces an explicit semi-algebraic partition of the complement of real hyperplane arrangements, providing a basis for Borel-Moore homology and linking it to de Rham cohomology.

Contribution

It presents a novel explicit semi-algebraic partition that forms a basis for Borel-Moore homology and establishes a correspondence with de Rham cohomology.

Findings

Partition consists of contractible pieces forming a basis for Borel-Moore homology

Explicit correspondence between de Rham cohomology and Borel-Moore homology

Provides tools for topological and algebraic analysis of hyperplane arrangements

Abstract

We describe an explicit semi-algebraic partition for the complement of a real hyperplane arrangement such that each piece is contractible and so that the pieces form a basis of Borel-Moore homology. We also give an explicit correspondence between the de Rham cohomology and the Borel-Moore homology.