The $\textbf{nbc}$ minimal complex of supersolvable arrangements

TL;DR

This paper explores the connections between the minimal CW-complex, the nbc basis, and chambers in supersolvable arrangements, providing new insights into their homotopy and algebraic structures.

Contribution

It offers a natural description of bijections among key combinatorial and topological objects associated with supersolvable arrangements, linking homotopy, algebra, and chambers.

Findings

Describes bijections between minimal CW-complex, nbc basis, and chambers.

Provides results on the (co)homology of the Milnor fiber.

Establishes a bijection between the symmetric group and the nbc basis of the braid arrangement.

Abstract

In this paper we give a very natural description of the bijections between the minimal CW-complex homotopy equivalent to the complement of a supersolvable arrangement $\mathcal{A}$, the $\textbf{nbc}$ basis of the Orlik-Solomon algebra associated to $\mathcal{A}$ and the set of chambers of $\mathcal{A}$. We use these bijections to get results on the first (co)homology group of the Milnor fiber of $\mathcal{A}$ and to describe a bijection between the symmetric group and the $\textbf{nbc}$ basis of the braid arrangement.