Limits with braid arrangements

TL;DR

This paper introduces a monoid structure on k-equal arrangements to define their limits, computes the cohomology of these limits, and constructs an exact complex from these models.

Contribution

It presents a novel monoid framework for k-equal arrangements and develops a new differential structure to analyze their cohomology.

Findings

Defined limits of braid arrangements using monoid structure

Computed cohomology of the arrangement limits

Constructed an exact complex from the models

Abstract

We define a monoid structure on the set of $k$-equal arrangements and use this structure to define limits of braid arrangements. We compute the cohomology of the associated limits of rational models of the arrangements complex complements. We collect these complexes together into one complex by creating a new differential and product on their direct sum and show that the resulting complex is exact.