Vertical configuration spaces and their homology

TL;DR

This paper studies special configuration spaces with a verticality condition in Euclidean space, computes their homology, and establishes homological stability for unordered configurations, advancing understanding of their topological properties.

Contribution

Introduces and analyzes vertical configuration spaces, providing homology computations and stability results that extend classical configuration space theory.

Findings

Homology computed for ordered vertical configuration spaces.

Homological stability proved for unordered vertical configuration spaces.

New topological invariants introduced for clusters with verticality condition.

Abstract

We introduce ordered and unordered configuration spaces of 'clusters' of points in an Euclidean space $\mathbb{R}^d$, where points in each cluster satisfy a 'verticality' condition, depending on a decomposition $d=p+q$. We compute the homology in the ordered case and prove homological stability in the unordered case.