Configuration spaces of clusters as $E_d$-algebras

TL;DR

This paper explores the structure of configuration spaces of labelled clusters of particles as $E_d$-algebras, providing geometric models for their bar constructions and computing their stable homology.

Contribution

It introduces new geometric models for configuration spaces of labelled clusters as $E_d$-algebras and analyzes their bar constructions and homology.

Findings

Configuration spaces of labelled clusters are $E_d$-algebras.

Provided geometric models for their iterated bar constructions.

Calculated stable homology of certain vertical configuration spaces.

Abstract

It is a classical result that configuration spaces of labelled particles in $\mathbb{R}^d$ are free $E_d$-algebras and that their $d$-fold bar construction is equivalent to the $d$-fold suspension of the labelling space. In this paper, we study a variation of these spaces, namely configuration spaces of labelled clusters of particles. These configuration spaces are again $E_d$-algebras, and we give geometric models for their iterated bar construction in two different ways: one establishes a description of these configuration spaces of clusters as cellular $E_1$-algebras, and the other one uses an additional verticality constraint. In the last section, we apply these results in order to calculate the stable homology of certain vertical configuration spaces.