Configuration spaces of clusters as $E_d$-algebras

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.

Keywords

configuration space, $E_d$-algebra, bar construction, stable homology

2010 Mathematics Subject Classification

55P35, 55P48, 55P65, 55R80, 57T30

Full Text (PDF format)

Received 6 October 2022

Received revised 23 June 2023

Accepted 27 June 2023

Published 29 May 2024