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Homology, Homotopy and Applications
Volume 26 (2024)
Number 1
Configuration spaces of clusters as $E_d$-algebras
Pages: 319 – 339
DOI: https://dx.doi.org/10.4310/HHA.2024.v26.n1.a19
Author
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.
Keywords
configuration space, $E_d$-algebra, bar construction, stable homology
2010 Mathematics Subject Classification
55P35, 55P48, 55P65, 55R80, 57T30
Received 6 October 2022
Received revised 23 June 2023
Accepted 27 June 2023
Published 29 May 2024