Configuration spaces of clusters as $E_d$algebras
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.
 Publication:

arXiv eprints
 Pub Date:
 April 2021
 DOI:
 10.48550/arXiv.2104.02729
 arXiv:
 arXiv:2104.02729
 Bibcode:
 2021arXiv210402729K
 Keywords:

 Mathematics  Algebraic Topology;
 55R80;
 55I42;
 55P35;
 55P48;
 55P65;
 18N40;
 57T30
 EPrint:
 22 pages, 6 figures