Configuration spaces of clusters as $E_d$-algebras
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866914815937609728 |
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| author | Kranhold, Florian |
| author_facet | Kranhold, Florian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_02729 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Configuration spaces of clusters as $E_d$-algebras Kranhold, Florian Algebraic Topology 55R80, 55I42, 55P35, 55P48, 55P65, 18N40, 57T30 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. |
| title | Configuration spaces of clusters as $E_d$-algebras |
| topic | Algebraic Topology 55R80, 55I42, 55P35, 55P48, 55P65, 18N40, 57T30 |
| url | https://arxiv.org/abs/2104.02729 |