Configuration spaces on a wedge of spheres and Hochschild-Pirashvili homology
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866918055796277248 |
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| author | Gadish, Nir Hainaut, Louis |
| author_facet | Gadish, Nir Hainaut, Louis |
| contents | We study the compactly supported rational cohomology of configuration spaces of points on wedges of spheres, equipped with natural actions of the symmetric group and the group $Out(F_g)$ of outer automorphisms of the free group. These representations show up in seemingly unrelated parts of mathematics, from cohomology of moduli spaces of curves to polynomial functors on free groups and Hochschild-Pirashvili cohomology.
We show that these cohomology representations form a polynomial functor, and use various geometric models to compute many of its composition factors. We further compute the composition factors completely for all configurations of $n\leq 10$ particles. An application of this analysis is a new super-exponential lower bound on the symmetric group action on the weight $0$ component of $H^*_c(M_{2,n})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_12494 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Configuration spaces on a wedge of spheres and Hochschild-Pirashvili homology Gadish, Nir Hainaut, Louis Algebraic Topology Representation Theory 55R80, 14H10, 14Q05, 13D03 We study the compactly supported rational cohomology of configuration spaces of points on wedges of spheres, equipped with natural actions of the symmetric group and the group $Out(F_g)$ of outer automorphisms of the free group. These representations show up in seemingly unrelated parts of mathematics, from cohomology of moduli spaces of curves to polynomial functors on free groups and Hochschild-Pirashvili cohomology. We show that these cohomology representations form a polynomial functor, and use various geometric models to compute many of its composition factors. We further compute the composition factors completely for all configurations of $n\leq 10$ particles. An application of this analysis is a new super-exponential lower bound on the symmetric group action on the weight $0$ component of $H^*_c(M_{2,n})$. |
| title | Configuration spaces on a wedge of spheres and Hochschild-Pirashvili homology |
| topic | Algebraic Topology Representation Theory 55R80, 14H10, 14Q05, 13D03 |
| url | https://arxiv.org/abs/2202.12494 |