Configuration spaces on a wedge of spheres and Hochschild-Pirashvili homology. (Espaces de configuration sur un bouquet de sphères et homologie de Hochschild-Pirashvili.) (English. French summary) Zbl 07914808
The authors study the compactly supported rational cohomology of configuration spaces of \(n\) points in wedges of spheres, equipped with an action of the symmetric group and of the group of outer morphisms of the free group.
The authors show that these cohomology representations form a polynomial functor, whose composition factors are completely computed for \(n\leq 10\).
The compactly supported cohomology of these configuration spaces is related to that of the moduli space \(\mathcal M_{2,n}\) of genus \(2\) curves with \(n\) markings. The relationship is formulated in [C. Bibby et al., “Homology representations of compactified configurations on graphs applied to \(\mathcal{M}_{2,n}\)”, Preprint, arXiv:2109.03302].
Indeed, with their computations, the authors recover the \(\mathfrak S_n\)-character of weight 0 cohomology groups \(\operatorname{gr}^W_0H^*_c(\mathcal M_{2,n};\mathbb Q),\) for \(n\leq 10\), and provide a new lower bound on its equivariant character.
The authors show that these cohomology representations form a polynomial functor, whose composition factors are completely computed for \(n\leq 10\).
The compactly supported cohomology of these configuration spaces is related to that of the moduli space \(\mathcal M_{2,n}\) of genus \(2\) curves with \(n\) markings. The relationship is formulated in [C. Bibby et al., “Homology representations of compactified configurations on graphs applied to \(\mathcal{M}_{2,n}\)”, Preprint, arXiv:2109.03302].
Indeed, with their computations, the authors recover the \(\mathfrak S_n\)-character of weight 0 cohomology groups \(\operatorname{gr}^W_0H^*_c(\mathcal M_{2,n};\mathbb Q),\) for \(n\leq 10\), and provide a new lower bound on its equivariant character.
Reviewer: Angelina Zheng (Roma)
MSC:
| 55R80 | Discriminantal varieties and configuration spaces in algebraic topology |
| 14H10 | Families, moduli of curves (algebraic) |
| 14Q05 | Computational aspects of algebraic curves |
| 13D03 | (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) |
Software:
SageMathReferences:
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