Extra structure on the cohomology of configuration spaces of closed orientable surfaces

The rational homology of unordered configuration spaces of points on any surface was studied by Drummond–Cole and Knudsen. We compute the rational cohomology of configuration spaces on a closed orientable surface, keeping track of the mixed Hodge numbers and the action of the symplectic group on the cohomology. We find a series with coefficients in the Grothendieck ring of $\mathfrak{sp}(2g)$ that describes explicitly the decomposition of the cohomology into irreducible representations. From that we deduce the mixed Hodge numbers and the Betti numbers, obtaining a new formula without cancellations.

Keywords

configuration space, surface, cohomology

2010 Mathematics Subject Classification

55R80

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Received 14 September 2021

Received revised 20 March 2022

Accepted 21 March 2022

Published 22 March 2023