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

## TL;DR

This paper computes the rational cohomology of configuration spaces on closed orientable surfaces, detailing the mixed Hodge structure and symplectic group action, providing explicit representation decompositions and new formulas for Betti numbers.

## Contribution

It introduces a detailed analysis of the cohomology with symplectic group action, including explicit decomposition into irreducible representations and formulas for mixed Hodge and Betti numbers.

## Key findings

- Explicit decomposition of cohomology into irreducible representations
- New formulas for mixed Hodge numbers and Betti numbers
- Series with coefficients in the Grothendieck ring of sp(2g)

## Abstract

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 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.