Computing cohomology of configuration spaces

Abstract

We give a concrete method to explicitly compute the rational cohomology of the unordered configuration spaces of connected, oriented, closed, even-dimensional manifolds of finite type which we have implemented in Sage [S+09]. As an application, we give acomplete computation of the stable and unstable rational cohomology of unordered configuration spaces in some cases, including that of ${\mathbb{CP}}^3$ and a genus 1 Riemann surface, which is equivalently the homology of the elliptic braid group. In an appendix, we also give large tables of unstable and stable Betti numbers of unordered configuration spaces. From these, we empirically observe stability phenomenon in the unstable cohomology of unordered configuration spaces of some manifolds, some of which we prove and some of which we state as conjecture.

55 pages, 2 figures

References in corpus (1)

• Betti numbers of configuration spaces of surfaces

Cited by in corpus (5)

• Betti numbers of configuration spaces of surfaces
• Local systems on complements of arrangements of smooth, complex algebraic hypersurfaces
• Extra structure on the cohomology of configuration spaces of closed orientable surfaces
• Strong stability and shifted stability for the cohomology of configuration spaces
• A Kuenneth theorem for configuration spaces