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 a complete computation of the stable and unstable rational cohomology of unordered configuration spaces in some cases, including that of 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 conjectures.

Chiral Homology of elliptic curves and the Zhu algebra

“…)) are easily computed for small values of n using the presentation above and a computer algebra system such as SAGE [40] (for a general computation see [18,35,39]). We are particularly interested in the top Betti number h n ( Ů (n) ), we compute:…”

“…To compute higher chiral homologies we use an explicit description of the de Rham cohomology classes of configuration spaces of an elliptic curve. It is surprising that until quite recently even the Betti numbers where not available (see for example [18,35,39]). It is relatively easy to find a quotient complex of the chiral chain complex isomorphic to the Bar complex of Zhu(V ).…”

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

“…The cases of genera 0,1 have already been studied in [S4,S1,S3] and in [S2,MCF,P1], respectively. The Euler characteristic of the configuration spaces of any evendimensional orientable closed manifold M was computed by Félix and Thomas in [FT2] and it is given by the formula:…”

A Künneth theorem for configuration spaces

“…Even the celebrated representation stability theorem of [8] does little to lighten this gloomy outlook; indeed, computing stable multiplicities of representations in 𝐻 * (Conf 𝑘 (𝑋); ℚ) is a difficult open problem in almost all cases [14,Problem 3.5]. Even the multiplicity of the trivial representation, corresponding to the homology of unordered configuration spaces, was unknown for closed, orientable surfaces of positive genus until very recently-see [10] for the general case and [25,32] for two concurrent computations in the case of the torus.…”