2016/12/19 by Megan Maguire, Maguire, Megan, with Appendix by Matthew Christie +3 · 1 citation
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Geometry and complex manifolds #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1612.06314
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 \mathbbCP3 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.