Configuration spaces on the sphere and higher loop spaces

Abstract

We show that the homology over a field of the space ΛnΣnX of free maps from the n-sphere to the n-fold suspension of X depends only on the cohomology algebra of X and compute it explicitly. We compute also the homology of the closely related labelled configuration space C(Sn,X) on the n-sphere with labels in X and of its completion, that depend only on the homology of X. In many but not all cases the homology of C(Sn,X) coincides with the homology of ΛnΣnX. In particular we obtain the homology of the unordered configuration spaces on a sphere.