Representation stability for homotopy groups of configuration spaces

Abstract

Abstract We prove that the dual rational homotopy groups of the configuration spaces of a 1-connected manifold of dimension at least 3 are uniformly representation stable in the sense of [6], and that their derived dual integral homotopy groups are finitely generated as 𝖥𝖨 {{\mathsf{FI}}} -modules in the sense of [4]. This is a consequence of a more general theorem relating properties of the cohomology groups of a 1-connected co- 𝖥𝖨 {{\mathsf{FI}}} -space to properties of its dual homotopy groups. We also discuss several other applications, including free Lie and Gerstenhaber algebras.