Representation stability for homotopy groups of configuration spaces

TL;DR

This paper proves that the dual rational homotopy groups of configuration spaces of certain manifolds exhibit uniform representation stability, and their derived dual integral homotopy groups are finitely generated as FI-modules, linking cohomology and homotopy properties.

Contribution

It establishes a general theorem connecting cohomology properties of co-FI-spaces to the stability of their dual homotopy groups, advancing understanding of representation stability in topology.

Findings

Dual rational homotopy groups are uniformly representation stable.

Derived dual integral homotopy groups are finitely generated FI-modules.

The results apply to configuration spaces of 1-connected manifolds of dimension at least 3.

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 Church, and that their derived dual integral homotopy groups are finitely-generated as FI-modules in the sense of Church-Ellenberg-Farb. This is a consequence of a more general theorem relating properties of the cohomology groups of a 1-connected co-FI-space to properties of its dual homotopy groups. We also discuss several other applications.