Representation stability for homotopy automorphisms

TL;DR

This paper proves that the rational homotopy groups of certain homotopy automorphism groups exhibit representation stability, advancing understanding of their algebraic and topological structure.

Contribution

It establishes that under specific conditions, these homotopy automorphism groups' rational homotopy groups form finitely generated FI-modules, demonstrating representation stability.

Findings

Rational homotopy groups form finitely generated FI-modules

Homotopy automorphisms satisfy representation stability

Applicable to iterated wedge sums and connected sums of manifolds

Abstract

We consider in parallel pointed homotopy automorphisms of iterated wedge sums of topological spaces and boundary relative homotopy automorphisms of iterated connected sums of manifolds minus a disk. Under certain conditions on the spaces and manifolds, we prove that the rational homotopy groups of these homotopy automorphisms form finitely generated FI-modules, and thus satisfy representation stability for symmetric groups, in the sense of Church and Farb.