Linear stable ranges for integral homotopy groups of configuration spaces

We prove explicit linear stable ranges for the $\mathsf{FI}$-modules $\mathrm{Hom}(π_p \mathrm{Conf} M, \mathbb Z)$ and $\mathrm{Ext}(π_p \mathrm{Conf} M, \mathbb Z)$ with $\mathrm{Conf} M$ being the configuration co$\mathsf{FI}$-space of a $d$-dimensional manifold with $d \geq 3$. The proof of this result uses a homotopy-theoretic approach to representation stability for $\mathsf{FI}$-modules. This allows us to derive representation stability results from homotopy-theoretical statements, in particular the generalized Blakers-Massey theorem. We also generalize to $\mathsf{FI}_G$-modules and orbit configuration spaces.

💡 Research Summary

The paper establishes explicit linear stability ranges for the integral homotopy groups of configuration spaces, using the language of FI‑modules. Let M be a connected d‑dimensional manifold with d ≥ 3, and let Conf M denote the ordered configuration co‑FI‑space of M. For each p ≥ 2 the groups πp(Conf M) acquire a co‑FI‑module structure; after dualising one obtains the FI‑modules Hom(πp(Conf M),ℤ) (the non‑torsion part) and Ext(πp(Conf M),ℤ) (the torsion part). The main result (Theorem A) shows that these FI‑modules are finitely generated and gives concrete bounds on their generation degree t0 and presentation degree t1: \