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Linear stable ranges for integral homotopy groups of configuration spaces

Nicolas Guès

TL;DR

This work develops a homotopy-theoretic approach to representation stability for FI-objects, deriving explicit linear stable ranges for the integral duals of π_p Conf M via FI-homology and cohomology. By interpreting FI-objects and their (co)homology through total cofibers of cubes and employing a cardinality filtration, the author provides going-down results that translate connectivity data from configuration cubes (via Blakers–Massey) into concrete generation/presentation bounds for Hom and Ext of π_p Conf M. The main outputs are sharp linear bounds on t_0 and t_1 for Hom(π_p Conf M, Z) and Ext(π_p Conf M, Z), as well as analogous bounds for cohomology H^p Conf M, all invariant under FI_G extensions and orbit configuration spaces, thus covering simply connected and non-simply connected cases. This framework yields integral stability results that align with and extend recent rational and integral stability bounds, highlighting a unifying homotopical method for FI-module stability in geometric contexts.

Abstract

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.

Linear stable ranges for integral homotopy groups of configuration spaces

TL;DR

This work develops a homotopy-theoretic approach to representation stability for FI-objects, deriving explicit linear stable ranges for the integral duals of π_p Conf M via FI-homology and cohomology. By interpreting FI-objects and their (co)homology through total cofibers of cubes and employing a cardinality filtration, the author provides going-down results that translate connectivity data from configuration cubes (via Blakers–Massey) into concrete generation/presentation bounds for Hom and Ext of π_p Conf M. The main outputs are sharp linear bounds on t_0 and t_1 for Hom(π_p Conf M, Z) and Ext(π_p Conf M, Z), as well as analogous bounds for cohomology H^p Conf M, all invariant under FI_G extensions and orbit configuration spaces, thus covering simply connected and non-simply connected cases. This framework yields integral stability results that align with and extend recent rational and integral stability bounds, highlighting a unifying homotopical method for FI-module stability in geometric contexts.

Abstract

We prove explicit linear stable ranges for the -modules and with being the configuration co-space of a -dimensional manifold with . The proof of this result uses a homotopy-theoretic approach to representation stability for -modules. This allows us to derive representation stability results from homotopy-theoretical statements, in particular the generalized Blakers-Massey theorem. We also generalize to -modules and orbit configuration spaces.

Paper Structure

This paper contains 14 sections, 32 theorems, 119 equations, 2 figures.

Table of Contents

  1. $\mathsf{FI}$-objects and $\mathsf{FI}$-homology
  2. $\mathsf{FI}$-objects in an $\infty$-category
  3. The $\mathsf{FI}$-homology functor $\mathbb{H}$
  4. Recovering usual $\mathsf{FI}$-homology
  5. Homotopical description of $\mathbb{H}$
  6. co$\mathsf{FI}$ and co$\mathsf{FB}$-objects
  7. Shift functor and $\mathsf{FI}$-derivative
  8. A complex computing $\mathsf{FI}$-homology
  9. The cardinality filtration
  10. Stability of a (co)$\mathsf{FI}$-object versus stability of its homotopy groups
  11. Representation stability and Blakers-Massey theorem
  12. Cartesianity of the cube of configuration spaces
  13. Recovering stability of the cohomology of configuration spaces
  14. $\mathsf{FI}_G$-modules and non-trivial fundamental groups

Key Result

Theorem A

Let $M$ be a connected manifold of dimension $d \geq 3$. Then for $p\geq 2$, the homotopy groups $\pi_p(\mathop{\mathrm{Conf}}\nolimits M)$ can be given a co$\mathsf{FI}$-module structure, and the dual homotopy groups: $\mathop{\mathrm{Hom}}\nolimits(\pi_p(\mathop{\mathrm{Conf}}\nolimits M), \mathbb

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (67)

  • Theorem A
  • Theorem B
  • Theorem 1
  • Proposition 1.1
  • proof
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • proof
  • Corollary 1.5
  • ...and 57 more