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Homological stability: a tool for computations

Nathalie Wahl

TL;DR

This work surveys homological stability as a universal tool for computing the homology of families of groups, introducing a general Quillen-style framework based on braided or symmetric monoidal groupoids and a space of destabilizations $W_n(A,X)$ to derive stability ranges via a spectral sequence. It then connects stability to group completion, identifying stable homology with loop-space homology and detailing concrete identifications for symmetric groups, braid groups, GL$_n(R)$, mapping class groups, and Aut$(F_n)$, with implications for rational and integral stability and K-theory. The Higman–Thompson groups are treated as a primary application: stability for $V_{k,n}$ yields an explicit stable homology description in terms of Moore spectra and, in particular, demonstrates vanishing higher homology for $k=2$. The perspectives section highlights the limits of braiding as a criterion for stability, and points to higher algebraic structures (E$_k$-algebras), categorical completions, and representation stability as avenues to push stability beyond traditional ranges and broader contexts.

Abstract

Homological stability has shown itself to be a powerful tool for the computation of homology of families of groups such as general linear groups, mapping class groups or automorphisms of free groups. We survey here tools and techniques for proving homological stability theorems and for computing the stable homology, and illustrate the method through the computation of the homology of Higman-Thompson groups.

Homological stability: a tool for computations

TL;DR

This work surveys homological stability as a universal tool for computing the homology of families of groups, introducing a general Quillen-style framework based on braided or symmetric monoidal groupoids and a space of destabilizations to derive stability ranges via a spectral sequence. It then connects stability to group completion, identifying stable homology with loop-space homology and detailing concrete identifications for symmetric groups, braid groups, GL, mapping class groups, and Aut, with implications for rational and integral stability and K-theory. The Higman–Thompson groups are treated as a primary application: stability for yields an explicit stable homology description in terms of Moore spectra and, in particular, demonstrates vanishing higher homology for . The perspectives section highlights the limits of braiding as a criterion for stability, and points to higher algebraic structures (E-algebras), categorical completions, and representation stability as avenues to push stability beyond traditional ranges and broader contexts.

Abstract

Homological stability has shown itself to be a powerful tool for the computation of homology of families of groups such as general linear groups, mapping class groups or automorphisms of free groups. We survey here tools and techniques for proving homological stability theorems and for computing the stable homology, and illustrate the method through the computation of the homology of Higman-Thompson groups.
Paper Structure (9 sections, 8 theorems, 31 equations, 2 figures)

This paper contains 9 sections, 8 theorems, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A general framework for Quillen's stability argument
  3. The space of destabilizations
  4. Homological stability
  5. Twisted coefficients
  6. Group completion and the stable homology
  7. Higman-Thomson groups
  8. Perspectives
  9. Adding complements categorically

Key Result

Theorem 1.2

Nak60HatVog98bBarPriGal11 For all $i\le \frac{n}{2}$, $H_i(\Sigma_n)\cong H_i(\Omega_0^\infty \mathbb{S}) \cong H_i(\operatorname{Aut}(F_{n+3})),$

Figures (2)

  • Figure 1: Block braid $b_{3,2}$
  • Figure 2: An element of Thompson's group $V=V_{2,1}$ obtained from a binary subdivision of the source and target interval, and a choice of permutation of the subintervals.

Theorems & Definitions (23)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3: Arn70CLM
  • Theorem 1.4
  • Theorem 1.5
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 13 more