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Periodic families in the homology of $GL_n(F_2)$

Kelly Wang

TL;DR

This work proves that the slope $\lambda=\frac{2}{3}$ in homological stability for $GL_n(\mathbb{F}_2)$ with $\mathbb{F}_2$-coefficients is optimal by constructing infinite families of nonzero classes lying on lines $d=\tfrac{2}{3}n+\mu$. It develops and exploits a stability Hopf algebra perspective to propagate stability data from $GL_n(\mathbb{F}_2)$ to $GL_n(\mathbb{Z})$ and $Aut(F_n)$, establishing that the same slope bounds hold and are optimal in those settings as well. The approach identifies a truncated detector via a map to the dual Steenrod algebra $A(1)_*$ and computes the stability Hopf algebra $\Delta_{\mathrm{CGL}(\mathbb{F}_2)}$ up to grading $5$, revealing a $(12,8)$-periodic structure after quotienting by $\sigma$. These constructions yield explicit periodic families, deduce vanishing line bounds for relative and absolute homology, and provide a practical framework for calculating stability Hopf algebras over $\mathbb{F}_2$, with applications to $GL_n(\mathscr{O})$ and $Aut(F_n)$ via the RW25 machinery.

Abstract

We construct infinite families of nonzero classes in $H_d(GL_n(F_2);F_2)$ along lines of the form $d =\frac{2}{3}n +$(constant), thereby showing that the known slope $\frac{2}{3}$-stability for these homology groups are optimal. Using the new stability Hopf algebra perspective of Randal-Williams, our computations in addition recover the slope-$\frac{2}{3}$ stability for $GL_n(Z)$ with coefficients in $F_2$, improve that for $Aut(F_n)$ to $\frac{2}{3}$, and demonstrate that those slopes are optimal. Perhaps of independent interest, we also provide a manual for computing stability Hopf algebras over $F_2$.

Periodic families in the homology of $GL_n(F_2)$

TL;DR

This work proves that the slope in homological stability for with -coefficients is optimal by constructing infinite families of nonzero classes lying on lines . It develops and exploits a stability Hopf algebra perspective to propagate stability data from to and , establishing that the same slope bounds hold and are optimal in those settings as well. The approach identifies a truncated detector via a map to the dual Steenrod algebra and computes the stability Hopf algebra up to grading , revealing a -periodic structure after quotienting by . These constructions yield explicit periodic families, deduce vanishing line bounds for relative and absolute homology, and provide a practical framework for calculating stability Hopf algebras over , with applications to and via the RW25 machinery.

Abstract

We construct infinite families of nonzero classes in along lines of the form (constant), thereby showing that the known slope -stability for these homology groups are optimal. Using the new stability Hopf algebra perspective of Randal-Williams, our computations in addition recover the slope- stability for with coefficients in , improve that for to , and demonstrate that those slopes are optimal. Perhaps of independent interest, we also provide a manual for computing stability Hopf algebras over .
Paper Structure (42 sections, 52 theorems, 130 equations, 10 figures, 3 tables)

This paper contains 42 sections, 52 theorems, 130 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Outline
  3. Acknowledgements
  4. Low degree group homology computations: $\mathop{\mathrm{GL}}\nolimits_g(\mathbb{F}_2)$
  5. Proof of \ref{['thm: additive basis']}
  6. $g=2$.
  7. The map $\mathop{\mathrm{GL}}\nolimits_3\to \mathop{\mathrm{GL}}\nolimits_4$.
  8. $g=3$.
  9. $g=4$.
  10. $g=5$.
  11. $g=6$.
  12. Proof of \ref{['thm: Winfty structure']} and \ref{['cor: identities']}
  13. Low degree group homology computations: $\mathop{\mathrm{GL}}\nolimits_g(\mathbb{Z})$
  14. Backgrounds
  15. Conventions
  16. ...and 27 more sections

Key Result

Theorem 1.1

For every $i\geqslant 0$, there are non-zero classes $u_{00}\in H_1(\mathop{\mathrm{GL}}\nolimits_2(\mathbb{F}_2);\mathbb{F}_2)$ is represented by the matrix $\smqty( 0 1 \\ 10 )$, and $u_{i1} = u_{00}u_{i0}$, $u_{i2} = u_{00}^2u_{i0}$ under the algebra structure of $H_*(\mathop{\mathrm{GL}}\nolimits_*(\mathbb{F}_2);\mathbb{F}_2)$.

Figures (10)

  • Figure 1: The non-zero classes of \ref{['intro thm: abs']}; $i\geqslant 0$. The orange line indicates an $u_{00}$-extension.
  • Figure 2: Vector space basis for $H_d(\mathop{\mathrm{GL}}\nolimits_g)$.
  • Figure 3: $\dim H_d(\mathop{\mathrm{GL}}\nolimits_g)=$ number of dots at $(g,d)$.
  • Figure 4: Basis for $H_d(\mathop{\mathrm{GL}}\nolimits_g(\mathbb{Z});\mathbb{F}_2)$.
  • Figure 5: $\dim H_d(\mathop{\mathrm{GL}}\nolimits_g(Z);\mathbb{F}_2)=$ number of dots at $(g,d)$.
  • ...and 5 more figures

Theorems & Definitions (75)

  • Theorem 1.1: \ref{['thm: absolute clases']}
  • Theorem 1.3: \ref{['prop: BGL/sigma classes']}
  • Theorem 1.5: \ref{['sec: stability for Z']}
  • Theorem 1.6: \ref{['prop: GLnZ and AutFn']}
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.4
  • Remark 2.5
  • Proposition 2.6
  • Proposition 2.7
  • ...and 65 more