Periodic families in the homology of $G{L}_n(F_2)$

Abstract

We construct infinite families of nonzero classes in $H_d(G{L}_n(F_2);F_2)$ 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 $G{L}_n(Z)$ 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 $F_2$.