Periodic families in the homology of $GL_n(F_2)$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909994240180224 |
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| author | Wang, Kelly |
| author_facet | Wang, Kelly |
| contents | 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 $F_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12431 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Periodic families in the homology of $GL_n(F_2)$ Wang, Kelly Algebraic Topology K-Theory and Homology 20G40, 55S12 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 $F_2$. |
| title | Periodic families in the homology of $GL_n(F_2)$ |
| topic | Algebraic Topology K-Theory and Homology 20G40, 55S12 |
| url | https://arxiv.org/abs/2601.12431 |