On the stable cohomology of the IA-automorphism groups of free groups
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911810988277760 |
|---|---|
| author | Habiro, Kazuo Katada, Mai |
| author_facet | Habiro, Kazuo Katada, Mai |
| contents | Borel's stability and vanishing theorem gives the stable cohomology of $\mathrm{GL}(n,\mathbb{Z})$ with coefficients in algebraic $\mathrm{GL}(n,\mathbb{Z})$-representations. By combining the Borel theorem with the Hochschild-Serre spectral sequence, we compute the twisted first cohomology of the automorphism group $\mathrm{Aut}(F_n)$ of the free group $F_n$ of rank $n$. We also study the stable rational cohomology of the IA-automorphism group $\mathrm{IA}_n$ of $F_n$. We propose a conjectural algebraic structure of the stable rational cohomology of $\mathrm{IA}_n$, and consider some relations to known results and conjectures. We also consider a conjectural structure of the stable rational cohomology of the Torelli groups of surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_13458 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the stable cohomology of the IA-automorphism groups of free groups Habiro, Kazuo Katada, Mai Algebraic Topology Group Theory 20F28, 20J06 Borel's stability and vanishing theorem gives the stable cohomology of $\mathrm{GL}(n,\mathbb{Z})$ with coefficients in algebraic $\mathrm{GL}(n,\mathbb{Z})$-representations. By combining the Borel theorem with the Hochschild-Serre spectral sequence, we compute the twisted first cohomology of the automorphism group $\mathrm{Aut}(F_n)$ of the free group $F_n$ of rank $n$. We also study the stable rational cohomology of the IA-automorphism group $\mathrm{IA}_n$ of $F_n$. We propose a conjectural algebraic structure of the stable rational cohomology of $\mathrm{IA}_n$, and consider some relations to known results and conjectures. We also consider a conjectural structure of the stable rational cohomology of the Torelli groups of surfaces. |
| title | On the stable cohomology of the IA-automorphism groups of free groups |
| topic | Algebraic Topology Group Theory 20F28, 20J06 |
| url | https://arxiv.org/abs/2211.13458 |