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Verification of the Quillen conjecture in the rank 2 imaginary quadratic case
Homology, Homotopy and Applications ( IF 0.8 ) Pub Date : 2020-01-01 , DOI: 10.4310/hha.2020.v22.n2.a17
Bui Anh Tuan 1 , Alexander D. Rahm 2
Affiliation  

We confirm a conjecture of Quillen in the case of the mod $2$ cohomology of arithmetic groups ${\rm SL}_2({\mathcal{O}}_{\mathbb{Q}(\sqrt{-m})}[\frac{1}{2}]\thinspace)$, where ${\mathcal{O}}_{\mathbb{Q}(\sqrt{-m}\thinspace)}$ is an imaginary quadratic ring of integers. To make explicit the free module structure on the cohomology ring conjectured by Quillen, we compute the mod $2$ cohomology of ${\rm SL}_2({\mathbb{Z}}[\sqrt{-2}\thinspace][\frac{1}{2}])$ via the amalgamated decomposition of the latter group.

中文翻译:

Quillen 猜想在 2 阶虚二次情况下的验证

我们在算术群 ${\rm SL}_2({\mathcal{O}}_{\mathbb{Q}(\sqrt{-m})}[ \frac{1}{2}]\thinspace)$,其中 ${\mathcal{O}}_{\mathbb{Q}(\sqrt{-m}\thinspace)}$ 是一个虚二次整数环。为了明确 Quillen 推测的上同调环上的自由模结构,我们计算了 ${\rm SL}_2({\mathbb{Z}}[\sqrt{-2}\thinspace][\ frac{1}{2}])$ 通过后一组的合并分解。
更新日期:2020-01-01
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