A new approach to twisted homological stability with applications to congruence subgroups

Pub Date : 2023-11-21 DOI:10.1112/topo.12316

Andrew Putman

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引用次数: 6

Abstract

We introduce a new method for proving twisted homological stability, and use it to prove such results for symmetric groups and general linear groups. In addition to sometimes slightly improving the stable range given by the traditional method (due to Dwyer), it is easier to adapt to nonstandard situations. As an illustration of this, we generalize to ${GL}_{n}$ of many rings $R$ a theorem of Borel that says that passing from ${GL}_{n}$ of a number ring to a finite-index subgroup does not change the rational cohomology. Charney proved this generalization for trivial coefficients, and we extend it to twisted coefficients.

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一种扭同调稳定性的新方法及其在同余子群上的应用

本文介绍了一种新的证明扭同调稳定性的方法，并用它证明了对称群和一般线性群的扭同调稳定性。除了有时会略微提高传统方法给出的稳定范围(由于Dwyer)外，它更容易适应非标准情况。为了说明这一点，我们将Borel的一个定理推广到GL n$ \operatorname{GL}_n$的多环R$ R$，该定理表明从一个数环的GL n$ \operatorname{GL}_n$传递到有限索引子群并不改变有理上同。Charney证明了对平凡系数的推广，我们把它推广到扭曲系数。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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