Coxeter群族的同调稳定性
Homological stability for families of Coxeter groups
摘要：
我们证明了某些家庭的Coxeter群和包含$W_1\hookrightarrow W_2\hookrightarrow...$满足同调稳定性，这意味着在每一个度的同源$H_\ast(BW_n)$最终是独立的$n$。这给出了$A_n$、$B_n$和$D_n$型考克斯特群族的同调稳定性的统一处理，恢复了前两种情况下的现有结果，并在第三种情况下给出了一个新结果。证明的关键是证明具有$W_n$作用的单纯复形是高度连通的。要做到这一点，我们表明，重心重分是一个实例的“基本建设”，然后使用戴维斯的描述的基本建设作为一个不断增加的工会商会推导出所需的连接。
英文摘要：
We prove that certain families of Coxeter groups and inclusions $W_1\hookrightarrow W_2\hookrightarrow...$ satisfy homological stability, meaning that in each degree the homology $H_\ast(BW_n)$ is eventually independent of $n$. This gives a uniform treatment of homological stability for the families of Coxeter groups of type $A_n$, $B_n$ and $D_n$, recovering existing results in the first two cases, and giving a new result in the third. The key step in our proof is to show that a certain simplicial complex with $W_n$-action is highly connected. To do this we show that the barycentric subdivision is an instance of the 'basic construction', and then use Davis's description of the basic construction as an increasing union of chambers to deduce the required connectivity.
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