发表机构
Purdue University（普渡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用Cohen–Pakianathan的成果与$\text{FI}$同调理论，将稳定性模式推广到球面$S^2$和射影平面$\rp^2$上的辫群，证明其等变同调分别具有表示稳定性与稳定周期性。
AI 中文摘要
McDuff和Segal证明了具有非空边界的连通流形的无序构形空间的同调稳定性；后续Church证明了紧流形上有序构形空间的表示稳定性。当相关构形空间是Eilenberg–Maclane空间时，这些稳定性模式可分别推广到曲面辫群和纯曲面辫群。但上述结果不适用于球面$S^2$和射影平面$\rp^2$上的辫群及纯辫群，因为这两类曲面上的有序与无序构形空间并非Eilenberg–Maclane空间。本文将利用Cohen–Pakianathan的成果与$\text{FI}$同调理论，将已知稳定性模式推广到这些未知情形，这是本文证明的更一般结果的特例：有序与无序构形空间的等变同调分别具有表示稳定性与稳定周期性。
英文摘要
McDuff and Segal proved homological stability for unordered configuration spaces of connected manifolds with non-empty boundary. Later, Church proved representation stability for ordered configuration spaces on compact manifolds. These stability patterns extend to surface braid groups and pure surface braid groups, respectively, whenever the associated configuration space is an Eilenberg--Maclane space. However, these results do not apply to braid groups and pure braid groups on $S^2$ and $\RP^2$ since ordered and unordered configuration spaces on these surfaces are not Eilenberg--Maclane spaces. This paper will use results by Cohen--Pakianathan and the theory of $\FI$-homology to extend known stability patterns to these unknown cases. This is a special case of a more general result proven in this paper: the equivariant homology of ordered and unordered configuration spaces exhibit representation stability and stable periodicity respectively.
