与有限群相关的亏格为零的轨道构形空间的代数不变式
Algebraic invariants of orbit configuration spaces in genus zero associated to finite groups
摘要：
我们考虑与有限群相关联的轨道构形空间，它们通过保向同胚在减去有限个点的$2$ -球面上自由作用。这种作用等价于复射影线$\mathbb{P}^1$上的有限子群$G\subset \mathrm{PGL}(\mathbb{C}^2)$减去在$G$下稳定的有限集$Z$的单应作用。计算了轨道位形空间$C_n^G(\mathbb{P}^1 \setminus Z)$的上同调环和Poincaré级数.这可以看作是Arnold对经典位形空间$C_n(\mathbb{C})$（$(G,Z)=(\{1\},\infty$）的工作的推广。它遵循的工作，$C_n^G(\mathbb{P}^1\setminus Z)$是正式的意义上的理性同伦理论。我们还证明了存在一个LCS公式有关的Poincaré系列的$C_n^G(\mathbb{P}^1\setminus Z)$的连续项的低中心系列的基本组的$C_n^G(\mathbb{P}^1 \setminus Z)$的秩。连续的同分元对应于作者在较早的工作中引入的分次李代数的齐次元素。这样的公式对于$\mathbb{C}$的经典配置空间也是已知的，其中基本群是Artin辫子群，并且秩对应于Kohno-Drinfeld李代数的齐次元素的维数。
英文摘要：
We consider orbit configuration spaces associated to finite groups acting freely by orientation preserving homeomorphisms on the $2$-sphere minus a finite number of points. Such action is equivalent to a homography action of a finite subgroup $G\subset \mathrm{PGL}(\mathbb{C}^2)$ on the complex projective line $\mathbb{P}^1$ minus a finite set $Z$ stable under $G$. We compute the cohomology ring and the Poincaré series of the orbit configuration space $C_n^G(\mathbb{P}^1 \setminus Z)$. This can be seen as a generalization of the work of Arnold for the classical configuration space $C_n(\mathbb{C})$ ($(G,Z)=(\{1\},\infty$)). It follows from the work that $C_n^G(\mathbb{P}^1\setminus Z)$ is formal in the sense of rational homotopy theory. We also prove the existence of an LCS formula relating the Poincaré series of $C_n^G(\mathbb{P}^1\setminus Z)$ to the ranks of quotients of successive terms of the lower central series of the fundamental group of $C_n^G(\mathbb{P}^1 \setminus Z)$. The successive quotients correspond to homogenous elements of graded Lie algebras introduced by the author in an earlier work. Such formula is also known for classical configuration spaces of $\mathbb{C}$, where fundamental groups are Artin braid groups and the ranks correspond to dimensions of homogenous elements of the Kohno-Drinfeld Lie algebras.
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