论文标题
与有限组相关的零属轨道配置空间的代数不变式
Algebraic invariants of orbit configuration spaces in genus zero associated to finite groups
论文作者
论文摘要
我们考虑与有限组相关的轨道配置空间，这些轨道由定向自由作用，以保存同构$ 2 $ -sphere上的同态，减去有限数量的积分。这种动作等同于有限的子组$ g \ subset \ mathrm {pgl}（\ mathbb {c}^2）$上的有限子组$ g \ subset \ subset \ mathrm {pgl} $ \ mathbb {c}^2）$ $ \ mathbb {p}^1 $ y Minus a Minus a有限套装$ Z $ stable $ g $。我们计算轨道配置空间的同时戒指和Poincaré系列$ C_N^g（\ Mathbb {p}^1 \ setMinus Z）$。这可以看作是对经典配置空间Arnold的工作的概括$ C_N（\ Mathbb {C}）$（$（g，z）=（\ {1 \}，\ infty $））。从工作中可以看出，$ c_n^g（\ mathbb {p}^1 \ setminus z）$在理性同义理论的意义上是正式的。我们还证明了LCS公式的存在，该公式将$ C_N^g（\ Mathbb {p}^1 \ setMinus Z）$与$ C_N^G（$ c_n^g（\ Mathbb {p} p}^1 \ setminus z）的连续术语相关的商等级。连续的商对应于作者在早期工作中引入的分级谎言代数的同质元素。这种公式也以$ \ mathbb {c} $的经典配置空间而闻名，其中基本组是Artin辫子组，等级对应于Kohno-Drinfeld Lie代数的同质元素的维度。
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.
