论文标题
在轨道表面的轨道配置空间上
On models of orbit configuration spaces of surfaces
论文作者
论文摘要
我们考虑轨道配置空间$ c_n^g（s）$，其中$ s $是通过删除有限数量的积分（最终无）和$ g $是一个有限的组，是从$ s $连续自由操作的有限群体，是从封闭的可定向表面$ \ bar {s} $中获得的表面。我们证明，通过在第一个$ k $坐标上投影获得的纤维$π_{n，k}：c_ {n}^g（s）\ to c_k^g（s）$是合理的振动。结果，空间$ c_ {n}^g（s）$具有sullivan型号$ a_ {n，k} =λv_{c_k^g（s）} \ otimesλv_{c_ {c_ {n-k}^g（s_}^g（s_ {g，k}） $ΛV_{C_k^G(S)}\to A_{n,k} \to ΛV_{C_{n-k}^G(S_{G,k})},$ where $ΛV_X$ denotes the minimal model of $X$, and $C_{n-k}^G(S_{G,k})$ is the fiber of $π_{n，k} $。我们表明，此模型是最小的，除了某些情况下，当$ s \ simeq s^2 $并在所有情况下计算$ c_n^g（s）$的较高$ψ$ -HOMOTOPY组（与最小模型的发电机相关）。我们从计算中得出，有限的betti数字是一个合理的$ k（π，1）$，即其最小型号和$ 1 $ -Minimal的模型是相同的（或等于$ -HOMOTOPY SPACE的$ 2 $ 2 $ 2 $），如果$ 2 $），如果$ 2 $），如果$ 2 $），则仅$ s $ s $ s $ s $ s $ s $ s^2 2 2 2.特别是，对于$ s $而非同型至$ s^2 $，最小型号（与$ 1 $ - 少数模型的同构）完全由$π_1c_n^g（s）$的malcev lie代数确定。当$ a_ {n，k} $很少时，我们得到了一个确切的malcev lie代数$ 0 \ to l_ {c_ {c_ {n-k}^g（s_ {g，k}）} \ to l_ {c_ {c_ {n}^g（s s） Malcev Lie代数为$π_1x$。对于$ s \ varsubsetneq \ bar {s} = s^2 $和$ g $，通过保存同质形态的定向行动，我们证明，$ c_n^g（s）$的共同体戒指是koszul的 工作。
We consider orbit configuration spaces $C_n^G(S)$, where $S$ is a surface obtained out of a closed orientable surface $\bar{S}$ by removing a finite number of points (eventually none) and $G$ is a finite group acting freely continuously on $S$. We prove that the fibration $π_{n,k} : C_{n}^G(S) \to C_k^G(S)$ obtained by projecting on the first $k$ coordinates is a rational fibration. As a consequence, the space $C_{n}^G(S)$ has a Sullivan model $A_{n,k}=ΛV_{C_k^G(S)}\otimes ΛV_{C_{n-k}^G(S_{G,k})}$ fitting in a cdga sequence: $ΛV_{C_k^G(S)}\to A_{n,k} \to ΛV_{C_{n-k}^G(S_{G,k})},$ where $ΛV_X$ denotes the minimal model of $X$, and $C_{n-k}^G(S_{G,k})$ is the fiber of $π_{n,k}$. We show that this model is minimal except for some cases when $S\simeq S^2$ and compute in all the cases the higher $ψ$-homotopy groups (related to the generators of the minimal model) of $C_n^G(S)$. We deduce from the computation that $C_n^G(S)$ having finite Betti numbers is a rational $K(π,1)$, i.e its minimal model and $1$-minimal model are the same (or equivalently the $ψ$-homotopy space vanishes in degree grater then $2$), if and only if $S$ is not homeomorphic to $S^2$. In particular, for $S$ not homeomorphic to $S^2$, the minimal model (isomorphic to the $1$-minimal model) is entirely determined by the Malcev Lie algebra of $π_1 C_n^G(S)$. When $A_{n,k}$ is minimal, we get an exact sequence of Malcev Lie algebras $0\to L_{C_{n-k}^G(S_{G,k})}\to L_{C_{n}^G(S)}\to L_{C_k^G(S)}\to 0$, where $L_X$ is the Malcev Lie algebra of $π_1X$. For $S \varsubsetneq \bar{S}=S^2$ and $G$ acting by orientation preserving homeomorphism, we prove that the cohomology ring of $C_n^G(S)$ is Koszul, and that for some of these spaces the minimal model can be obtained out of a Cartan-Chevally-Eilenberg construction applied to graded Lie algebra computed in an earlier work.
