群在同伦$2n$ -球面上的自由和真不连续作用
Free and properly discontinuous actions of groups on homotopy $2n$-spheres
摘要：
设$G$是自由地、真不连续地和胞腔地作用在有限维$C$ W-复形$Σ(2n)$上的群，该复形具有$2n$ -球面$\mathbb{S}^{2n}$的同伦类型.然后，这个作用诱导了群$G$对$Σ(2n)$的上同调的作用。对于虚循环群族，我们对作用于$Σ(2n)$的群进行了分类，给出了所有可能轨道空间的同伦类型，以及所有作用于上同调的群。在\par的假设下，我们研究了具有虚上同调维数$\mbox{vcd}\,G<\infty$的群在$Σ(2n)$上的作用.结果表明，它们由自由群和某些半直积$F\rtimes \mathbb{Z}_2$组成，其中$F$是自由群.对于这些群$G$和$G$在$\mbox{Aut}(\mathbb{Z})$上的给定作用，我们给出了一个等价于作用$G$在$Σ(2n)$上的可实现性的代数判据，它诱导给定作用在其上同调上.然后，我们得到了这些群的分类以及对$Σ(2n)$的上同调的作用。
英文摘要：
Let $G$ be a group acting freely, properly discontinuously and cellularly on a finite dimensional $C$W-complex $Σ(2n)$ which has the homotopy type of the $2n$- sphere $\mathbb{S}^{2n}$. Then, this action induces an action of the group $G$ on the top cohomology of $Σ(2n)$. For the family of virtually cyclic groups, we classify all groups which act on $Σ(2n)$, the homotopy type of all possible orbit spaces and all actions on the top cohomology as well. \par Under the hypothesis that $\mbox{dim}\,Σ(2n)\leq 2n+1$, we study the groups with the virtual cohomological dimension $\mbox{vcd}\,G<\infty$ which act as above on $Σ(2n)$. It turns out that they consist of free groups and certain semi-direct products $F\rtimes \mathbb{Z}_2$ with $F$ a free group. For those groups $G$ and a given action of $G$ on $\mbox{Aut}(\mathbb{Z})$, we present an algebraic criterion equivalent to the realizability of an action $G$ on $Σ(2n)$ which induces the given action on its top cohomology. Then, we obtain a classification of those groups together with actions on the top cohomology of $Σ(2n)$.
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