论文标题
非操作行动和装配图的拓扑K-词素学
Topological K-homology for non-proper actions and assembly maps
论文作者
论文摘要
对于一个可数的离散$ g $，我们为$ g $的Equivariant拓扑$ k $ - 知识理论构建了一种新的和具体的模型，该理论定义为所有$ g $ actions，而不仅仅是适用于$ g $ actions。我们的模型的构建结合了一些想法，从定位代数方法到粗糙的鲍姆 - 康涅狄格州的猜想和受控代数方法到法雷尔·琼斯的猜想。它为鲍姆 - 康涅狄格州的猜想带来了新的想法。首先，由于为所有$ g $ actions定义了该模型，因此我们能够定义相对组装图。通过使用理论的诱导结构，我们证明了一个可用于验证相对组装图是同构的传递原理。这促进了相对于一个组的有限亚组家族的原始组装图相对于组的任何一个子组家族的研究。其次，我们的新模型适合使用受控拓扑和受控代数的方法，该方法在证明粗大的鲍姆 - 康涅狄格州的猜想和Farrell-Jones的猜想中起着重要作用。
For a countable discrete group $G$, we construct a new and concrete model for the equivariant topological $K$-homology theory of $G$, which is defined for all $G$-actions, not just for proper $G$-actions. The construction of our model combines some of the ideas from the localization algebra approach to the coarse Baum-Connes conjecture and the controlled algebra approach to the Farrell-Jones conjecture. It brings new ideas into the study of the Baum-Connes conjecture. First, as the model is defined for all $G$-actions, we are able to define relative assembly maps. We prove, by using an induction structure of our theory, a transitivity principle that can be used to verify when a relative assembly map is an isomorphism. This promotes the study of the original assembly map relative to the family of finite subgroups to assembly maps relative to any family of subgroups of a group. Second, our new model is suitable for the use of the method of controlled topology and controlled algebra, which plays an important role in proving the coarse Baum-Connes conjecture and the Farrell-Jones conjecture for a large class of groups.
