粗糙几何中下降的一般概念
The General Notion of Descent in Coarse Geometry
摘要：
在这篇文章中，我们引入了粗空间上的函子是粗切除的概念--一个粗类似于拓扑空间上的函子是切除的概念。此外，取锥，一个粗切除函子产生一个拓扑切除函子，并且对于粗拓扑空间，存在一个从拓扑切除函子到粗切除函子的相关粗组装映射。我们猜想这个粗装配映射是具有有界几何的一致可缩空间的同构，并证明了粗同构猜想，以及一些温和的技术条件，意味着相应的等变装配映射是单射的.这种等变集合映射的特殊例子是法雷尔-琼斯猜想和鲍姆-康纳斯猜想中的映射。
英文摘要：
In this article, we introduce the notion of a functor on coarse spaces being coarsely excisive- a coarse analogue of the notion of a functor on topological spaces being excisive. Further, taking cones, a coarsely excisive functor yields a topologically excisive functor, and for coarse topological spaces there is an associated coarse assembly map from the topologically exicisive functor to the coarsely excisive functor. We conjecture that this coarse assembly map is an isomorphism for uniformly contractible spaces with bounded geometry, and show that the coarse isomorphism conjecture, along with some mild technical conditions, implies that a correspoding equivariant assembly map is injective. Particular instances of this equivariant assembly map are the maps in the Farrell-Jones conjecture, and in the Baum-Connes conjecture.
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