论文标题
对成对和应用的粗糙亚历山大二元性
Coarse Alexander duality for pairs and applications
论文作者
论文摘要
对于一个$ g $（类型$ f $）的组，在粗糙的Poincaré二元空间上正确起作用$ x $，Kapovich-Kleiner在$ g $和$ x $中的补充之间引入了Alexander二元的粗糙版本。更确切地说，$ g $与集体环系数的共同体对于$ x $ $ g $ - 摩尔比特的越来越多的社区的某个家族的某个家族的同源群是双重的。这种双重性更普遍地适用于某些合同的简单复合物的粗嵌入到粗$ pd（n）$空间中。在本文中，我们介绍了此čech同源性的相对版本，该版本满足了Eilenberg-Steenrod精确性公理，我们证明了粗糙的Alexander Duality的相对版本。 作为应用程序，我们提供了以下结果的详细证明，首先由Kapovich-Kleiner表示。鉴于$ 2 $ - 复杂的沿边界线粘合$ k $的半平面形成，将粗嵌入到合同的$ 3 $ manifold中，补充由$ k $的深层组成组成，这些组件以称为乔丹周期的模式周期性地安排。我们将约旦周期用作不变的，以证明存在一个$ 3 $的manifold群体，该组实际上是克莱尼人，但本身并非克莱琳。
For a group $G$ (of type $F$) acting properly on a coarse Poincaré duality space $X$, Kapovich-Kleiner introduced a coarse version of Alexander duality between $G$ and its complement in $X$. More precisely, the cohomology of $G$ with group ring coefficients is dual to a certain Čech homology group of the family of increasing neighborhoods of a $G$-orbit in $X$. This duality applies more generally to coarse embeddings of certain contractible simplicial complexes into coarse $PD(n)$ spaces. In this paper we introduce a relative version of this Čech homology that satisfies the Eilenberg-Steenrod Exactness Axiom, and we prove a relative version of coarse Alexander duality. As an application we provide a detailed proof of the following result, first stated by Kapovich-Kleiner. Given a $2$-complex formed by gluing $k$ halfplanes along their boundary lines and a coarse embedding into a contractible $3$-manifold, the complement consists of $k$ deep components that are arranged cyclically in a pattern called a Jordan cycle. We use the Jordan cycle as an invariant in proving the existence of a $3$-manifold group that is virtually Kleinian but not itself Kleinian.
