Coarse Alexander duality for pairs and applications
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2020
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909741709524992 |
|---|---|
| author | Hruska, G. Christopher Stark, Emily Tran, Hung Cong |
| author_facet | Hruska, G. Christopher Stark, Emily Tran, Hung Cong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_00059 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Coarse Alexander duality for pairs and applications Hruska, G. Christopher Stark, Emily Tran, Hung Cong Geometric Topology Algebraic Topology Group Theory 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. |
| title | Coarse Alexander duality for pairs and applications |
| topic | Geometric Topology Algebraic Topology Group Theory |
| url | https://arxiv.org/abs/2011.00059 |