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Coarse Alexander
duality for pairs and applications
G Christopher Hruska, Emily Stark and Hùng Công Trần |
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Algebraic & Geometric Topology 25 (2025) 1999–2035
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Abstract |
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For a group (of type ) acting properly on a coarse Poincaré duality space , Kapovich and Kleiner introduced a coarse version of Alexander duality between and its complement in . More precisely, the cohomology of with group ring coefficients is dual to a certain Čech homology group of the family of increasing neighborhoods of a -orbit in . This duality applies more generally to coarse embeddings of certain contractible simplicial complexes into coarse 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 and Kleiner. Given a -complex formed by gluing halfplanes along their boundary lines and a coarse embedding into a contractible -manifold, the complement consists of 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 -manifold group that is virtually Kleinian but not itself Kleinian. |
Keywords
Alexander duality, coarse $\mathrm{PD}(n)$ space, Kleinian
group
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Mathematical Subject Classification
Primary: 20F65, 55M05, 55N05
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References |
Publication
Received: 5 May 2022
Revised: 26 January 2024
Accepted: 21 April 2024
Published: 11 August 2025
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Authors |
| G Christopher Hruska | |
| Department of Mathematical
Sciences University of Wisconsin–Milwaukee Milwaukee, WI United States |
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| Emily Stark | |
| Department of Mathematics Wesleyan University Middletown, CT United States |
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| Hùng Công Trần | |
| FCCI Insurance Group Sarasota, FL United States |
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| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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