A topological splitting theorem for Poincaré duality groups and high dimensional manifolds
A topological splitting theorem for Poincaré duality groups and high dimensional manifolds
Waldhausen's celebrated torus theorem plays a central role in the classification of topological 3-manifolds. It also led to a number of algebraic splitting theorems for discrete groups including Kropholler's algebraic torus theorem for Poincaré duality groups and to the algebraic annulus theorems of Dunwoody/Sageev and Scott/Swarup. Here, in the same spirit, we offer topological and algebraic decomposition theorems in the context of high dimensional aspherical manifolds, providing an algebraic splitting theorem for Poincar\'e duality groups and exploiting Cappell's splitting theory to extract the required topological splittings. As a result we show that for a wide class of manifold pairs $N,M$ with $\dim(M)=\dim(N)+1$, every, $\pi_1$-injective map f$N\rightarrow M$ factorises up to homotopy as a finite cover of an embedding. As an application of this we show that under certain circumstances the vanishing of the first Betti number for $M$ is an obstruction to the existence of such maps.
Torus theorem, Poincaré duality group, Bass-Serre theory, Kazhdan's property (T), Borel conjecture, surgery, Cappell's splitting theorem, embeddings, rigidity, geometric group theory, quaternionic hyperbolic manifolds
2203-2221
Niblo, Graham
43fe9561-c483-4cdf-bee5-0de388b78944
Kar, Aditi
af416da5-c2ae-4f05-b692-758dc4a9bf69
26 July 2013
Niblo, Graham
43fe9561-c483-4cdf-bee5-0de388b78944
Kar, Aditi
af416da5-c2ae-4f05-b692-758dc4a9bf69
Niblo, Graham and Kar, Aditi
(2013)
A topological splitting theorem for Poincaré duality groups and high dimensional manifolds.
Geometry & Topology, 17, .
(doi:10.2140/gt.2013.17.2203).
Abstract
Waldhausen's celebrated torus theorem plays a central role in the classification of topological 3-manifolds. It also led to a number of algebraic splitting theorems for discrete groups including Kropholler's algebraic torus theorem for Poincaré duality groups and to the algebraic annulus theorems of Dunwoody/Sageev and Scott/Swarup. Here, in the same spirit, we offer topological and algebraic decomposition theorems in the context of high dimensional aspherical manifolds, providing an algebraic splitting theorem for Poincar\'e duality groups and exploiting Cappell's splitting theory to extract the required topological splittings. As a result we show that for a wide class of manifold pairs $N,M$ with $\dim(M)=\dim(N)+1$, every, $\pi_1$-injective map f$N\rightarrow M$ factorises up to homotopy as a finite cover of an embedding. As an application of this we show that under certain circumstances the vanishing of the first Betti number for $M$ is an obstruction to the existence of such maps.
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1110.2041v3.pdf
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gt-v17-n4-p06-s
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1110.2041v3.pdf
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SuperRigidity11102011(ArXiv_updated_version).pdf
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Submitted date: 28 July 2010
Accepted/In Press date: 26 April 2013
Published date: 26 July 2013
Keywords:
Torus theorem, Poincaré duality group, Bass-Serre theory, Kazhdan's property (T), Borel conjecture, surgery, Cappell's splitting theorem, embeddings, rigidity, geometric group theory, quaternionic hyperbolic manifolds
Organisations:
Pure Mathematics
Identifiers
Local EPrints ID: 161381
URI: http://eprints.soton.ac.uk/id/eprint/161381
ISSN: 1465-3060
PURE UUID: a971c878-b841-4f85-9d94-624680eaef9a
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Date deposited: 28 Jul 2010 18:59
Last modified: 14 Mar 2024 02:36
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Author:
Aditi Kar
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