# A topological splitting theorem for Poincar\'e duality groups and high dimensional manifolds

Niblo, Graham and Kar, Aditi (2013) A topological splitting theorem for Poincar\'e duality groups and high dimensional manifolds. Geometry & Topology, 17, 2203-2221.

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Original Publication URL: http://arxiv.org/abs/1110.2041

## Description/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 Poincar\'e 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.

Item Type: Article 1465-3060 (print)1364-0380 (electronic) Topological superrigidty http://arxiv.org/abs/1110.2041 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 Q Science > QA Mathematics University Structure - Pre August 2011 > School of Mathematics > Pure MathematicsFaculty of Social and Human Sciences > Mathematical Sciences > Pure Mathematics 161381 28 Jul 2010 18:59 27 Mar 2014 19:16 Analysis and geometry of metric spaces with applications in geometric group theory and topology. Funded by: EPSRC (EP/F031947/1) Led by: Jacek Brodzki 1 October 2008 to 31 January 2012 http://eprints.soton.ac.uk/id/eprint/161381