Niblo, Graham and Kar, Aditi (2013) Topological superrigidty. Geometry & Topology (arXiv:1110.2041v2). (In Press).
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.
|Alternative titles:||A topological splitting theorem for Poincar\'e duality groups and high dimensional manifolds|
|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|
|Subjects:||Q Science > QA Mathematics|
|Divisions:||University Structure - Pre August 2011 > School of Mathematics > Pure Mathematics
Faculty of Social and Human Sciences > Mathematics > Pure Mathematics
|Date Deposited:||28 Jul 2010 18:59|
|Last Modified:||30 Apr 2013 11:20|
|Contributors:||Niblo, Graham (Author)
Kar, Aditi (Author)
|Date:||26 April 2013|
|RDF:||RDF+N-Triples, RDF+N3, RDF+XML, Browse.|
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