#### Volume 17, issue 4 (2013)

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 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
A topological splitting theorem for Poincaré duality groups and high-dimensional manifolds

### Aditi Kar and Graham A Niblo

Geometry & Topology 17 (2013) 2203–2221
##### Abstract

We show that for a wide class of manifold pairs $N,M$ with $dim\left(M\right)=dim\left(N\right)+1$, every ${\pi }_{1}$–injective map $f:\phantom{\rule{0.3em}{0ex}}N\to M$ factorises up to homotopy as a finite cover of an embedding. This result, in the spirit of Waldhausen’s torus theorem, is derived using Cappell’s surgery methods from a new algebraic splitting theorem for Poincaré duality groups. As an application we derive a new obstruction to the existence of ${\pi }_{1}$–injective maps.

##### 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
##### Mathematical Subject Classification 2000
Primary: 20F65, 57N35
Secondary: 57R67, 57Q20, 57P10