A topological splitting theorem for Poincaré duality groups and high-dimensional manifolds

Kar, Aditi¹ ; Niblo, Graham²

¹ Mathematical Institute, University of Oxford, 24–29 St Giles’, Oxford, OX1 3LB, UK
² Mathematical Sciences, University of Southampton, Highfield, Southampton, SO17 1BJ, UK

Geometry & topology, Tome 17 (2013) no. 4, pp. 2203-2221.

Résumé

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 $f:N\to M$ 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.

Détail

DOI : 10.2140/gt.2013.17.2203

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

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