Volume 17, issue 4 (2013)

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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(M) = dim(N) + 1, every π1–injective map f : N 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 π1–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
References
Publication
Received: 10 October 2011
Revised: 19 October 2011
Accepted: 25 April 2013
Published: 26 July 2013
Proposed: Martin R Bridson
Seconded: Walter Neumann, Benson Farb
Authors
Aditi Kar
Mathematical Institute
University of Oxford
24–29 St Giles’
Oxford
OX1 3LB
UK
http://people.maths.ox.ac.uk/kar/
Graham A Niblo
Mathematical Sciences
University of Southampton
Highfield
Southampton
SO17 1BJ
UK
http://www.personal.soton.ac.uk/gan/