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

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 $f:N\to M$ 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.