  We show that for a wide class of manifold pairs N, M satisfying dim(M) =
dim(N) + 1, every \pi_1-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 Poincare duality groups. As an application we derive a
new obstruction to the existence of \pi_1-injective maps.
