Filling Riemannian manifolds

n is said to be 1 -essential (or, for brevity, essential) if for some map into an aspherical space, /: V - K, the induced top dimensional homomorphism on homology does not vanish, i.e., /JF] 0. Here V stands for the integral fundamental class, i.e., V e H n (V\ Z) Z, in case the manifold Fis oriented.