Realisation of Cycles by Aspherical Manifolds

In the late 1940s N. Steenrod posed the following problem, which is now familiar as the problem on realisation of cycles. For a given homology class z ∈ Hn(X;Z), do there exist an oriented manifold N and a mapping f : N → X such that f∗[N ] = z? A famous theorem of R. Thom claims that each integral homology class is realisable in sense of Steenrod with some multiplicity. A classical problem is the problem of realisation of cycles by images of spheres, that is, the problem of the description for the image of the Hurewicz homomorphism. In this case not every homology class can be realised with multiplicity. It is interesting to find a class Mn of smooth n-dimensional manifolds sufficient for realisation with multiplicities of all integral n-dimensional homology classes of every space X. The following theorem is the main result of this paper.