Geometric representatives of homology classes of manifolds

Is it true that for even dimensional differentiable manifold $M^{2n}$ all singular homology classes in dimension less than $n$ can be represented by a submanifold?

• Look at the two answers to this duplicate question:mathoverflow.net/questions/21171/… – Achim Krause Mar 12 '15 at 0:02
• In particular, there are obstructions for an integral homology class to be represented by a manifold map altogether (you can get some odd multiple in general), but as soon as that part works you can turn such a map into an embedding in the dimensions you're asking for. – Achim Krause Mar 12 '15 at 0:04