  We show that in every codimension greater than one there exists a mod 2
homology class in some closed manifold (of sufficiently high dimension) which
cannot be realized by an immersion of closed manifolds. The proof gives
explicit obstructions (in terms of cohomology operations) for realizability of
mod 2 homology classes by immersions. We also prove the corresponding result in
which the word `immersion' is replaced by `map with some restricted set of
multi-singularities'.
