Immersions in the metastable dimension range via the normal bordism approach

Let f :M→N be a continuous map between two closed n-manifolds such that f∗ :H∗(M,Z2)→H∗(N,Z2) is an isomorphism. Suppose that M immerses in Rn+k for 5⩽n<2k. Then N also immerses in Rn+k. We use techniques of normal bordism theory to prove this result and we show that for a large family of spaces we can replace the homology condition by the corresponding one in homotopy.