Abstract
Let f :M→N be a continuous map between two closed n-manifolds such that f ∗ :H ∗(M, Z 2)→H ∗(N, Z 2) is an isomorphism. Suppose that M immerses in R n+k for 5⩽ n<2 k. Then N also immerses in R n+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.
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