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We show that in every codimension greater than 1, there exists a mod 2 homology class in some closed manifold (of sufficiently high dimension) that 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’.

On realizing homology classes by maps of restricted complexity