Let be a closed, oriented manifold, and let be its orientation class. Here denotes the integral, -dimensional homology group of . Any continuous map defines an induced homomorphism . A homology class of is called realisable if it is of the form where . The Steenrod problem is concerned with describing the realisable homology classes of .
All elements of are realisable by smooth manifolds provided . Any elements of are realisable by a mapping of a Poincaré complex provided n ≠ 3. Moreover, any cycle can be realisable by the mapping of a pseudo-manifold.
The assumption that M be orientable can be relaxed. In the case of non-orientable manifolds, every homology class of Hn(X,Z2), where Z2 denotes the integers modulo 2, can be realised by a non-oriented manifold ƒ : Mn → X.
For smooth manifolds M the problem reduces to finding the form of the homomorphism Ωn(X) → Hn(X), where Ωn(X) is the oriented bordism group of X. The connection between the bordisms Ω* and the Thom spaces MSO(k) clarified the Steenrod problem by reducing it to the study of the mappings H*(MSO(k)) → H*(X). A non-realisable class, [M] ∈ H7(X), has been found where M is the Eilenberg–MacLane space: K(Z3⊕Z3,1).
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