In mathematics, and particularly homology theory, Steenrod's Problem (named after mathematician Norman Steenrod) is a problem concerning the realisation of homology classes by singular manifolds.
Contents
Formulation
Let M be a closed, oriented manifold, and let [M] ∈ Hn(M) be its orientation. Here Hn(M) denotes the n-dimensional homology group of M. Any continuous map ƒ : M → X defines an induced homomorphism ƒ* : Hn(M) → Hn(X). A homology class of Hn(X) is called realisable if it is of the form ƒ* [M] where [M] ∈ Hn(M). The Steenrod problem is concerned with describing the realisable homology classes of Hn(X).
Results
All elements of Hk(X) are realisable by smooth manifolds provided k ≤ 6. Any elements of Hn(X) 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.
Conclusions
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).