Steenrod problem

Let ${\displaystyle M}$ be a closed, oriented manifold, and let ${\displaystyle [M]\in H_{n}(M)}$ be its orientation class. Here ${\displaystyle H_{n}(M)}$ denotes the integral, ${\displaystyle n}$-dimensional homology group of ${\displaystyle M}$. Any continuous map ${\displaystyle f\colon M\to X}$ defines an induced homomorphism ${\displaystyle f_{*}\colon H_{n}(M)\to H_{n}(X)}$.[2] A homology class of ${\displaystyle H_{n}(X)}$ is called realisable if it is of the form ${\displaystyle f_{*}[M]}$ where ${\displaystyle [M]\in H_{n}(M)}$. The Steenrod problem is concerned with describing the realisable homology classes of ${\displaystyle H_{n}(X)}$.[3]

All elements of ${\displaystyle H_{k}(X)}$ are realisable by smooth manifolds provided ${\displaystyle k\leq 6}$. Any elements of ${\displaystyle H_{n}(X)}$ 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.[3]

The assumption that M be orientable can be relaxed. In the case of non-orientable manifolds, every homology class of H_n(X,Z₂), where Z₂ denotes the integers modulo 2, can be realised by a non-oriented manifold ƒ : Mⁿ → X.[3]

For smooth manifolds M the problem reduces to finding the form of the homomorphism Ω_n(X) → H_n(X), where Ω_n(X) is the oriented bordism group of X.[4] 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).[3][5] A non-realisable class, [M] ∈ H₇(X), has been found where M is the Eilenberg–MacLane space: K(Z₃⊕Z₃,1).

• Singular homology
• Pontryagin-Thom construction
• Cobordism

• Thom construction and the Steenrod problem on MathOverflow
• Explanation for the Pontryagin-Thom construction