Steenrod problem

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.

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

Results
All elements of $$H_k(X)$$ are realisable by smooth manifolds provided $$k\le 6$$. Moreover, any cycle can be realized 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 $$H_n(X,\Z_2)$$, where $$\Z_2$$ denotes the integers modulo 2, can be realized by a non-oriented manifold, $$f\colon M^n\to X$$.

Conclusions
For smooth manifolds M the problem reduces to finding the form of the homomorphism $$\Omega_n(X) \to H_n(X)$$, where $$\Omega_n(X)$$ is the oriented bordism group of X. The connection between the bordism groups $$\Omega_*$$ and the Thom spaces MSO(k) clarified the Steenrod problem by reducing it to the study of the homomorphisms $$H_*(\operatorname{MSO}(k)) \to H_*(X)$$. In his landmark paper from 1954, René Thom produced an example of a non-realisable class, $$[M] \in H_7(X)$$, where M is the Eilenberg–MacLane space $$K(\Z_3\oplus \Z_3,1)$$.