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 formula_1 be a closed, oriented manifold of dimension formula_2, and let formula_3 be its orientation class. Here formula_4 denotes the integral, formula_2-dimensional homology group of formula_1. Any continuous map formula_7 defines an induced homomorphism formula_8. A homology class of formula_9 is called realisable if it is of the form formula_10 where formula_3. The Steenrod problem is concerned with describing the realisable homology classes of formula_9.
Results.
All elements of formula_13 are realisable by smooth manifolds provided formula_14. Any elements of formula_9 are realisable by a mapping of a Poincaré complex provided formula_16. 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 formula_17, where formula_18 denotes the integers modulo 2, can be realized by a non-oriented manifold, formula_19.
Conclusions.
For smooth manifolds "M" the problem reduces to finding the form of the homomorphism formula_20, where formula_21 is the oriented bordism group of "X". The connection between the bordism groups formula_22 and the Thom spaces MSO("k") clarified the Steenrod problem by reducing it to the study of the homomorphisms formula_23. In his landmark paper from 1954, René Thom produced an example of a non-realisable class, formula_24, where "M" is the Eilenberg–MacLane space formula_25.