Steenrod problem

In mathematics, and particularly homology theory, Steenrod's Problem is a problem concerning the realisation of homology classes by singular manifolds.

Formulation

Let be a closed, oriented manifold of dimension, 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 for some manifold and map. The Steenrod problem is concerned with describing the realisable homology classes of.

All elements of are realisable by smooth manifolds provided. 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, where denotes the integers modulo 2, can be realized by a non-oriented manifold,.

For smooth manifolds M the problem reduces to finding the form of the homomorphism, where is the oriented bordism group of X. The connection between the bordism groups and the Thom spaces MSO clarified the Steenrod problem by reducing it to the study of the homomorphisms. In his landmark paper from 1954, René Thom produced an example of a non-realisable class,, where M is the Eilenberg–MacLane space.