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.
[1] Let
{\displaystyle M}
be a closed, oriented manifold of dimension
n
be its orientation class.
denotes the integral,
-dimensional homology group of
Any continuous map
defines an induced homomorphism
[2] A homology class of
is called realisable if it is of the form
The Steenrod problem is concerned with describing the realisable homology classes of
[3] All elements of
are realisable by smooth manifolds provided
Moreover, any cycle can be realized 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
denotes the integers modulo 2, can be realized by a non-oriented manifold,
[3] For smooth manifolds M the problem reduces to finding the form of the homomorphism
is the oriented bordism group of X.
[4] The connection between the bordism groups
and the Thom spaces MSO(k) clarified the Steenrod problem by reducing it to the study of the homomorphisms
{\displaystyle H_{*}(\operatorname {MSO} (k))\to H_{*}(X)}
[3][5] In his landmark paper from 1954,[5] René Thom produced an example of a non-realisable class,
, where M is the Eilenberg–MacLane space