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 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 . [2] A homology class of is called realisable if it is of the form where . 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 , where 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 , where 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 . [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 .
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 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 . [2] A homology class of is called realisable if it is of the form where . 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 , where 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 , where 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 . [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 .