Steenrod problem explained

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]

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)} .^[2]

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.^[2]

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 .^[2]

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.^[3] 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)} .^{[2] [4]} In his landmark paper from 1954,^[4] 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 ⊕} \Z _{3,1)} .

See also

• Singular homology
• Pontryagin-Thom construction
• Cobordism

External links

• Thom construction and the Steenrod problem on MathOverflow
• Explanation for the Pontryagin-Thom construction

Notes and References

1. Eilenberg. Samuel. Samuel Eilenberg. 1949. On the problems of topology. Annals of Mathematics. 50. 2 . 247–260. 10.2307/1969448. 1969448 .
2. Web site: Steenrod Problem. Encyclopedia of Mathematics. October 29, 2020.
3. Rudyak. Yuli B.. 1987. Realization of homology classes of PL-manifolds with singularities. Mathematical Notes. 41. 5. 417–421. 10.1007/bf01159869. 122228542 .
4. Thom. René . René Thom. 1954. Quelques propriétés globales des variétés differentiable. Commentarii Mathematici Helvetici. 28. 17–86. French. 10.1007/bf02566923. 120243638 .