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 $M$ be a closed, oriented manifold of dimension $n$, and let \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_* /math> where \in H_n(M). The Steenrod problem is concerned with describing the realisable homology classes of $H_n(X)$.

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.

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, $f\colon M^n\to X$.

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''. 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 $H_*(\operatorname(k)) \to H_*(X)$. In his landmark paper from 1954, René Thom produced an example of a non-realisable class, \in H_7(X), where ''M'' is the Eilenberg–MacLane space $K(\Z_3\oplus \Z_3,1)$.

See also

* Singular homology
* Pontryagin-Thom construction
* Cobordism

References

{{Reflist

External links

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

Homology theory
Manifolds
Geometric topology