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 $[M]\in H_n(M)$ be its orientation class. Here $H_n(M)$ denotes the integral, $n$-dimensional homology group of $M$. Any continuous map $f:M\to X$ defines an induced homomorphism $f_{*}:H_n(M)\to H_n(X)$. 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 $H_n(X)$.

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 $H_n(X,{\mathbb{\mathbb{Z}}}_2)$, where ${\mathbb{\mathbb{Z}}}_2$ denotes the integers modulo 2, can be realized by a non-oriented manifold, $f:M^n\to X$.

Conclusions

For smooth manifolds M the problem reduces to finding the form of the homomorphism ${\mathrm{\Omega }}_n(X)\to H_n(X)$, where ${\mathrm{\Omega }}_n(X)$ is the oriented bordism group of X. The connection between the bordism groups ${\mathrm{\Omega }}_{*}$ and the Thom spaces MSO(k) clarified the Steenrod problem by reducing it to the study of the homomorphisms $H_{*}(\mathrm{MSO}(k))\to H_{*}(X)$. In his landmark paper from 1954, René Thom produced an example of a non-realisable class, $[M]\in H_7(X)$, where M is the Eilenberg–MacLane space $K({\mathbb{\mathbb{Z}}}_3\oplus {\mathbb{\mathbb{Z}}}_3,1)$.

See also

• Singular homology
• Pontryagin-Thom construction
• Cobordism

External links

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