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 $f\colon M\to X$ 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 $f_{*}[M]$ for some manifold $M$ and map $f:M\to X$. 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 $k\leq 6$. 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 $H_{n}(X,\mathbb {Z} _{2})$, where $\mathbb {Z} _{2}$ 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 $H_{*}(\operatorname {MSO} (k))\to H_{*}(X)$. 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 $K(\mathbb {Z} _{3}\oplus \mathbb {Z} _{3},1)$.

See also

• Singular homology
• Pontryagin-Thom construction
• Cobordism

External links

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