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.¹

Let ${\displaystyle M}$ be a closed, oriented manifold of dimension ${\displaystyle n}$, and let ${\displaystyle [M]\in H_{n}(M)}$ be its orientation class. Here ${\displaystyle H_{n}(M)}$ denotes the integral, ${\displaystyle n}$-dimensional homology group of ${\displaystyle M}$. Any continuous map ${\displaystyle f\colon M\to X}$ defines an induced homomorphism ${\displaystyle f_{*}\colon H_{n}(M)\to H_{n}(X)}$.² A homology class of ${\displaystyle H_{n}(X)}$ is called realisable if it is of the form ${\displaystyle f_{*}[M]}$ for some manifold ${\displaystyle M}$ and map ${\displaystyle f:M\to X}$. The Steenrod problem is concerned with describing the realisable homology classes of ${\displaystyle H_{n}(X)}$.³

All elements of ${\displaystyle H_{k}(X)}$ are realisable by smooth manifolds provided ${\displaystyle k\leq 6}$. 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 ${\displaystyle H_{n}(X,\mathbb {Z} _{2})}$, where ${\displaystyle \mathbb {Z} _{2}}$ denotes the integers modulo 2, can be realized by a non-oriented manifold, ${\displaystyle f\colon M^{n}\to X}$.³

For smooth manifolds M the problem reduces to finding the form of the homomorphism ${\displaystyle \Omega _{n}(X)\to H_{n}(X)}$, where ${\displaystyle \Omega _{n}(X)}$ is the oriented bordism group of X.⁴ The connection between the bordism groups ${\displaystyle \Omega _{*}}$ and the Thom spaces MSO(k) clarified the Steenrod problem by reducing it to the study of the homomorphisms ${\displaystyle H_{*}(\operatorname {MSO} (k))\to H_{*}(X)}$.³⁵ In his landmark paper from 1954,⁵ René Thom produced an example of a non-realisable class, ${\displaystyle [M]\in H_{7}(X)}$, where M is the Eilenberg–MacLane space ${\displaystyle K(\mathbb {Z} _{3}\oplus \mathbb {Z} _{3},1)}$.