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]

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 $f_{*}\colon H_{n}(M)\to H_{n}(X)$ .^[2] 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 $H_{n}(X)$ .^[3]

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.^[3]

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, $f\colon M^{n}\to X$ .^[3]

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.^[4] 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 $H_{*}(\operatorname {MSO} (k))\to H_{*}(X)$ .^[3]^[5] In his landmark paper from 1954,^[5] René Thom produced an example of a non-realisable class, $[M]\in H_{7}(X)$ , where M is the Eilenberg–MacLane space $K(\mathbb {Z} _{3}\oplus \mathbb {Z} _{3},1)$ .

References

^ Eilenberg, Samuel (1949). "On the problems of topology". Annals of Mathematics. 50 (2): 247–260. doi:10.2307/1969448. JSTOR 1969448.
^ Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0
^ ^a ^b ^c ^d Encyclopedia of Mathematics. "Steenrod Problem". Retrieved October 29, 2020.
^ Rudyak, Yuli B. (1987). "Realization of homology classes of PL-manifolds with singularities". Mathematical Notes. 41 (5): 417–421. doi:10.1007/bf01159869. S2CID 122228542.
^ ^a ^b Thom, René (1954). "Quelques propriétés globales des variétés differentiable". Commentarii Mathematici Helvetici (in French). 28: 17–86. doi:10.1007/bf02566923. S2CID 120243638.