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 $f_{*} [M]$ 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 $H_{n} (X, ℤ_{2})$ , where $ℤ_{2}$ 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 $Ω_{n} (X) \to H_{n} (X)$ , where $Ω_{n} (X)$ is the oriented bordism group of X. The connection between the bordism groups $Ω_{*}$ and the Thom spaces MSO(k) clarified the Steenrod problem by reducing it to the study of the homomorphisms $H_{*} (MSO (k)) \to H_{*} (X)$ . 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 $K (ℤ_{3} \oplus ℤ_{3}, 1)$ .

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
1 2 3 4 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.
1 2 Thom, René (1954). "Quelques propriétés globales des variétés differentiable". Commentarii Mathematici Helvetici (in French). 28: 17–86. doi:10.1007/bf02566923. S2CID 120243638.

External links

[1] Eilenberg, Samuel (1949). "On the problems of topology". Annals of Mathematics. 50 (2): 247–260. doi:10.2307/1969448. JSTOR 1969448.

[TOP-2] Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0

[YUB-3] 1 2 3 4 Encyclopedia of Mathematics. "Steenrod Problem". Retrieved October 29, 2020.

[4] 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.

[Thom-5] 1 2 Thom, René (1954). "Quelques propriétés globales des variétés differentiable". Commentarii Mathematici Helvetici (in French). 28: 17–86. doi:10.1007/bf02566923. S2CID 120243638.