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 [math]\displaystyle{ M }[/math] be a closed, oriented manifold of dimension [math]\displaystyle{ n }[/math], and let [math]\displaystyle{ [M] \in H_n(M) }[/math] be its orientation class. Here [math]\displaystyle{ H_n(M) }[/math] denotes the integral, [math]\displaystyle{ n }[/math]-dimensional homology group of [math]\displaystyle{ M }[/math]. Any continuous map [math]\displaystyle{ f\colon M\to X }[/math] defines an induced homomorphism [math]\displaystyle{ f_*\colon H_n(M)\to H_n(X) }[/math].[2] A homology class of [math]\displaystyle{ H_n(X) }[/math] is called realisable if it is of the form [math]\displaystyle{ f_*[M] }[/math] where [math]\displaystyle{ [M] \in H_n(M) }[/math]. The Steenrod problem is concerned with describing the realisable homology classes of [math]\displaystyle{ H_n(X) }[/math].[3]
Results
All elements of [math]\displaystyle{ H_k(X) }[/math] are realisable by smooth manifolds provided [math]\displaystyle{ k\le 6 }[/math]. 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 [math]\displaystyle{ H_n(X,\Z_2) }[/math], where [math]\displaystyle{ \Z_2 }[/math] denotes the integers modulo 2, can be realized by a non-oriented manifold, [math]\displaystyle{ f\colon M^n\to X }[/math].[3]
Conclusions
For smooth manifolds M the problem reduces to finding the form of the homomorphism [math]\displaystyle{ \Omega_n(X) \to H_n(X) }[/math], where [math]\displaystyle{ \Omega_n(X) }[/math] is the oriented bordism group of X.[4] The connection between the bordism groups [math]\displaystyle{ \Omega_* }[/math] and the Thom spaces MSO(k) clarified the Steenrod problem by reducing it to the study of the homomorphisms [math]\displaystyle{ H_*(\operatorname{MSO}(k)) \to H_*(X) }[/math].[3][5] In his landmark paper from 1954,[5] René Thom produced an example of a non-realisable class, [math]\displaystyle{ [M] \in H_7(X) }[/math], where M is the Eilenberg–MacLane space [math]\displaystyle{ K(\Z_3\oplus \Z_3,1) }[/math].
See also
- Singular homology
- Pontryagin-Thom construction
- Cobordism
References
- ↑ Eilenberg, Samuel (1949). "On the problems of topology". Annals of Mathematics 50 (2): 247–260. doi:10.2307/1969448.
- ↑ Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0
- ↑ 3.0 3.1 3.2 3.3 Encyclopedia of Mathematics. "Steenrod Problem". https://encyclopediaofmath.org/wiki/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.
- ↑ 5.0 5.1 Thom, René (1954). "Quelques propriétés globales des variétés differentiable" (in French). Commentarii Mathematici Helvetici 28: 17–86. doi:10.1007/bf02566923.
External links
- Thom construction and the Steenrod problem on MathOverflow
- Explanation for the Pontryagin-Thom construction
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