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.{{Cite journal|last=Eilenberg|first=Samuel|authorlink=Samuel Eilenberg|year=1949|title=On the problems of topology|journal=Annals of Mathematics|volume=50|issue=2 | pages=247–260|doi=10.2307/1969448|jstor=1969448 }}

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).{{Citation|first=Allen|last=Hatcher| authorlink=Allen Hatcher|title=Algebraic Topology|publisher=Cambridge University Press|year=2001|isbn=0-521-79540-0}} 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).{{Cite web|url=https://encyclopediaofmath.org/wiki/Steenrod_problem|author=Encyclopedia of Mathematics|title=Steenrod Problem|accessdate=October 29, 2020}}

Results

All elements of H_k(X) are realisable by smooth manifolds provided k\le 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\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.{{Cite journal|last=Rudyak|first=Yuli B.|year=1987|title=Realization of homology classes of PL-manifolds with singularities|journal=Mathematical Notes|volume=41|issue=5|pages=417–421|doi=10.1007/bf01159869|s2cid=122228542 }} 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).{{Cite journal|last=Thom|first=René |authorlink=René Thom|year=1954|title=Quelques propriétés globales des variétés differentiable|journal=Commentarii Mathematici Helvetici|volume=28|pages=17–86|language=French|doi=10.1007/bf02566923|s2cid=120243638 }} 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).

See also

References

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