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 be a closed, oriented manifold of dimension , and let be its orientation class. Here denotes the integral, -dimensional homology group of . Any continuous map defines an induced homomorphism .{{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 is called realisable if it is of the form where . The Steenrod problem is concerned with describing the realisable homology classes of .{{Cite web|url=https://encyclopediaofmath.org/wiki/Steenrod_problem|author=Encyclopedia of Mathematics|title=Steenrod Problem|accessdate=October 29, 2020}}
Results
All elements of are realisable by smooth manifolds provided . 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 , where denotes the integers modulo 2, can be realized by a non-oriented manifold, .
Conclusions
For smooth manifolds M the problem reduces to finding the form of the homomorphism , where 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 and the Thom spaces MSO(k) clarified the Steenrod problem by reducing it to the study of the homomorphisms .{{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, , where M is the Eilenberg–MacLane space .
See also
References
{{Reflist}}
External links
- [https://mathoverflow.net/q/48814 Thom construction and the Steenrod problem] on MathOverflow
- [https://mathoverflow.net/q/32828 Explanation for the Pontryagin-Thom construction]