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 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 .[2] 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 .[3]

Results

All elements of are realisable by smooth manifolds provided . Any elements of are realisable by a mapping of a Poincaré complex provided . 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 , where denotes the integers modulo 2, can be realized by a non-oriented manifold, .[3]

Conclusions

For smooth manifolds M the problem reduces to finding the form of the homomorphism , where is the oriented bordism group of X.[4] 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 .[3][5] In his landmark paper from 1954,[5] René Thom produced an example of a non-realisable class, , where M is the Eilenberg–MacLane space .

See also

References

  1. Eilenberg, Samuel (1949). "On the problems of topology". Annals of Mathematics. 50: 247–260. doi:10.2307/1969448.
  2. Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0
  3. 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.
  5. Thom, René (1954). "Quelques propriétés globales des variétés differentiable". Commentarii Mathematici Helvetici (in French). 28: 17–86. doi:10.1007/bf02566923.
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