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 M be a closed, oriented manifold, and let [M] ∈ Hn(M) be its orientation. Here Hn(M) denotes the n-dimensional homology group of M. Any continuous map ƒ : M → X defines an induced homomorphism ƒ* : Hn(M) → Hn(X).[2] A homology class of Hn(X) is called realisable if it is of the form ƒ* [M] where [M] ∈ Hn(M). The Steenrod problem is concerned with describing the realisable homology classes of Hn(X).[3]

Results

All elements of Hk(X) are realisable by smooth manifolds provided k ≤ 6. Any elements of Hn(X) are realisable by a mapping of a Poincaré complex provided n ≠ 3. Moreover, any cycle can be realisable 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 Hn(X,Z2), where Z2 denotes the integers modulo 2, can be realised by a non-oriented manifold ƒ : Mn → X.[3]

Conclusions

For smooth manifolds M the problem reduces to finding the form of the homomorphism Ωn(X) → Hn(X), where Ωn(X) is the oriented bordism group of X.[4] The connection between the bordisms Ω* and the Thom spaces MSO(k) clarified the Steenrod problem by reducing it to the study of the mappings H*(MSO(k)) → H*(X).[3][5] A non-realisable class, [M] ∈ H7(X), has been found where M is the Eilenberg–MacLane space: K(Z3⊕Z3,1).

See also

References

  1. ↑ Eilenberg, S. (1949). "On the problems of topology". Ann. of Math. 50: 247 − 260. doi:10.2307/1969448.
  2. ↑ Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0
  3. 1 2 3 4 Yu. B. Rudyak. "Steenrod Problem". Retrieved August 6, 2010.
  4. ↑ Rudyak, Yu. B. (1987). "Realization of homology classes of PL-manifolds with singularities". Math. Notes 41 (5): 417 − 421. doi:10.1007/bf01159869.
  5. ↑ Thom, R. (1954). "Quelques propriétés globales des variétés differentiable". Comm. Math. Helv. (in French) 28: 17 − 86. doi:10.1007/bf02566923.
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