# Steenrod problem

The problem of the realization of cycles (homology classes) by singular manifolds; formulated by N. Steenrod, cf. [1]. Let $ M $
be a closed oriented manifold (topological, piecewise-linear, smooth, etc.) and let $ [ M] \in H _ {n} ( M) $
be its orientation (here $ H _ {n} ( M) $
is the $ n $-
dimensional homology group of $ M $).
Any continuous mapping $ f: M\rightarrow X $
defines an element $ f _ \star [ M] \in H _ {n} ( X) $.
The Steenrod problem consists of describing those homology classes of $ X $,
called realizable, which are obtained in this way, i.e. which take the form $ f _ \star [ M] $
for a certain $ M $
from the given class. All elements of the groups $ H _ {i} ( X) $,
$ i \leq 6 $,
are realizable by a smooth manifold. Any element of the group $ H _ {n} ( X) $,
$ n \neq 3 $,
is realizable by a mapping of a Poincaré complex $ P $.
Moreover, any cycle can be realized by a pseudo-manifold. Non-orientable manifolds can also be considered, and every homology class modulo $ 2 $(
i.e. element of $ H _ {n} ( X , \mathbf Z / 2 ) $)
can be realized by a non-oriented smooth singular manifold $ f : M ^ {n} \rightarrow X $.

Thus, for smooth $ M $ the Steenrod problem consists of describing the form of the homomorphism $ \Omega _ {n} ( X) \rightarrow H _ {n} ( X) $, where $ \Omega _ {n} ( X) $ is the oriented bordism group of the space. The connection between the bordisms $ \Omega _ \star $ and the Thom spaces (cf. Thom space) $ \mathop{\rm MSO} ( k) $, discovered by R. Thom [2], clarified the Steenrod problem by reducing it to the study of the mappings $ H ^ \star ( \mathop{\rm MSO} ( k)) \rightarrow H ^ \star ( X) $. A non-realizable class $ x \in H _ {7} ( X) $ has been exhibited, where $ X $ is the Eilenberg–MacLane space $ K( \mathbf Z _ {3} \oplus \mathbf Z _ {3} , 1) $. For any class $ x $, some multiple $ nx $ is realizable (by a smooth manifold); moreover, $ n $ can be chosen odd.

#### References

[1] | S. Eilenberg, "On the problems of topology" Ann. of Math. , 50 (1949) pp. 247–260 MR0030189 Zbl 0034.25304 |

[2] | R. Thom, "Quelques propriétés globales des variétés differentiables" Comm. Math. Helv. , 28 (1954) pp. 17–86 MR0061823 Zbl 0057.15502 |

[3] | P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) MR0176478 Zbl 0125.40103 |

[4] | R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) MR0248858 Zbl 0181.26604 |

[5] | Yu.B. Rudyak, "Realization of homology classes of PL-manifolds with singularities" Math. Notes , 41 : 5 (1987) pp. 417–421 Mat. Zametki , 41 : 5 (1987) pp. 741–749 MR898135 Zbl 0632.57020 |

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Steenrod problem.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Steenrod_problem&oldid=48825