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