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