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The problem of the realization of cycles (homology classes) by singular manifolds; formulated by N. Steenrod, cf. [[#References|[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|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|Poincaré complex]]  $  P $. 
Moreover, any cycle can be realized by a [[Pseudo-manifold|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|bordism]] group of the space. The connection between the bordisms  $  \Omega _  \star  $
and the Thom spaces (cf. [[Thom space|Thom space]])  $   \mathop{\rm MSO} ( k) $, 
discovered by R. Thom [[#References|[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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Eilenberg, "On the problems of topology" ''Ann. of Math.'' , '''50''' (1949) pp. 247–260 {{MR|0030189}} {{ZBL|0034.25304}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Thom, "Quelques propriétés globales des variétés differentiables" ''Comm. Math. Helv.'' , '''28''' (1954) pp. 17–86 {{MR|0061823}} {{ZBL|0057.15502}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) {{MR|0176478}} {{ZBL|0125.40103}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) {{MR|0248858}} {{ZBL|0181.26604}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> 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 {{MR|898135}} {{ZBL|0632.57020}} </TD></TR></table>