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The problem of the realization of cycles (homology classes) by singular manifolds; formulated by N. Steenrod, cf. [[#References|[1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s0875401.png" /> be a closed oriented manifold (topological, piecewise-linear, smooth, etc.) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s0875402.png" /> be its orientation (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s0875403.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s0875404.png" />-dimensional [[Homology group|homology group]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s0875405.png" />). Any continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s0875406.png" /> defines an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s0875407.png" />. The Steenrod problem consists of describing those homology classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s0875408.png" />, called realizable, which are obtained in this way, i.e. which take the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s0875409.png" /> for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754010.png" /> from the given class. All elements of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754012.png" />, are realizable by a smooth manifold. Any element of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754014.png" />, is realizable by a mapping of a [[Poincaré complex|Poincaré complex]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754015.png" />. Moreover, any cycle can be realized by a [[Pseudo-manifold|pseudo-manifold]]. Non-orientable manifolds can also be considered, and every homology class modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754016.png" /> (i.e. element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754017.png" />) can be realized by a non-oriented smooth singular manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754018.png" />.
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Thus, for smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754019.png" /> the Steenrod problem consists of describing the form of the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754021.png" /> is the oriented [[Bordism|bordism]] group of the space. The connection between the bordisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754022.png" /> and the Thom spaces (cf. [[Thom space|Thom space]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754023.png" />, discovered by R. Thom [[#References|[2]]], clarified the Steenrod problem by reducing it to the study of the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754024.png" />. A non-realizable class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754025.png" /> has been exhibited, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754026.png" /> is the [[Eilenberg–MacLane space|Eilenberg–MacLane space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754027.png" />. For any class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754028.png" />, some multiple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754029.png" /> is realizable (by a smooth manifold); moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754030.png" /> can be chosen odd.
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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====
 
====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>
 
<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>

Latest revision as of 08:23, 6 June 2020


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
How to Cite This Entry:
Steenrod problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steenrod_problem&oldid=24570
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article