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The general name for the stable cohomology operations (cf. [[Cohomology operation|Cohomology operation]]) created by N.E. Steenrod for every prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087530/s0875301.png" />. The first example is contained in [[#References|[1]]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087530/s0875302.png" /> this is the [[Steenrod square|Steenrod square]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087530/s0875303.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087530/s0875304.png" /> the [[Steenrod reduced power|Steenrod reduced power]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087530/s0875305.png" />. The operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087530/s0875306.png" /> multiplicatively generate the [[Steenrod algebra|Steenrod algebra]] modulo 2, while the operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087530/s0875307.png" /> together with the Bockstein homomorphism generate the Steenrod algebra modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087530/s0875308.png" />.
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====References====
 
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.E. Steenrod,  "Products of cocycles and extensions of mappings"  ''Ann. of Math.'' , '''48'''  (1947)  pp. 290–320</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.E. Steenrod,  D.B.A. Epstein,  "Cohomology operations" , Princeton Univ. Press  (1962)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.K. Tangora,  "Cohomology operations and applications in homotopy theory" , Harper &amp; Row  (1968)</TD></TR></table>
 
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====Comments====
 
  
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The general name for the stable cohomology operations (cf. [[Cohomology operation|Cohomology operation]]) created by N.E. Steenrod for every prime number $p$. The first example is contained in {{Cite|St}}. For $p=2$ this is the [[Steenrod square|Steenrod square]] $Sq^i$, for $p>2$ the [[Steenrod reduced power|Steenrod reduced power]] $\mathcal{P}^i$. The operations $Sq^i$ multiplicatively generate the [[Steenrod algebra|Steenrod algebra]] modulo 2, while the operations $\mathcal{P}^i$ together with the Bockstein homomorphism generate the Steenrod algebra modulo $p$.
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.M. Switzer,  "Algebraic topology - homotopy and homology" , Springer  (1975)  pp. Chapt. 18</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.F. Adams,  "Stable homotopy and generalized homology" , Univ. Chicago Press (1974) pp. Part III, Chapt. 12</TD></TR></table>
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|valign="top"|{{Ref|Ad}}||valign="top"|  J.F. Adams,  "Stable homotopy and generalized homology", Univ. Chicago Press  (1974)  pp. Part III, Chapt. 12
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|valign="top"|{{Ref|St}}||valign="top"|  N.E. Steenrod,  "Products of cocycles and extensions of mappings"  ''Ann. of Math.'', '''48'''  (1947)  pp. 290–320
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|valign="top"|{{Ref|StEp}}||valign="top"|  N.E. Steenrod,  D.B.A. Epstein,  "Cohomology operations", Princeton Univ. Press  (1962)
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|valign="top"|{{Ref|Sw}}||valign="top"| R.M. Switzer,  "Algebraic topology - homotopy and homology", Springer  (1975)  pp. Chapt. 18
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|valign="top"|{{Ref|Ta}}||valign="top"| M.K. Tangora,  "Cohomology operations and applications in homotopy theory", Harper &amp; Row (1968)
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Latest revision as of 22:34, 19 April 2012


The general name for the stable cohomology operations (cf. Cohomology operation) created by N.E. Steenrod for every prime number $p$. The first example is contained in [St]. For $p=2$ this is the Steenrod square $Sq^i$, for $p>2$ the Steenrod reduced power $\mathcal{P}^i$. The operations $Sq^i$ multiplicatively generate the Steenrod algebra modulo 2, while the operations $\mathcal{P}^i$ together with the Bockstein homomorphism generate the Steenrod algebra modulo $p$.

References

[Ad] J.F. Adams, "Stable homotopy and generalized homology", Univ. Chicago Press (1974) pp. Part III, Chapt. 12
[St] N.E. Steenrod, "Products of cocycles and extensions of mappings" Ann. of Math., 48 (1947) pp. 290–320
[StEp] N.E. Steenrod, D.B.A. Epstein, "Cohomology operations", Princeton Univ. Press (1962)
[Sw] R.M. Switzer, "Algebraic topology - homotopy and homology", Springer (1975) pp. Chapt. 18
[Ta] M.K. Tangora, "Cohomology operations and applications in homotopy theory", Harper & Row (1968)
How to Cite This Entry:
Steenrod operation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steenrod_operation&oldid=18169
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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