Namespaces
Variants
Actions

Difference between revisions of "Steenrod operation"

From Encyclopedia of Mathematics
Jump to: navigation, search
(TeX)
 
Line 1: Line 1:
 
{{TEX|done}}
 
{{TEX|done}}
−
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 [[#References|[1]]]. 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====
 
−
<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>
 
−
 
−
 
−
 
−
====Comments====
 
  
 +
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====
−
<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>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Ad}}||valign="top"|  J.F. Adams,  "Stable homotopy and generalized homology", Univ. Chicago Press  (1974)  pp. Part III, Chapt. 12
 +
|-
 +
|valign="top"|{{Ref|St}}||valign="top"|  N.E. Steenrod,  "Products of cocycles and extensions of mappings"  ''Ann. of Math.'', '''48'''  (1947)  pp. 290–320
 +
|-
 +
|valign="top"|{{Ref|StEp}}||valign="top"|  N.E. Steenrod,  D.B.A. Epstein,  "Cohomology operations", Princeton Univ. Press  (1962)
 +
|-
 +
|valign="top"|{{Ref|Sw}}||valign="top"| R.M. Switzer,  "Algebraic topology - homotopy and homology", Springer  (1975)  pp. Chapt. 18
 +
|-
 +
|valign="top"|{{Ref|Ta}}||valign="top"| M.K. Tangora,  "Cohomology operations and applications in homotopy theory", Harper &amp; Row (1968)
 +
|-
 +
|}

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=24815
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Steenrod_operation&oldid=24818"