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The graded algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s0875001.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s0875002.png" /> of all stable cohomology operations (cf. [[Cohomology operation|Cohomology operation]]) modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s0875003.png" />. For any space ([[Spectrum of spaces|spectrum of spaces]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s0875004.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s0875005.png" /> is a module over the Steenrod algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s0875006.png" />.
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The Steenrod algebra is multiplicatively generated by the Steenrod operations (cf. [[Steenrod operation|Steenrod operation]]). Thus, the Steenrod algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s0875007.png" /> is a graded associative algebra, multiplicatively generated by the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s0875008.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s0875009.png" />, which satisfy the Adem relation:
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750010.png" /></td> </tr></table>
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The graded algebra  $  A _ {p} $
 +
over the field  $  \mathbf Z _ {p} $
 +
of all stable cohomology operations (cf. [[Cohomology operation|Cohomology operation]]) modulo  $  p $.  
 +
For any space ([[Spectrum of spaces|spectrum of spaces]])  $  X $,
 +
the group  $  H  ^  \star  ( X;  \mathbf Z _ {p} ) $
 +
is a module over the Steenrod algebra  $  A _ {p} $.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750011.png" />, so that an additive basis (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750012.png" />) of the Steenrod algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750013.png" /> consists of the operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750015.png" /> (the so-called Cartan–Serre basis). Similar results are true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750016.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750017.png" />. Furthermore,
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The Steenrod algebra is multiplicatively generated by the Steenrod operations (cf. [[Steenrod operation|Steenrod operation]]). Thus, the Steenrod algebra $  A _ {2} $
 +
is a graded associative algebra, multiplicatively generated by the symbols  $  Sq  ^ {i} $
 +
with  $  \mathop{\rm deg}  Sq  ^ {i} = i $,  
 +
which satisfy the Adem relation:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750018.png" /></td> </tr></table>
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$$
 +
Sq  ^ {a} Sq  ^ {b}  = \
 +
\sum _ { t } \left ( \begin{array}{c}
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b- t- 1 \\
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a- 2t
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\end{array}
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\right ) Sq  ^ {a+} b- t Sq  ^ {t} ,
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750019.png" /> is an [[Eilenberg–MacLane space|Eilenberg–MacLane space]]. The multiplication
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$  a < 2b $,
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so that an additive basis (over  $  \mathbf Z _ {2} $)
 +
of the Steenrod algebra  $  A _ {2} $
 +
consists of the operations  $  Sq ^ {i _ {1} } \dots Sq ^ {i _ {r} } $,
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$  i _ {k} \geq  2i _ {k+} 1 $(
 +
the so-called Cartan–Serre basis). Similar results are true for  $  A _ {p} $
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with  $  p > 2 $.  
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Furthermore,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750020.png" /></td> </tr></table>
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$$
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( A _ {p} )  ^ {i}  \cong  H  ^ {i+} n ( K( \mathbf Z _ {p} , n); \
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\mathbf Z _ {p} ) ,\ \
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n  \textrm{ large } ,
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$$
  
induces the diagonal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750021.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750022.png" />, which is a homomorphism of algebras, and, consequently, turns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750023.png" /> into a [[Hopf algebra|Hopf algebra]].
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where  $  K( \mathbf Z _ {p} , n) $
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is an [[Eilenberg–MacLane space|Eilenberg–MacLane space]]. The multiplication
 +
 
 +
$$
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K( \mathbf Z _ {p} , m) \wedge K( \mathbf Z _ {p} , n )  \rightarrow  K( \mathbf Z _ {p} , m+ n)
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$$
 +
 
 +
induces the diagonal $  \Delta : A _ {p} \rightarrow A _ {p} \otimes A _ {p} $
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in $  A _ {p} $,  
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which is a homomorphism of algebras, and, consequently, turns $  A _ {p} $
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into a [[Hopf algebra|Hopf algebra]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.E. Steenrod,  D.B.A. Epstein,  "Cohomology operations" , Princeton Univ. Press  (1962)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Milnor,  "The Steenrod algebra and its dual"  ''Ann. of Math.'' , '''67'''  (1958)  pp. 150–171</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>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.E. Steenrod,  D.B.A. Epstein,  "Cohomology operations" , Princeton Univ. Press  (1962)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Milnor,  "The Steenrod algebra and its dual"  ''Ann. of Math.'' , '''67'''  (1958)  pp. 150–171</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====
 
====Comments====
The analogue of the Steenrod algebra for a cohomology theory defined by a spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750024.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750025.png" />; cf. [[Generalized cohomology theories|Generalized cohomology theories]] and [[Spectrum of spaces|Spectrum of spaces]]. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750026.png" />-term of the Adams spectral sequence, cf. [[Spectral sequence|Spectral sequence]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750027.png" /> is a purely (homological) algebra construct obtained by regarding the homology groups as modules over the Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750028.png" />.
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The analogue of the Steenrod algebra for a cohomology theory defined by a spectrum $  E $
 +
is $  E _  \star  ( E) $;  
 +
cf. [[Generalized cohomology theories|Generalized cohomology theories]] and [[Spectrum of spaces|Spectrum of spaces]]. The $  E _ {2} $-
 +
term of the Adams spectral sequence, cf. [[Spectral sequence|Spectral sequence]], $  \mathop{\rm Ext} _ {E _  \star  ( E) } ^ {s, t } ( E _  \star  ( X), E _  \star  ( Y) \Rightarrow [ X, Y] _  \star  ) $
 +
is a purely (homological) algebra construct obtained by regarding the homology groups as modules over the Hopf algebra $  E _  \star  ( E) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dieudonné,  "A history of algebraic and differential topology 1900–1960" , Birkhäuser  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.M. Switzer,  "Algebraic topology - homotopy and homology" , Springer  (1975)  pp. Chapts. 18–19</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.F. Adams,  "Stable homotopy and generalized homology" , Univ. Chicago Press  (1974)  pp. Part III, Chapts. 12, 15</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dieudonné,  "A history of algebraic and differential topology 1900–1960" , Birkhäuser  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.M. Switzer,  "Algebraic topology - homotopy and homology" , Springer  (1975)  pp. Chapts. 18–19</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.F. Adams,  "Stable homotopy and generalized homology" , Univ. Chicago Press  (1974)  pp. Part III, Chapts. 12, 15</TD></TR></table>

Latest revision as of 08:23, 6 June 2020


The graded algebra $ A _ {p} $ over the field $ \mathbf Z _ {p} $ of all stable cohomology operations (cf. Cohomology operation) modulo $ p $. For any space (spectrum of spaces) $ X $, the group $ H ^ \star ( X; \mathbf Z _ {p} ) $ is a module over the Steenrod algebra $ A _ {p} $.

The Steenrod algebra is multiplicatively generated by the Steenrod operations (cf. Steenrod operation). Thus, the Steenrod algebra $ A _ {2} $ is a graded associative algebra, multiplicatively generated by the symbols $ Sq ^ {i} $ with $ \mathop{\rm deg} Sq ^ {i} = i $, which satisfy the Adem relation:

$$ Sq ^ {a} Sq ^ {b} = \ \sum _ { t } \left ( \begin{array}{c} b- t- 1 \\ a- 2t \end{array} \right ) Sq ^ {a+} b- t Sq ^ {t} , $$

$ a < 2b $, so that an additive basis (over $ \mathbf Z _ {2} $) of the Steenrod algebra $ A _ {2} $ consists of the operations $ Sq ^ {i _ {1} } \dots Sq ^ {i _ {r} } $, $ i _ {k} \geq 2i _ {k+} 1 $( the so-called Cartan–Serre basis). Similar results are true for $ A _ {p} $ with $ p > 2 $. Furthermore,

$$ ( A _ {p} ) ^ {i} \cong H ^ {i+} n ( K( \mathbf Z _ {p} , n); \ \mathbf Z _ {p} ) ,\ \ n \textrm{ large } , $$

where $ K( \mathbf Z _ {p} , n) $ is an Eilenberg–MacLane space. The multiplication

$$ K( \mathbf Z _ {p} , m) \wedge K( \mathbf Z _ {p} , n ) \rightarrow K( \mathbf Z _ {p} , m+ n) $$

induces the diagonal $ \Delta : A _ {p} \rightarrow A _ {p} \otimes A _ {p} $ in $ A _ {p} $, which is a homomorphism of algebras, and, consequently, turns $ A _ {p} $ into a Hopf algebra.

References

[1] N.E. Steenrod, D.B.A. Epstein, "Cohomology operations" , Princeton Univ. Press (1962)
[2] J. Milnor, "The Steenrod algebra and its dual" Ann. of Math. , 67 (1958) pp. 150–171
[3] M.K. Tangora, "Cohomology operations and applications in homotopy theory" , Harper & Row (1968)

Comments

The analogue of the Steenrod algebra for a cohomology theory defined by a spectrum $ E $ is $ E _ \star ( E) $; cf. Generalized cohomology theories and Spectrum of spaces. The $ E _ {2} $- term of the Adams spectral sequence, cf. Spectral sequence, $ \mathop{\rm Ext} _ {E _ \star ( E) } ^ {s, t } ( E _ \star ( X), E _ \star ( Y) \Rightarrow [ X, Y] _ \star ) $ is a purely (homological) algebra construct obtained by regarding the homology groups as modules over the Hopf algebra $ E _ \star ( E) $.

References

[a1] J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989)
[a2] R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) pp. Chapts. 18–19
[a3] J.F. Adams, "Stable homotopy and generalized homology" , Univ. Chicago Press (1974) pp. Part III, Chapts. 12, 15
How to Cite This Entry:
Steenrod algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steenrod_algebra&oldid=12348
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article