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A stable [[Cohomology operation|cohomology operation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s0875501.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s0875502.png" />, of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s0875503.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s0875504.png" /> is a fixed odd prime number, which is the analogue modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s0875505.png" /> of the [[Steenrod square|Steenrod square]], and which is a homomorphism
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<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/s087550/s0875506.png" /></td> </tr></table>
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defined for every pair of topological spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s0875507.png" /> and any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s0875508.png" />. The Steenrod reduced powers possess the following properties (apart from naturality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s0875509.png" /> and stability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755011.png" /> is the coboundary homomorphism):
+
A stable [[Cohomology operation|cohomology operation]]  $  {\mathcal P}  ^ {i} $,
 +
$  i \geq  0 $,
 +
of the type  $  ( \mathbf Z _ {p} , \mathbf Z _ {p} ) $,  
 +
where $  p $
 +
is a fixed odd prime number, which is the analogue modulo  $  p $
 +
of the [[Steenrod square|Steenrod square]], and which is a homomorphism
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755012.png" />;
+
$$
 +
{\mathcal P}  ^ {i} :  H  ^ {n} ( X, Y;  \mathbf Z _ {p} )  \rightarrow  H  ^ {n+} 2i( p- 1) ( X, Y; \mathbf Z _ {p} ),
 +
$$
  
2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755014.png" />;
+
defined for every pair of topological spaces  $  ( X, Y) $
 +
and any integer  $  n $.
 +
The Steenrod reduced powers possess the following properties (apart from naturality  $  f ^ { * } {\mathcal P}  ^ {i} = {\mathcal P}  ^ {i} f ^ { * } $
 +
and stability  $  \delta {\mathcal P}  ^ {i} = {\mathcal P}  ^ {i} \delta $,
 +
where  $  \delta : H  ^ {q} ( Y;  \mathbf Z _ {p} ) \rightarrow H  ^ {q+} 1 ( X, Y;  \mathbf Z _ {p} ) $
 +
is the coboundary homomorphism):
  
3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755015.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755016.png" />;
+
1) $  {\mathcal P}  ^ {0} = \mathop{\rm id} $;
  
4) (Cartan's formula) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755017.png" />;
+
2) if  $  2i =  \mathop{\rm dim}  x $,
 +
then  $  {\mathcal P}  ^ {i} x = x  ^ {p} $;
 +
 
 +
3) if  $  2i >  \mathop{\rm dim}  x $,
 +
then  $  {\mathcal P}  ^ {i} x = 0 $;
 +
 
 +
4) (Cartan's formula) $  {\mathcal P}  ^ {i} ( x, y) = \sum _ {j=} 0 ^ {i} ( {\mathcal P}  ^ {i} x) \cdot ( {\mathcal P}  ^ {i-} j x) $;
  
 
5) (Adem's relation)
 
5) (Adem's 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/s087550/s08755018.png" /></td> </tr></table>
+
$$
 +
{\mathcal P}  ^ {a} {\mathcal P}  ^ {b}  = \sum _ { t= } 0 ^ { [ }  v/p] (- 1)  ^ {a+} t \left ( \begin{array}{c}
 +
( p- 1)( b- t)- 1 \\
 +
a- pt
 +
\end{array}
 +
\right ) _ {p} {\mathcal P}  ^ {a+} b- t
 +
$$
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755019.png" />,
+
if $  a < pb $,
  
<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/s087550/s08755020.png" /></td> </tr></table>
+
$$
 +
{\mathcal P}  ^ {a} \beta {\mathcal P}  ^ {b}  = \
 +
\sum _ { t= } 0 ^ { [ }  a/p] (- 1)  ^ {a+} t \left ( \begin{array}{c}
 +
( p- 1)( b- t) \\
 +
a- pt
 +
\end{array}
 +
\right
 +
) _ {p} \beta {\mathcal P}  ^ {a+} b- t {\mathcal P}  ^ {t} +
 +
$$
  
<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/s087550/s08755021.png" /></td> </tr></table>
+
$$
 +
+
 +
\sum _ { t= } 0 ^ { [( }  a- 1)/p] (- 1)  ^ {a+} t- 1 \left (
 +
\begin{array}{c}
 +
( p- 1)( b- t)- 1 \\
 +
a- pt- 1
 +
\end{array}
 +
\right ) _ {p} {\mathcal P}  ^ {a+} b- t \beta {\mathcal P}  ^ {t}
 +
$$
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755023.png" /> is the Bockstein homomorphism associated with the short exact sequence of coefficient groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755024.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755025.png" /> are the binomial coefficients reduced modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755026.png" />.
+
if $  a \leq  pb $,  
 +
where $  \beta $
 +
is the Bockstein homomorphism associated with the short exact sequence of coefficient groups 0 \rightarrow \mathbf Z _ {p} \rightarrow \mathbf Z _ {p  ^ {2}  } \rightarrow \mathbf Z _ {p} \rightarrow 0 $,  
 +
while $  ( \cdot ) _ {p} $
 +
are the binomial coefficients reduced modulo $  p $.
  
These properties are analogous to the corresponding properties of Steenrod squares, whereby the operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755027.png" /> corresponds to the operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755028.png" />. Just as for Steenrod squares, the multiplication in 4) can be considered to be both exterior (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755029.png" />-multiplication) and interior (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755030.png" />-multiplication). Steenrod reduced powers commute with [[Suspension|suspension]] and [[Transgression|transgression]].
+
These properties are analogous to the corresponding properties of Steenrod squares, whereby the operation $  Sq  ^ {2i} $
 +
corresponds to the operation $  {\mathcal P}  ^ {i} $.  
 +
Just as for Steenrod squares, the multiplication in 4) can be considered to be both exterior ( $  \times $-
 +
multiplication) and interior ( $  \cup $-
 +
multiplication). Steenrod reduced powers commute with [[Suspension|suspension]] and [[Transgression|transgression]].
  
The properties 1)–3) uniquely characterize <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755031.png" />, and can be constructed in the same way as Steenrod squares using the minimal acyclic free chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755032.png" />-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087550/s08755033.png" />.
+
The properties 1)–3) uniquely characterize $  {\mathcal P}  ^ {i} $,  
 +
and can be constructed in the same way as Steenrod squares using the minimal acyclic free chain $  \mathbf Z _ {p} $-
 +
complex $  W $.
  
 
====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">  ''Matematika'' , '''5''' :  2  (1961)  pp. 3–11; 11–30; 30–49; 50–102</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">  ''Matematika'' , '''5''' :  2  (1961)  pp. 3–11; 11–30; 30–49; 50–102</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
For more references see [[Steenrod algebra|Steenrod algebra]].
 
For more references see [[Steenrod algebra|Steenrod algebra]].

Latest revision as of 08:23, 6 June 2020


A stable cohomology operation $ {\mathcal P} ^ {i} $, $ i \geq 0 $, of the type $ ( \mathbf Z _ {p} , \mathbf Z _ {p} ) $, where $ p $ is a fixed odd prime number, which is the analogue modulo $ p $ of the Steenrod square, and which is a homomorphism

$$ {\mathcal P} ^ {i} : H ^ {n} ( X, Y; \mathbf Z _ {p} ) \rightarrow H ^ {n+} 2i( p- 1) ( X, Y; \mathbf Z _ {p} ), $$

defined for every pair of topological spaces $ ( X, Y) $ and any integer $ n $. The Steenrod reduced powers possess the following properties (apart from naturality $ f ^ { * } {\mathcal P} ^ {i} = {\mathcal P} ^ {i} f ^ { * } $ and stability $ \delta {\mathcal P} ^ {i} = {\mathcal P} ^ {i} \delta $, where $ \delta : H ^ {q} ( Y; \mathbf Z _ {p} ) \rightarrow H ^ {q+} 1 ( X, Y; \mathbf Z _ {p} ) $ is the coboundary homomorphism):

1) $ {\mathcal P} ^ {0} = \mathop{\rm id} $;

2) if $ 2i = \mathop{\rm dim} x $, then $ {\mathcal P} ^ {i} x = x ^ {p} $;

3) if $ 2i > \mathop{\rm dim} x $, then $ {\mathcal P} ^ {i} x = 0 $;

4) (Cartan's formula) $ {\mathcal P} ^ {i} ( x, y) = \sum _ {j=} 0 ^ {i} ( {\mathcal P} ^ {i} x) \cdot ( {\mathcal P} ^ {i-} j x) $;

5) (Adem's relation)

$$ {\mathcal P} ^ {a} {\mathcal P} ^ {b} = \sum _ { t= } 0 ^ { [ } v/p] (- 1) ^ {a+} t \left ( \begin{array}{c} ( p- 1)( b- t)- 1 \\ a- pt \end{array} \right ) _ {p} {\mathcal P} ^ {a+} b- t $$

if $ a < pb $,

$$ {\mathcal P} ^ {a} \beta {\mathcal P} ^ {b} = \ \sum _ { t= } 0 ^ { [ } a/p] (- 1) ^ {a+} t \left ( \begin{array}{c} ( p- 1)( b- t) \\ a- pt \end{array} \right ) _ {p} \beta {\mathcal P} ^ {a+} b- t {\mathcal P} ^ {t} + $$

$$ + \sum _ { t= } 0 ^ { [( } a- 1)/p] (- 1) ^ {a+} t- 1 \left ( \begin{array}{c} ( p- 1)( b- t)- 1 \\ a- pt- 1 \end{array} \right ) _ {p} {\mathcal P} ^ {a+} b- t \beta {\mathcal P} ^ {t} $$

if $ a \leq pb $, where $ \beta $ is the Bockstein homomorphism associated with the short exact sequence of coefficient groups $ 0 \rightarrow \mathbf Z _ {p} \rightarrow \mathbf Z _ {p ^ {2} } \rightarrow \mathbf Z _ {p} \rightarrow 0 $, while $ ( \cdot ) _ {p} $ are the binomial coefficients reduced modulo $ p $.

These properties are analogous to the corresponding properties of Steenrod squares, whereby the operation $ Sq ^ {2i} $ corresponds to the operation $ {\mathcal P} ^ {i} $. Just as for Steenrod squares, the multiplication in 4) can be considered to be both exterior ( $ \times $- multiplication) and interior ( $ \cup $- multiplication). Steenrod reduced powers commute with suspension and transgression.

The properties 1)–3) uniquely characterize $ {\mathcal P} ^ {i} $, and can be constructed in the same way as Steenrod squares using the minimal acyclic free chain $ \mathbf Z _ {p} $- complex $ W $.

References

[1] N.E. Steenrod, D.B.A. Epstein, "Cohomology operations" , Princeton Univ. Press (1962)
[2] Matematika , 5 : 2 (1961) pp. 3–11; 11–30; 30–49; 50–102

Comments

For more references see Steenrod algebra.

How to Cite This Entry:
Steenrod reduced power. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steenrod_reduced_power&oldid=48826
This article was adapted from an original article by S.N. MalyginM.M. Postnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article