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Difference between revisions of "Steenrod reduced power"

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$$  
 
$$  
{\mathcal P}  ^ {i} :  H  ^ {n} ( X, Y;  \mathbf Z _ {p} )  \rightarrow  H  ^ {n+} 2i( p- 1) ( X, Y;  \mathbf Z _ {p} ),
+
{\mathcal P}  ^ {i} :  H  ^ {n} ( X, Y;  \mathbf Z _ {p} )  \rightarrow  H  ^ {n+2i}( p- 1) ( X, Y;  \mathbf Z _ {p} ),
 
$$
 
$$
  
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The Steenrod reduced powers possess the following properties (apart from naturality  $  f ^ { * } {\mathcal P}  ^ {i} = {\mathcal P}  ^ {i} f ^ { * } $
 
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 $,  
 
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} ) $
+
where  $  \delta :  H  ^ {q} ( Y;  \mathbf Z _ {p} ) \rightarrow H  ^ {q+1} ( X, Y;  \mathbf Z _ {p} ) $
 
is the coboundary homomorphism):
 
is the coboundary homomorphism):
  
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then  $  {\mathcal P}  ^ {i} x = 0 $;
 
then  $  {\mathcal P}  ^ {i} x = 0 $;
  
4) (Cartan's formula)  $  {\mathcal P}  ^ {i} ( x, y) = \sum _ {j=} ^ {i} ( {\mathcal P}  ^ {i} x) \cdot ( {\mathcal P}  ^ {i-} j x) $;
+
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)
  
 
$$  
 
$$  
{\mathcal P}  ^ {a} {\mathcal P}  ^ {b}  =  \sum _ { t= } 0 ^ { [ v/p] (- 1)  ^ {a+} t \left ( \begin{array}{c}
+
{\mathcal P}  ^ {a} {\mathcal P}  ^ {b}  =  \sum_{t=0}^ { [v/p]} (- 1)  ^ {a+t} \left ( \begin{array}{c}
 
( p- 1)( b- t)- 1 \\
 
( p- 1)( b- t)- 1 \\
 
  a- pt  
 
  a- pt  
 
\end{array}
 
\end{array}
  \right ) _ {p} {\mathcal P}  ^ {a+} b- t
+
  \right ) _ {p} {\mathcal P}  ^ {a+b-t}
 
$$
 
$$
  
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$$  
 
$$  
 
{\mathcal P}  ^ {a} \beta {\mathcal P}  ^ {b}  = \  
 
{\mathcal P}  ^ {a} \beta {\mathcal P}  ^ {b}  = \  
\sum _ { t= } 0 ^ { [ }  a/p] (- 1)  ^ {a+} t \left ( \begin{array}{c}
+
\sum_{t=0}^ { [ }  a/p] (- 1)  ^ {a+t} \left ( \begin{array}{c}
 
( p- 1)( b- t) \\
 
( p- 1)( b- t) \\
 
  a- pt  
 
  a- pt  
 
\end{array}
 
\end{array}
 
  \right
 
  \right
) _ {p} \beta {\mathcal P}  ^ {a+} b- t {\mathcal P}  ^ {t} +
+
) _ {p} \beta {\mathcal P}  ^ {a+b-t} {\mathcal P}  ^ {t} +
 
$$
 
$$
  
 
$$  
 
$$  
 
+  
 
+  
\sum _ { t= } 0 ^ { [( a- 1)/p] (- 1)  ^ {a+} t- 1 \left (
+
\sum_{t=0}^ { [(a- 1)/p]}(- 1)  ^ {a+t-1} \left (
 
\begin{array}{c}
 
\begin{array}{c}
 
( p- 1)( b- t)- 1 \\
 
( p- 1)( b- t)- 1 \\
 
  a- pt- 1  
 
  a- pt- 1  
 
\end{array}
 
\end{array}
  \right ) _ {p} {\mathcal P}  ^ {a+} b- t \beta {\mathcal P}  ^ {t}
+
  \right ) _ {p} {\mathcal P}  ^ {a+b-t} \beta {\mathcal P}  ^ {t}
 
$$
 
$$
  

Latest revision as of 19:54, 18 January 2024


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