Steenrod reduced power

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.