Steenrod reduced power

The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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) ${\mathcal P} ^ {i} ( x, y) = \sum_{j=0}^ {i} ( {\mathcal P} ^ {i} x) \cdot ( {\mathcal P} ^ {i-j} x)$;

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

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