cohomology

# Contents

## Definition

For $n\in ℕ$ write ${B}^{n}{ℤ}_{2}$ for the classifying space of ordinary cohomology in degree $n$ with coefficients in ${ℤ}_{2}$ (the Eilenberg-MacLane space $K\left({ℤ}_{2},n\right)$), regarded as an object in the homotopy category $H$ of topological spaces.

Notice that for $X$ any topological space (CW-complex),

${H}^{n}\left(X,{ℤ}_{2}\right):=H\left(X,{B}^{n}{ℤ}_{2}\right)$H^n(X, \mathbb{Z}_2) := H(X, B^n \mathbb{Z}_2)

is the ordinary cohomology of $X$ in degree $n$ with coefficients in ${ℤ}_{2}$. Therefore, by the Yoneda lemma, natural transformations

${H}^{k}\left(-,{ℤ}_{2}\right)\to {H}^{l}\left(-,{ℤ}_{2}\right)$H^{k}(-, \mathbb{Z}_2) \to H^l(-, \mathbb{Z}_2)

correspond bijectively to morphisms ${B}^{k}{ℤ}_{2}\to {B}^{l}{ℤ}_{2}$.

The following characterization is due to (SteenrodEpstein).

###### Definition

The Steenrod squares are a family of cohomology operations

${\mathrm{Sq}}^{n}:{H}^{k}\left(-,{ℤ}_{2}\right)\to {H}^{k+n}\left(-,{ℤ}_{2}\right)\phantom{\rule{thinmathspace}{0ex}},$Sq^n : H^k(-, \mathbb{Z}_2)\to H^{k+n}(-, \mathbb{Z}_2) \,,

hence of morphisms in the homotopy category

${\mathrm{Sq}}^{n}:{B}^{k}{ℤ}_{2}\to {B}^{k+n}{ℤ}_{2}$Sq^n : B^k \mathbb{Z}_2 \to B^{k + n} \mathbb{Z}_2

for all $n,k\in ℕ$ satisfying the following conditions:

1. for $n=0$ it is the identity;

2. if $X$ is a manifold of dimension $\mathrm{dim}X then ${\mathrm{Sq}}^{n}=0$;

3. for $k=n$ the morphism ${\mathrm{Sq}}^{n}:{B}^{n}{ℤ}_{2}\to {B}^{2n}{ℤ}_{2}$ is the cup product $x↦x\cup x$;

4. ${\mathrm{Sq}}^{n}\left(x\cup y\right)={\sum }_{i+j=n}\left({\mathrm{Sq}}^{i}x\right)\cup \left({\mathrm{Sq}}^{j}y\right)$;

An analogous definition works for coefficients in ${ℤ}_{p}$ for any $p>2$. The corresponding oerations are usually denoted

${P}^{n}:{B}^{k}{ℤ}_{p}\to {B}^{k+n}{ℤ}_{p}\phantom{\rule{thinmathspace}{0ex}}.$P^n : B^k \mathbb{Z}_p \to B^{k+n} \mathbb{Z}_{p} \,.

## Properties

### Relation to Bockstein homomorphism

${\mathrm{Sq}}^{1}$ is the Bockstein homomorphism of the short exact sequence ${ℤ}_{2}\to {ℤ}_{4}\to {ℤ}_{2}$.

(…)

${\mathrm{Sq}}^{i}\circ {\mathrm{Sq}}^{j}=\sum _{k=0}^{\left[i/2\right]}{\left(\begin{array}{c}j-k-1\\ i-2k\end{array}\right)}_{\mathrm{mod}2}{\mathrm{Sq}}^{i+j-k}\circ {\mathrm{Sq}}^{k}$Sq^i \circ Sq^j = \sum_{k = 0}^{[i/2]} \left( \array{ j - k - 1 \\ i - 2k } \right)_{mod 2} Sq^{i + j -k} \circ Sq^k

for all $0.

(…)

## References

The operations were first defined in

• Norman Steenrod, Products of cocycles and extensions of mappings, Annals of mathematics (1947)

The axiomatic definition appears in