nLab
Steenrod square

Definition

For $n\in \mathbb{N}$ write $B^n{\mathbb{Z}}_2$ for the classifying space of ordinary cohomology in degree $n$ with coefficients in ${\mathbb{Z}}_2$ (the Eilenberg-MacLane space $K({\mathbb{Z}}_2,n)$), regarded as an object in the homotopy category $H$ of topological spaces.

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

$$H^n(X,{\mathbb{Z}}_2):=H(X,B^n{\mathbb{Z}}_2)$$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 ${\mathbb{Z}}_2$. Therefore, by the Yoneda lemma, natural transformations

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

correspond bijectively to morphisms $B^k{\mathbb{Z}}_2\to B^l{\mathbb{Z}}_2$.

The following characterization is due to (SteenrodEpstein).

Definition

The Steenrod squares are a family of cohomology operations

$${\mathrm{Sq}}^n:H^k(-,{\mathbb{Z}}_2)\to H^{k+n}(-,{\mathbb{Z}}_2),$$Sq^n : H^k(-, \mathbb{Z}_2)\to H^{k+n}(-, \mathbb{Z}_2) \,,

hence of morphisms in the homotopy category

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

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

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

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

3. for $k=n$ the morphism ${\mathrm{Sq}}^n:B^n{\mathbb{Z}}_2\to B^{2n}{\mathbb{Z}}_2$ is the cup product $x\mapsto x\cup x$;

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

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

$$P^n:B^k{\mathbb{Z}}_p\to B^{k+n}{\mathbb{Z}}_p.$$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 ${\mathbb{Z}}_2\to {\mathbb{Z}}_4\to {\mathbb{Z}}_2$.

Adem relations

(…)

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

(…)

Related concepts

• Bockstein homomorphism,

• Wu class

• Steenrod algebra?

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

• Norman Steenrod, David Epstein, Cohomology operations, Annals of mathematics studies, Princeton University Press (1962)

See also

• Wen-Tsun Wu, Sur les puissances de Steenrod, Colloque de Topologie de Strasbourg, IX, La Bibliothèque Nationale et Universitaire de Strasbourg, (1952)