Contents

cohomology

# Contents

## Idea

What are called the Steenrod squares is the system of cohomology operations on cohomology with coefficients in $\mathbb{Z}_2$ (the cyclic group of order 2) which is compatible with suspension (the “stable cohomology operations”). They are special examples of power operations.

The Steenrod squares together form the Steenrod algebra, see there for more.

## Definition

### Construction in terms of extended squares

We discuss the explicit construction of the Steenrod-operations in terms of chain maps of chain complexes of $\mathbb{F}_2$-vector spaces equipped with a suitable product. We follow (Lurie 07, lecture 2).

Write $\mathbb{F}_2 \coloneqq \mathbb{Z}/2\mathbb{Z}$ for the field with two elements.

For $V$ an $\mathbb{F}_2$-module, hence an $\mathbb{F}_2$-vector space, and for $n \in \mathbb{N}$, write

$V^{\otimes n}_{h \Sigma_n} \in \mathbb{F}_2 Mod$

for the homotopy quotient of the $n$-fold tensor product of $V$ with itself by the action of the symmetric group. Explicitly this is presented, up to quasi-isomorphism by the ordinary coinvariants $D_n(V)$ of the tensor product of $V^{\otimes n}$ with a free resolution $E \Sigma_n^\bullet$ of $\mathbb{F}_2$:

$V^{\otimes n}_{h \Sigma_n} \simeq D_n(V) \coloneqq (V^{\otimes n} \otimes E\Sigma_n)_{\Sigma_n} \,.$

This is called the $n$th extended power of $V$.

For instance

$D_2( \mathbb{F}_2[-n]) \simeq \mathbb{F}_2[-2n] \otimes C^\bullet(B \Sigma_2) \,,$

where on the right we have the, say, singular cohomology cochain complex of the homotopy quotient $\ast //\Sigma_2 \simeq B \Sigma_2 \simeq \mathbb{R}P^\infty$, which is the homotopy type of the classifying space for $\Sigma_2$.

$D_2(V) \longrightarrow V$

is called a symmetric multiplication on $V$ (a shadow of an E-infinity algebra structure). The archetypical class of examples of these are given by the singular cohomology $V = C^\bullet(X, \mathbb{F}_2)$ of any topological space $X$, for instance of $B \Sigma_2$.

Therefore there is a canonical isomorphism

$H^k(D_2(\mathbb{F}_2[-n])) \simeq H_{2n - k}(B \Sigma_2, \mathbb{F}_2) e_{2n}$

of the cochain cohomology of the extended square of the chain compplex concentrated on $\mathbb{F}_2$ in degree $n$ with the singular homology of this classifying space shifted by $2 n$.

Using this one gets for general $V$ and for each $i \leq n$ a map that sends an element in the $n$th cochain cohomology

$[v] \in H^n(V)$

represented by a morphism of chain complexes

$v \;\colon\; \mathbb{F}_2[-n] \longrightarrow V$

to the element

$\overline{Sq}^i(v) \in H^{n+1}(D_2(V))$

represented by the chain map

$\mathbb{F}_2[-n-i] \stackrel{1}{\longrightarrow} C^\bullet(B \Sigma_2, \mathbb{F}_2) \stackrel{\simeq}{\longrightarrow} D_2(\mathbb{F}_2[-n]) \stackrel{D_2(v)}{\longrightarrow} D_2(V) \,.$

If moreover $V$ is equipped with a symmetric product $D_2(V) \longrightarrow V$ as above, then one can further compose and form the element

${Sq}^i(v) \in H^{n+1}(V)$

represented by the chain map

$\mathbb{F}_2[-n-i] \stackrel{1}{\longrightarrow} C^\bullet(B \Sigma_2, \mathbb{F}_2) \stackrel{\simeq}{\longrightarrow} D_2(\mathbb{F}_2[-n]) \stackrel{D_2(v)}{\longrightarrow} D_2(V) \longrightarrow V \,.$

This linear map

$Sq^i \;\colon\; H^\bullet(V) \longrightarrow H^{\bullet + i}(V)$

is called the $i$th Steenrod operation or the $i$th Steenrod square on $V$. By default this is understood for $V = C^\bullet(X,\mathbb{F}_2)$ the $\mathbb{F}_2$-singular cochain complex of some topological space $X$, as in the above examples, in which case it has the form

$Sq^i \;\colon\; H^\bullet(X, \mathbb{F}_2) \longrightarrow H^{\bullet+i}(X,\mathbb{F}_2) \,.$

### Axiomatic characterization

For $n \in \mathbb{N}$ write $B^n \mathbb{Z}_2$ for the classifying space of ordinary cohomology in degree $n$ with coefficients in the group of order 2 $\mathbb{Z}_2$ (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),

$H^n(X, \mathbb{Z}_2) \coloneqq H(X, B^n \mathbb{Z}_2)$

is the ordinary cohomology of $X$ in degree $n$ with coefficients in $\mathbb{Z}_2$. Therefore, by the Yoneda lemma, natural transformations

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

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 collection of cohomology operations

$Sq^n \;\colon\; H^k(-, \mathbb{Z}_2) \longrightarrow H^{k+n}(-, \mathbb{Z}_2) \,,$

hence of morphisms in the homotopy category

$Sq^n \;\colon\; B^k \mathbb{Z}_2 \longrightarrow B^{k + n} \mathbb{Z}_2$

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

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

2. if $n \gt deg(x)$ then $Sq^n(x) = 0$;

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

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

An analogous definition works for coefficients in $\mathbb{Z}_p$ for any prime number $p \gt 2$. The corresponding operations are then usually denoted

$P^n \;\colon\; B^k \mathbb{Z}_p \longrightarrow B^{k+n} \mathbb{Z}_{p} \,.$

Under composition, the Steenrod squares form an associative algebra over $\mathbb{F}_2$, called the Steenrod algebra. See there for more.

## Properties

### Relation to Bockstein homomorphism

$Sq^1$ is the Bockstein homomorphism of the short exact sequence $\mathbb{Z}_2 \to \mathbb{Z}_4 \to \mathbb{Z}_2$.

### Compatibility with suspension

The Steenrod squares are compatible with the suspension isomorphism.

Therefore the Steenrod squares are often also referred to as the stable cohomology operations

### Relation to Massey products

###### Proposition

The composition of Steenrod square operations satisfies the following relations

$Sq^i \circ Sq^j = \sum_{0 \leq k \leq i/2} \left( { { j - k - 1 } \atop { i - 2k } } \right)_{mod 2} Sq^{i + j -k} \circ Sq^k$

for all $0 \lt i \lt 2 j$.

Here $\left( a \atop b \right) \coloneqq 0$ if $a \lt b$.

###### Example

(Adem relation for postcomposition with the Bockstein homomorphism $Sq^1 = \beta$)

For $j \geq 2$ and $i =1$, the Adem relations (prop. ) say that:

\begin{aligned} Sq^1 \circ Sq^j & = \underset{ (j-1)_{mod 2} }{ \underbrace{ \left( { {j - 1 } \atop 1 } \right)_{mod 2} }} Sq^{j + 1} \\ & = \left\{ \array{ Sq^{j+1} &\vert& j \, \text{even} \\ 0 &\vert& j \, \text{odd} } \right. \end{aligned}

This gives rise to:

###### Example

(integral Steenrod squares)

For odd $2n + 1 \in \mathbb{N}$ defines the integral Steenrod squares to be

$Sq^{2n + 1}_{\mathbb{Z}} \;\coloneqq\; \beta \circ Sq^{2n} \,.$

By example and by this example these indeed are lifts of the odd Steenrod squares:

$(mod\, 2) \circ Sq^{2n + 1}_{\mathbb{Z}} \;=\; Sq^{2n+1} \,,$

in that we have

$\array{ Sq^{2n+1}_{\mathbb{Z}} &\colon& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\overset{Sq^{2n}}{\longrightarrow}& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\overset{ \beta }{\longrightarrow}& B^{\bullet + 2n + 1} \mathbb{Z} \\ && \downarrow^{ id } && \downarrow^{ id } && \downarrow^{\mathrlap{B^{k + 2 n + 1}(mod\, 2)}} \\ Sq^{2n+1} &\colon& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\underset{Sq^{2n}}{\longrightarrow}& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\underset{ Sq^1 }{\longrightarrow}& B^{\bullet + 2n + 1} (\mathbb{Z}/2\mathbb{Z}) }$

## Examples

### Hopf invariant

###### Proposition

For $\phi \colon S^{k+n-1} \to S^k$, a map of spheres, the Steenrod square

$Sq^n \colon H^k(cofib(\phi), \mathbb{F}_2) \longrightarrow H^{k+n}(cofib(\phi),\mathbb{F}_2)$

(on the homotopy cofiber $cofib(\phi)\simeq S^k \underset{S^{k+n-1}}{\cup} D^{k+n}$)

is non-vanishing exactly for $n \in \{1,2,4,8\}$.

See at Hopf invariant one theorem.

## 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

Lecture notes on Steenrod squares and the Steenrod algebra include