cohomology

# Contents

## Idea

What are called the Steenrod squares is the system of cohomology operations on cohomology with coefficients in $\mathbb{Z}_2$ 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 $X$ is a manifold of dimension $dim X \lt n$ 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

(…)

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

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

(…)

## 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