cohomology

# Contents

## Idea

What are called the Steenrod squares is the system of cohomology operations on cohomology with coefficients in ${ℤ}_{2}$ which is compatible with suspension (the “stable cohomology 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 ${𝔽}_{2}$-vector spaces equipped with a suitable product. We follow (Lurie 07, lecture 2).

Write ${𝔽}_{2}≔ℤ/2ℤ$ for the field with two elements.

For $V$ an ${𝔽}_{2}$-module, hence an ${𝔽}_{2}$-vector space, and for $n\in ℕ$, write

${V}_{h{\Sigma }_{n}}^{\otimes n}\in {𝔽}_{2}\mathrm{Mod}$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 ${D}_{n}\left(V\right)$ of the tensor product of ${V}^{\otimes n}$ with a free resolution $E{\Sigma }_{n}^{•}$ of ${𝔽}_{2}$:

${V}_{h{\Sigma }_{n}}^{\otimes n}\simeq {D}_{n}\left(V\right)≔\left({V}^{\otimes n}\otimes E{\Sigma }_{n}{\right)}_{{\Sigma }_{n}}\phantom{\rule{thinmathspace}{0ex}}.$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

${D}_{2}\left({𝔽}_{2}\left[-n\right]\right)\simeq {𝔽}_{2}\left[-2n\right]\otimes {C}^{•}\left(B{\Sigma }_{2}\right)\phantom{\rule{thinmathspace}{0ex}},$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 $*//{\Sigma }_{2}\simeq B{\Sigma }_{2}\simeq ℝ{P}^{\infty }$, which is the homotopy type of the classifying space for ${\Sigma }_{2}$.

${D}_{2}\left(V\right)⟶V$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 $V={C}^{•}\left(X,{𝔽}_{2}\right)$ of any topological space $X$, for instance of $B{\Sigma }_{2}$.

Therefore there is a canonical isomorphism

${H}^{k}\left({D}_{2}\left({𝔽}_{2}\left[-n\right]\right)\right)\simeq {H}_{2n-k}\left(B{\Sigma }_{2},{𝔽}_{2}\right){e}_{2n}$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 ${𝔽}_{2}$ in degree $n$ with the singular homology of this classifying space shifted by $2n$.

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

$\left[v\right]\in {H}^{n}\left(V\right)$[v] \in H^n(V)

represented by a morphism of chain complexes

$v\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}{𝔽}_{2}\left[-n\right]⟶V$v \;\colon\; \mathbb{F}_2[-n] \longrightarrow V

to the element

${\overline{\mathrm{Sq}}}^{i}\left(v\right)\in {H}^{n+1}\left({D}_{2}\left(V\right)\right)$\overline{Sq}^i(v) \in H^{n+1}(D_2(V))

represented by the chain map

${𝔽}_{2}\left[-n-i\right]\stackrel{1}{⟶}{C}^{•}\left(B{\Sigma }_{2},{𝔽}_{2}\right)\stackrel{\simeq }{⟶}{D}_{2}\left({𝔽}_{2}\left[-n\right]\right)\stackrel{{D}_{2}\left(v\right)}{⟶}{D}_{2}\left(V\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 ${D}_{2}\left(V\right)⟶V$ as above, then one can further compose and form the element

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

represented by the chain map

${𝔽}_{2}\left[-n-i\right]\stackrel{1}{⟶}{C}^{•}\left(B{\Sigma }_{2},{𝔽}_{2}\right)\stackrel{\simeq }{⟶}{D}_{2}\left({𝔽}_{2}\left[-n\right]\right)\stackrel{{D}_{2}\left(v\right)}{⟶}{D}_{2}\left(V\right)⟶V\phantom{\rule{thinmathspace}{0ex}}.$\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

${\mathrm{Sq}}^{i}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}{H}^{•}\left(V\right)⟶{H}^{•+i}\left(V\right)$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 $V={C}^{•}\left(X,{𝔽}_{2}\right)$ the ${𝔽}_{2}$-singular cochain complex of some topological space $X$, as in the above examples, in which case it has the form

${\mathrm{Sq}}^{i}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}{H}^{•}\left(X,{𝔽}_{2}\right)⟶{H}^{•+i}\left(X,{𝔽}_{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$Sq^i \;\colon\; H^\bullet(X, \mathbb{F}_2) \longrightarrow H^{\bullet+i}(X,\mathbb{F}_2) \,.

### Axiomatic characterization

For $n\in ℕ$ write ${B}^{n}{ℤ}_{2}$ for the classifying space of ordinary cohomology in degree $n$ with coefficients in the group of order 2 ${ℤ}_{2}$ (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) \coloneqq 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 collection of cohomology operations

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

hence of morphisms in the homotopy category

${\mathrm{Sq}}^{n}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}{B}^{k}{ℤ}_{2}⟶{B}^{k+n}{ℤ}_{2}$Sq^n \;\colon\; B^k \mathbb{Z}_2 \longrightarrow 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 ${\mathrm{Sq}}^{n}\left(X\right)=0$;

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 prime number $p>2$. The corresponding operations are then usually denoted

${P}^{n}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}{B}^{k}{ℤ}_{p}⟶{B}^{k+n}{ℤ}_{p}\phantom{\rule{thinmathspace}{0ex}}.$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 ${𝔽}_{2}$, called the Steenrod algebra. See there for more.

## Properties

### Relation to Bockstein homomorphism

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

### 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

(…)

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

Lecture notes on Steenrod squares and the Steenrod algebra include