Steenrod algebra

A cohomology operation is a natural transformation between cohomology functors. For example, if we take cohomology with coefficients in a ring ${\displaystyle R}$, the cup product squaring operation yields a family of cohomology operations:

${\displaystyle H^{n}(X;R)\to H^{2n}(X;R)}$

${\displaystyle x\mapsto x\smile x.}$

Cohomology operations need not be homomorphisms of graded rings; see the Cartan formula below.

These operations do not commute with suspension—that is, they are unstable. (This is because if ${\displaystyle Y}$ is a suspension of a space ${\displaystyle X}$, the cup product on the cohomology of ${\displaystyle Y}$ is trivial.) Steenrod constructed stable operations

${\displaystyle Sq^{i}\colon H^{n}(X;\mathbb {Z} /2)\to H^{n+i}(X;\mathbb {Z} /2)}$

for all ${\displaystyle i}$ greater than zero. The notation ${\displaystyle Sq}$ and their name, the Steenrod squares, comes from the fact that ${\displaystyle Sq^{n}}$ restricted to classes of degree ${\displaystyle n}$ is the cup square. There are analogous operations for odd primary coefficients, usually denoted ${\displaystyle P^{i}}$ and called the reduced ${\displaystyle p}$-th power operations:

${\displaystyle P^{i}\colon H^{n}(X;\mathbb {Z} /p)\to H^{n+2i(p-1)}(X;\mathbb {Z} /p)}$

The ${\displaystyle Sq^{i}}$ generate a connected graded algebra over ${\displaystyle \mathbb {Z} /2}$, where the multiplication is given by composition of operations. This is the mod 2 Steenrod algebra. In the case ${\displaystyle p>2}$, the mod ${\displaystyle p}$ Steenrod algebra is generated by the ${\displaystyle P^{i}}$ and the Bockstein operation ${\displaystyle \beta }$ associated to the short exact sequence

${\displaystyle 0\to \mathbb {Z} /p\to \mathbb {Z} /p^{2}\to \mathbb {Z} /p\to 0}$.

In the case ${\displaystyle p=2}$, the Bockstein element is ${\displaystyle Sq^{1}}$ and the reduced ${\displaystyle p}$-th power ${\displaystyle P^{i}}$ is ${\displaystyle Sq^{2i}}$.

Remark about generalizing to generalized cohomology theories

It should be observed if the Eilenberg–Maclane spectrum ${\displaystyle H\mathbb {F} _{p}}$ is replaced by an arbitrary spectrum ${\displaystyle E}$, then there are many challenges for studying the cohomology ring ${\displaystyle E^{*}(E)}$. In this case, the generalized dual Steenrod algebra ${\displaystyle E_{*}(E)}$ should be considered instead because it has much better properties and can be tractably studied in many cases (such as ${\displaystyle KO,KU,MO,MU,MSp,\mathbb {S} ,H\mathbb {F} _{p}}$).[4] In fact, these ring spectra are commutative and the ${\displaystyle \pi _{*}(E)}$ bimodules ${\displaystyle E_{*}(E)}$ are flat. In this case, these is a canonical coaction of ${\displaystyle E_{*}(E)}$ on ${\displaystyle E_{*}(X)}$ for any space ${\displaystyle X}$, such that this action behaves well with respect to the stable homotopy category, i.e., there is an isomorphism

$${\displaystyle E_{*}(E)\otimes _{\pi _{*}(E)}E_{*}(X)\to [\mathbb {S} ,E\wedge E\wedge X]_{*}}$$

hence we can use the unit the ring spectrum ${\displaystyle E}$

$${\displaystyle \eta$$ :\mathbb {S} \to E}

to get a coaction of ${\displaystyle E_{*}(E)}$ on ${\displaystyle E_{*}(X)}$.

Norman Steenrod and David B. A. Epstein (1962) showed that the Steenrod squares ${\displaystyle Sq^{n}\colon H^{m}\to H^{m+n}}$ are characterized by the following 5 axioms:

1. Naturality: ${\displaystyle Sq^{n}\colon H^{m}(X;\mathbb {Z} /2)\to H^{m+n}(X;\mathbb {Z} /2)}$ is an additive homomorphism and is natural with respect to any ${\displaystyle f\colon X\to Y}$, so ${\displaystyle f^{*}(Sq^{n}(x))=Sq^{n}(f^{*}(x))}$.
2. ${\displaystyle Sq^{0}}$ is the identity homomorphism.
3. ${\displaystyle Sq^{n}(x)=x\smile x}$ for ${\displaystyle x\in H^{n}(X;\mathbb {Z} /2)}$.
4. If ${\displaystyle n>\deg(x)}$ then ${\displaystyle Sq^{n}(x)=0}$
5. Cartan Formula: ${\displaystyle Sq^{n}(x\smile y)=\sum _{i+j=n}(Sq^{i}x)\smile (Sq^{j}y)}$

In addition the Steenrod squares have the following properties:

• ${\displaystyle Sq^{1}}$ is the Bockstein homomorphism ${\displaystyle \beta }$ of the exact sequence ${\displaystyle 0\to \mathbb {Z} /2\to \mathbb {Z} /4\to \mathbb {Z} /2\to 0.}$
• ${\displaystyle Sq^{i}}$ commutes with the connecting morphism of the long exact sequence in cohomology. In particular, it commutes with respect to suspension ${\displaystyle H^{k}(X;\mathbb {Z} /2)\cong H^{k+1}(\Sigma X;\mathbb {Z} /2)}$
• They satisfy the Adem relations, described below

Similarly the following axioms characterize the reduced ${\displaystyle p}$-th powers for ${\displaystyle p>2}$.

1. Naturality: ${\displaystyle P^{n}\colon H^{m}(X,\mathbb {Z} /p\mathbb {Z} )\to H^{m+2n(p-1)}(X,\mathbb {Z} /p\mathbb {Z} )}$ is an additive homomorphism and natural.
2. ${\displaystyle P^{0}}$ is the identity homomorphism.
3. ${\displaystyle P^{n}}$ is the cup ${\displaystyle p}$-th power on classes of degree ${\displaystyle 2n}$.
4. If ${\displaystyle 2n>\deg(x)}$ then ${\displaystyle P^{n}(x)=0}$
5. Cartan Formula: ${\displaystyle P^{n}(x\smile y)=\sum _{i+j=n}(P^{i}x)\smile (P^{j}y)}$

As before, the reduced p-th powers also satisfy the Adem relations and commute with the suspension and boundary operators.

The Adem relations for ${\displaystyle p=2}$ were conjectured by Wen-tsün Wu (1952) and established by José Adem (1952). They are given by

${\displaystyle Sq^{i}Sq^{j}=\sum _{k=0}^{\lfloor i/2\rfloor }{j-k-1 \choose i-2k}Sq^{i+j-k}Sq^{k}}$

for all ${\displaystyle i,j>0}$ such that ${\displaystyle i<2j}$. (The binomial coefficients are to be interpreted mod 2.) The Adem relations allow one to write an arbitrary composition of Steenrod squares as a sum of Serre–Cartan basis elements.

For odd ${\displaystyle p}$ the Adem relations are

${\displaystyle P^{a}P^{b}=\sum _{i}(-1)^{a+i}{(p-1)(b-i)-1 \choose a-pi}P^{a+b-i}P^{i}}$

for a<pb and

${\displaystyle P^{a}\beta P^{b}=\sum _{i}(-1)^{a+i}{(p-1)(b-i) \choose a-pi}\beta P^{a+b-i}P^{i}+\sum _{i}(-1)^{a+i+1}{(p-1)(b-i)-1 \choose a-pi-1}P^{a+b-i}\beta P^{i}}$

for ${\displaystyle a\leq pb}$.

There is a nice straightforward geometric interpretation of the Steenrod squares using manifolds representing cohomology classes. Suppose ${\displaystyle X}$ is a smooth manifold and consider a cohomology class ${\displaystyle \alpha \in H^{*}(X)}$ represented geometrically as a smooth submanifold ${\displaystyle f\colon Y\hookrightarrow X}$. Cohomologically, if we let ${\displaystyle 1=[Y]\in H^{0}(Y)}$ represent the fundamental class of ${\displaystyle Y}$ then the pushforward map

${\displaystyle f_{*}(1)=\alpha }$

gives a representation of ${\displaystyle \alpha }$. In addition, associated to this immersion is a real vector bundle call the normal bundle ${\displaystyle \nu _{Y/X}\to Y}$. The Steenrod squares of ${\displaystyle \alpha }$ can now be understood — they are the pushforward of the Stiefel–Whitney class of the normal bundle

${\displaystyle Sq^{i}(\alpha )=f_{*}(w_{i}(\nu _{Y/X})),}$

which gives a geometric reason for why the Steenrod products eventually vanish. Note that because the Steenrod maps are group homomorphisms, if we have a class ${\displaystyle \beta }$ which can be represented as a sum

${\displaystyle \beta =\alpha _{1}+\cdots +\alpha _{n},}$

where the ${\displaystyle \alpha _{k}}$ are represented as manifolds, we can interpret the squares of the classes as sums of the pushforwards of the normal bundles of their underlying smooth manifolds, i.e.,

${\displaystyle Sq^{i}(\beta )=\sum _{k=1}^{n}f_{*}(w_{i}(\nu _{Y_{k}/X})).}$

Also, this equivalence is strongly related to the Wu formula.

Suppose that ${\displaystyle \pi }$ is any degree ${\displaystyle n}$ subgroup of the symmetric group on ${\displaystyle n}$ points, ${\displaystyle u}$ a cohomology class in ${\displaystyle H^{q}(X,B)}$, ${\displaystyle A}$ an abelian group acted on by ${\displaystyle \pi }$, and ${\displaystyle c}$ a cohomology class in ${\displaystyle H_{i}(\pi ,A)}$. Steenrod (1953a, 1953b) showed how to construct a reduced power ${\displaystyle u^{n}/c}$ in ${\displaystyle H^{nq-i}(X,(A\otimes B\otimes \cdots \otimes B)/\pi )}$, as follows.

1. Taking the external product of ${\displaystyle u}$ with itself ${\displaystyle n}$ times gives an equivariant cocycle on ${\displaystyle X^{n}}$ with coefficients in ${\displaystyle B\otimes \cdots \otimes B}$.
2. Choose ${\displaystyle E}$ to be a contractible space on which ${\displaystyle \pi }$ acts freely and an equivariant map from ${\displaystyle E\times X}$ to ${\displaystyle X^{n}.}$ Pulling back ${\displaystyle u^{n}}$ by this map gives an equivariant cocycle on ${\displaystyle E\times X}$ and therefore a cocycle of ${\displaystyle E/\pi \times X}$ with coefficients in ${\displaystyle B\otimes \cdots \otimes B}$.
3. Taking the slant product with ${\displaystyle c}$ in ${\displaystyle H_{i}(E/\pi ,A)}$ gives a cocycle of ${\displaystyle X}$ with coefficients in ${\displaystyle H_{0}(\pi ,A\otimes B\otimes \cdots \otimes B)}$.

The Steenrod squares and reduced powers are special cases of this construction where ${\displaystyle \pi }$ is a cyclic group of prime order ${\displaystyle p=n}$ acting as a cyclic permutation of ${\displaystyle n}$ elements, and the groups ${\displaystyle A}$ and ${\displaystyle B}$ are cyclic of order ${\displaystyle p}$, so that ${\displaystyle H_{0}(\pi ,A\otimes B\otimes \cdots \otimes B)}$ is also cyclic of order ${\displaystyle p}$.

In addition to the axiomatic structure the Steenrod algebra satisfies, it has a number of additional useful properties.

The ${\displaystyle p=2}$ Steenrod algebra admits a filtration by finite sub-Hopf algebras. As ${\displaystyle {\mathcal {A}}_{2}}$ is generated by the elements [5]

${\displaystyle Sq^{2^{i}}}$,

we can form subalgebras ${\displaystyle {\mathcal {A}}_{2}(n)}$ generated by the Steenrod squares

${\displaystyle Sq^{1},Sq^{2},\ldots ,Sq^{2^{n}}}$,

giving the filtration

${\displaystyle {\mathcal {A}}_{2}(1)\subset {\mathcal {A}}_{2}(2)\subset \cdots \subset {\mathcal {A}}_{2}.}$

These algebras are significant because they can be used to simplify many Adams spectral sequence computations, such as for ${\displaystyle \pi _{*}(ko)}$, and ${\displaystyle \pi _{*}(tmf)}$.[6]

Larry Smith (2007) gave the following algebraic construction of the Steenrod algebra over a finite field ${\displaystyle \mathbb {F} _{q}}$ of order q. If V is a vector space over ${\displaystyle \mathbb {F} _{q}}$ then write SV for the symmetric algebra of V. There is an algebra homomorphism

${\displaystyle {\begin{cases}P(x)\colon SV[[x]]\to SV[[x]]\\P(x)(v)=v+F(v)x=v+v^{q}x&v\in V\end{cases}}}$

where F is the Frobenius endomorphism of SV. If we put

${\displaystyle P(x)(f)=\sum P^{i}(f)x^{i}\qquad p>2}$

or

${\displaystyle P(x)(f)=\sum Sq^{2i}(f)x^{i}\qquad p=2}$

for ${\displaystyle f\in SV}$ then if V is infinite dimensional the elements ${\displaystyle P^{I}}$ generate an algebra isomorphism to the subalgebra of the Steenrod algebra generated by the reduced p′th powers for p odd, or the even Steenrod squares ${\displaystyle Sq^{2i}}$ for ${\displaystyle p=2}$.

Early applications of the Steenrod algebra were calculations by Jean-Pierre Serre of some homotopy groups of spheres, using the compatibility of transgressive differentials in the Serre spectral sequence with the Steenrod operations, and the classification by René Thom of smooth manifolds up to cobordism, through the identification of the graded ring of bordism classes with the homotopy groups of Thom complexes, in a stable range. The latter was refined to the case of oriented manifolds by C. T. C. Wall. A famous application of the Steenrod operations, involving factorizations through secondary cohomology operations associated to appropriate Adem relations, was the solution by J. Frank Adams of the Hopf invariant one problem. One application of the mod 2 Steenrod algebra that is fairly elementary is the following theorem.

Theorem. If there is a map ${\displaystyle S^{2n-1}\to S^{n}}$ of Hopf invariant one, then n is a power of 2.

The proof uses the fact that each ${\displaystyle Sq^{k}}$ is decomposable for k which is not a power of 2; that is, such an element is a product of squares of strictly smaller degree.

Michael A. Mandell gave a proof of the following theorem by studying the Steenrod algebra (with coefficients in the algebraic closure of ${\displaystyle \mathbb {F} _{p}}$):

Theorem. The singular cochain functor with coefficients in the algebraic closure of ${\displaystyle \mathbb {F} _{p}}$ induces a contravariant equivalence from the homotopy category of connected ${\displaystyle p}$-complete nilpotent spaces of finite ${\displaystyle p}$-type to a full subcategory of the homotopy category of [[${\displaystyle E_{\infty }}$-algebras]] with coefficients in the algebraic closure of ${\displaystyle \mathbb {F} _{p}}$.

The cohomology of the Steenrod algebra is the ${\displaystyle E_{2}}$ term for the (p-local) Adams spectral sequence, whose abutment is the p-component of the stable homotopy groups of spheres. More specifically, the ${\displaystyle E_{2}}$ term of this spectral sequence may be identified as

${\displaystyle \mathrm {Ext} _{A}^{s,t}(\mathbb {F} _{p},\mathbb {F} _{p}).}$

This is what is meant by the aphorism "the cohomology of the Steenrod algebra is an approximation to the stable homotopy groups of spheres."

• Pontryagin cohomology operation
• Dual Steenrod algebra
• Cohomology operation