Inalgebraic topology, aSteenrod algebra was defined byHenri Cartan (1955) to be the algebra of stablecohomology operations for mod${\displaystyle p}$cohomology.

For a givenprime number${\displaystyle p}$, the Steenrod algebra${\displaystyle A_p}$ is the gradedHopf algebra over thefield${\displaystyle {\mathbb{F}}_p}$ of order${\displaystyle p}$, consisting of all stable cohomology operations for mod${\displaystyle p}$ cohomology. It is generated by theSteenrod squares introduced by Norman Steenrod (1947) for${\displaystyle p=2}$, and by theSteenrod reduced${\displaystyle p}$th powers introduced in Steenrod (1953a,1953b) and theBockstein homomorphism for${\displaystyle p>2}$.

The term "Steenrod algebra" is also sometimes used for the algebra of cohomology operations of ageneralized cohomology theory.

A cohomology operation is anatural transformation between cohomologyfunctors. For example, if we take cohomology with coefficients in aring${\displaystyle R}$, thecup product squaring operation yields a family of cohomology operations:

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

${\displaystyle x\mapsto x⌣x.}$

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

These operations do not commute withsuspension—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

${\displaystyle S{q}^i:H^n(X;\mathbb{Z}/2)\to H^{n+i}(X;\mathbb{Z}/2)}$

for all${\displaystyle i}$ greater than zero. The notation${\displaystyle Sq}$ and their name, the Steenrod squares, comes from the fact that${\displaystyle S{q}^n}$ 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:

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

The${\displaystyle S{q}^i}$ 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 theBockstein operation${\displaystyle \beta }$ associated to theshort 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${\displaystyle S{q}^1}$ and the reduced${\displaystyle p}$-th power${\displaystyle P^i}$ is${\displaystyle S{q}^{2i}}$.

We can summarize the properties of the Steenrod operations as generators in the cohomology ring ofEilenberg–Maclane spectra

${\displaystyle {\mathcal{A}}_p=H{\mathbb{F}}_p^{\ast }(H{\mathbb{F}}_p)}$,

since there is anisomorphism

giving a direct sum decomposition of all possible cohomology operations with coefficients in${\displaystyle {\mathbb{F}}_p}$. Note the inverse limit of cohomology groups appears because it is a computation in thestable range of cohomology groups of Eilenberg–Maclane spaces. This result^[1] was originally computed^[2] byCartan (1954–1955, p. 7) andSerre (1953).

Note there is a dual characterization^[3] using homology for thedual Steenrod algebra.

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${\displaystyle E^{\ast }(E)}$. In this case, the generalized dual Steenrod algebra${\displaystyle E_{\ast }(E)}$ 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, thesering spectra are commutative and the${\displaystyle {\pi }_{\ast }(E)}$ bimodules${\displaystyle E_{\ast }(E)}$ are flat. In this case, these is a canonical coaction of${\displaystyle E_{\ast }(E)}$ on${\displaystyle E_{\ast }(X)}$ for any space${\displaystyle X}$, such that this action behaves well with respect to the stable homotopy category, i.e., there is an isomorphism$${\displaystyle E_{\ast }(E){\otimes }_{{\pi }_{\ast }(E)}{E}_{\ast }(X)\to [\mathbb{S},E\wedge E\wedge X{]}_{\ast }}$$ hence we can use the unit the ring spectrum${\displaystyle E}$$${\displaystyle \eta :\mathbb{S}\to E}$$ to get a coaction of${\displaystyle E_{\ast }(E)}$ on${\displaystyle E_{\ast }(X)}$.

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

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

In addition the Steenrod squares have the following properties:

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

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

1. Naturality:${\displaystyle P^n:H^m(X,\mathbb{Z}/p\mathbb{Z})\to H^{m+2n(p-1)}(X,\mathbb{Z}/p\mathbb{Z})}$ 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${\displaystyle 2n>\mathrm{deg}(x)}$ then${\displaystyle P^n(x)=0}$
5. Cartan Formula:${\displaystyle P^n(x⌣y)=\sum _{i+j=n}(P^{i}x)⌣(P^{j}y)}$

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

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

${\displaystyle S{q}^{i}S{q}^j=\sum _{k=0}^{\lfloor i/2\rfloor }(\genfrac{}{}{0ex}{}{j-k-1}{i-2k})S{q}^{i+j-k}S{q}^k}$

for all${\displaystyle i,j>0}$ such that. (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

${\displaystyle P^a{P}^b=\sum _i(-1{)}^{a+i}(\genfrac{}{}{0ex}{}{(p-1)(b-i)-1}{a-pi})P^{a+b-i}{P}^i}$

fora<pb and

${\displaystyle P^a\beta P^b=\sum _i(-1{)}^{a+i}(\genfrac{}{}{0ex}{}{(p-1)(b-i)}{a-pi})\beta P^{a+b-i}{P}^i+\sum _i(-1{)}^{a+i+1}(\genfrac{}{}{0ex}{}{(p-1)(b-i)-1}{a-pi-1})P^{a+b-i}\beta P^i}$

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

Shaun R. Bullett and Ian G. Macdonald (1982) reformulated the Adem relations as the following identities.

For${\displaystyle p=2}$ put

${\displaystyle P(t)=\sum _{i\ge 0}{t}^i{\text{Sq}}^i}$

then the Adem relations are equivalent to

${\displaystyle P(s^2+st)\cdot P(t^2)=P(t^2+st)\cdot P(s^2)}$

For${\displaystyle p>2}$ put

${\displaystyle P(t)=\sum _{i\ge 0}{t}^i{\text{P}}^i}$

then the Adem relations are equivalent to the statement that

${\displaystyle (1+s\mathrm{Ad}\beta )P(t^p+t^{p-1}s+\cdots +t{s}^{p-1})P(s^p)}$

is symmetric in${\displaystyle s}$ and${\displaystyle t}$. Here${\displaystyle \beta }$ is the Bockstein operation and${\displaystyle (\mathrm{Ad}\beta )P=\beta P-P\beta }$.

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${\displaystyle \alpha \in H^{\ast }(X)}$ represented geometrically as a smooth submanifold${\displaystyle f:Y\hookrightarrow X}$. Cohomologically, if we let${\displaystyle 1=[Y]\in H^0(Y)}$ represent the fundamental class of${\displaystyle Y}$ then thepushforward map

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

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 theStiefel–Whitney class of the normal bundle

${\displaystyle S{q}^i(\alpha )=f_{\ast }(w_i({\nu }_{Y/X})),}$

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.,

${\displaystyle S{q}^i(\beta )=\sum _{k=1}^n{f}_{\ast }(w_i({\nu }_{Y_k/X})).}$

Also, this equivalence is strongly related to theWu formula.

On thecomplex projective plane${\displaystyle {\mathbf{C}\mathbf{P}}^2}$, there are only the following non-trivial cohomology groups,

${\displaystyle H^0({\mathbf{C}\mathbf{P}}^2)\cong H^2({\mathbf{C}\mathbf{P}}^2)\cong H^4({\mathbf{C}\mathbf{P}}^2)\cong \mathbb{Z}}$,

as can be computed using a cellular decomposition. This implies that the only possible non-trivial Steenrod product is${\displaystyle S{q}^2}$ on${\displaystyle H^2({\mathbf{C}\mathbf{P}}^2;\mathbb{Z}/2)}$ since it gives thecup product on cohomology. As the cup product structure on${\displaystyle H^{\ast }({\mathbf{C}\mathbf{P}}^2;\mathbb{Z}/2)}$ is nontrivial, this square is nontrivial. There is a similar computation on thecomplex projective space${\displaystyle {\mathbf{C}\mathbf{P}}^6}$, where the only non-trivial squares are${\displaystyle S{q}^0}$ and the squaring operations${\displaystyle S{q}^{2i}}$ on the cohomology groups${\displaystyle H^{2i}}$ representing thecup product. In${\displaystyle {\mathbf{C}\mathbf{P}}^8}$ the square

${\displaystyle S{q}^2:H^4({\mathbf{C}\mathbf{P}}^8;\mathbb{Z}/2)\to H^6({\mathbf{C}\mathbf{P}}^8;\mathbb{Z}/2)}$

can be computed using the geometric techniques outlined above and the relation between Chern classes and Stiefel–Whitney classes; note that${\displaystyle f:{\mathbf{C}\mathbf{P}}^4\hookrightarrow {\mathbf{C}\mathbf{P}}^8}$ represents the non-zero class in${\displaystyle H^4({\mathbf{C}\mathbf{P}}^8;\mathbb{Z}/2)}$. It can also be computed directly using the Cartan formula since${\displaystyle x^2\in H^4({\mathbf{C}\mathbf{P}}^8)}$ and

The Steenrod operations for real projective spaces can be readily computed using the formal properties of the Steenrod squares. Recall that

${\displaystyle H^{\ast }({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }};\mathbb{Z}/2)\cong \mathbb{Z}/2[x],}$

where${\displaystyle \mathrm{deg}(x)=1.}$ For the operations on${\displaystyle H^1}$ we know that

The Cartan relation implies that the total square

${\displaystyle Sq:=S{q}^0+S{q}^1+S{q}^2+\cdots }$

is a ring homomorphism

${\displaystyle Sq:H^{\ast }(X)\to H^{\ast }(X).}$

Hence

${\displaystyle Sq(x^n)=(Sq(x){)}^n=(x+x^2{)}^n=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})x^{n+i}}$

Since there is only one degree${\displaystyle n+i}$ component of the previous sum, we have that

${\displaystyle S{q}^i(x^n)=(\genfrac{}{}{0ex}{}{n}{i})x^{n+i}.}$

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 acontractible 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 theslant 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.

Jean-Pierre Serre (1953) (for${\displaystyle p=2}$) and Henri Cartan (1954,1955) (for${\displaystyle p>2}$) described the structure of the Steenrod algebra of stable mod${\displaystyle p}$ cohomology operations, showing that it is generated by the Bockstein homomorphism together with the Steenrod reduced powers, and the Adem relations generate the ideal of relations between these generators. In particular they found an explicit basis for the Steenrod algebra. This basis relies on a certain notion of admissibility for integer sequences. We say a sequence

${\displaystyle i_1,i_2,\dots ,i_n}$

isadmissible if for each${\displaystyle j}$, we have that${\displaystyle i_j\ge 2i_{j+1}}$. Then the elements

${\displaystyle S{q}^I=S{q}^{i_1}\cdots S{q}^{i_n},}$

where${\displaystyle I}$ is an admissible sequence, form a basis (the Serre–Cartan basis) for the mod 2 Steenrod algebra, called theadmissible basis. There is a similar basis for the case${\displaystyle p>2}$ consisting of the elements

${\displaystyle S{q}_p^I=S{q}_p^{i_1}\cdots S{q}_p^{i_n}}$,

such that

${\displaystyle i_j\ge p{i}_{j+1}}$

${\displaystyle i_j\equiv 0,1mod2(p-1)}$

${\displaystyle S{q}_p^{2k(p-1)}=P^k}$

${\displaystyle S{q}_p^{2k(p-1)+1}=\beta P^k}$

The Steenrod algebra has more structure than a graded${\displaystyle {\mathbf{F}}_p}$-algebra. It is also aHopf algebra, so that in particular there is a diagonal orcomultiplication map

${\displaystyle \psi :A\to A\otimes A}$

induced by the Cartan formula for the action of the Steenrod algebra on the cup product. This map is easier to describe than the product map, and is given by

${\displaystyle \psi (S{q}^k)=\sum _{i+j=k}S{q}^i\otimes S{q}^j}$

${\displaystyle \psi (P^k)=\sum _{i+j=k}{P}^i\otimes P^j}$

${\displaystyle \psi (\beta )=\beta \otimes 1+1\otimes \beta }$.

These formulas imply that the Steenrod algebra isco-commutative.

The linear dual of${\displaystyle \psi }$ makes the (graded)linear dual${\displaystyle A_{\ast }}$ ofA into an algebra.John Milnor (1958) proved, for${\displaystyle p=2}$, that${\displaystyle A_{\ast }}$ is apolynomial algebra, with one generator${\displaystyle {\xi }_k}$ of degree${\displaystyle 2^k-1}$, for everyk, and for${\displaystyle p>2}$ the dual Steenrod algebra${\displaystyle A_{\ast }}$ is the tensor product of the polynomial algebra in generators${\displaystyle {\xi }_k}$ of degree${\displaystyle 2p^k-2}$${\displaystyle (k\ge 1)}$ and the exterior algebra in generators τ_k of degree${\displaystyle 2p^k-1}$${\displaystyle (k\ge 0)}$. The monomial basis for${\displaystyle A_{\ast }}$ then gives another choice of basis forA, called the Milnor basis. The dual to the Steenrod algebra is often more convenient to work with, because the multiplication is (super) commutative. The comultiplication for${\displaystyle A_{\ast }}$ is the dual of the product onA; it is given by

${\displaystyle \psi ({\xi }_n)=\sum _{i=0}^n{\xi }_{n-i}^{p^i}\otimes {\xi }_i.}$ where${\displaystyle {\xi }_0=1}$, and

${\displaystyle \psi ({\tau }_n)={\tau }_n\otimes 1+\sum _{i=0}^n{\xi }_{n-i}^{p^i}\otimes {\tau }_i}$ if${\displaystyle p>2}$.

The onlyprimitive elements of${\displaystyle A_{\ast }}$ for${\displaystyle p=2}$ are the elements of the form${\displaystyle {\xi }_1^{2^i}}$, and these are dual to the${\displaystyle S{q}^{2^i}}$ (the only indecomposables ofA).

Thedual Steenrod algebras are supercommutative Hopf algebras, so their spectra are algebra supergroup schemes. These group schemes are closely related to the automorphisms of 1-dimensional additive formal groups. For example, if${\displaystyle p=2}$ then the dual Steenrod algebra is the group scheme of automorphisms of the 1-dimensional additive formal group scheme${\displaystyle x+y}$ that are the identity to first order. These automorphisms are of the form

${\displaystyle x\to x+{\xi }_1{x}^2+{\xi }_2{x}^4+{\xi }_3{x}^8+\cdots }$

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]

${\displaystyle S{q}^{2^i}}$,

we can form subalgebras${\displaystyle {\mathcal{A}}_2(n)}$ generated by the Steenrod squares

${\displaystyle S{q}^1,S{q}^2,\dots ,S{q}^{2^n}}$,

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${\displaystyle {\pi }_{\ast }(ko)}$, and${\displaystyle {\pi }_{\ast }(tmf)}$.^[6]

Larry Smith (2007) gave the following algebraic construction of the Steenrod algebra over afinite field${\displaystyle {\mathbb{F}}_q}$ of orderq. IfV is avector space over${\displaystyle {\mathbb{F}}_q}$ then writeSV for thesymmetric algebra ofV. There is analgebra homomorphism

whereF is theFrobenius endomorphism ofSV. If we put

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

or

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

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

Early applications of the Steenrod algebra were calculations byJean-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 byRené 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 byC. T. C. Wall. A famous application of the Steenrod operations, involving factorizations through secondary cohomology operations associated to appropriate Adem relations, was the solution byJ. Frank Adams of theHopf 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${\displaystyle S^{2n-1}\to S^n}$ ofHopf invariant one, thenn is a power of 2.

The proof uses the fact that each${\displaystyle S{q}^k}$ is decomposable fork 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. Thesingular cochain functor with coefficients in the algebraic closure of${\displaystyle {\mathbb{F}}_p}$ induces acontravariant 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 [[${\displaystyle E_{\mathrm{\infty }}}$-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 thep-component of the stable homotopy groups of spheres. More specifically, the${\displaystyle E_2}$ term of this spectral sequence may be identified as

${\displaystyle {\mathrm{E}\mathrm{x}\mathrm{t}}_A^{s,t}({\mathbb{F}}_p,{\mathbb{F}}_p).}$

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

• Malkiewich, Cary,The Steenrod Algebra(PDF),archived(PDF) from the original on 2017-08-15
• Characteristic classes – contains more calculations, such as for Wu manifolds
• Steenrod squares in Adams spectral sequence – contains interpretations of Ext terms and Streenrod squares

• Reduced power operations in motivic cohomology
• Motivic cohomology withZ/2-coefficients
• Motivic Eilenberg–Maclane spaces
• The homotopy of${\displaystyle \mathbb{C}}$-motivic modular forms – relates${\displaystyle \mathcal{A}//\mathcal{A}(2)}$ to motivic tmf

• Adams, J. Frank (1974).Stable homotopy and generalised homology. Chicago: University of Chicago Press.ISBN 0-226-00523-2.OCLC 1083550.
• Adem, José (1952), "The iteration of the Steenrod squares in algebraic topology",Proceedings of the National Academy of Sciences of the United States of America,38 (8):720–726,Bibcode:1952PNAS...38..720A,doi:10.1073/pnas.38.8.720,ISSN 0027-8424,JSTOR 88494,MR 0050278,PMC 1063640,PMID 16589167
• Bullett, Shaun R.;Macdonald, Ian G. (1982), "On the Adem relations",Topology,21 (3):329–332,doi:10.1016/0040-9383(82)90015-5,ISSN 0040-9383,MR 0649764
• Cartan, Henri (1954), "Sur les groupes d'Eilenberg–Mac Lane. II",Proceedings of the National Academy of Sciences of the United States of America,40 (8):704–707,Bibcode:1954PNAS...40..704C,doi:10.1073/pnas.40.8.704,ISSN 0027-8424,JSTOR 88981,MR 0065161,PMC 534145,PMID 16589542
• Cartan, Henri (1955), "Sur l'itération des opérations de Steenrod",Commentarii Mathematici Helvetici,29 (1):40–58,doi:10.1007/BF02564270,ISSN 0010-2571,MR 0068219,S2CID 124558011
• Cartan, Henri (1954–1955)."Détermination des algèbres${\displaystyle H_{\ast }(\pi ,n;Z_2)}$ et${\displaystyle H^{\ast }(\pi ,n;Z_2)}$ ; groupes stables modulo${\displaystyle p}$"(PDF).Séminaire Henri Cartan (in French).7 (1):1–8.
• Allen Hatcher,Algebraic Topology. Cambridge University Press, 2002. Available free online from theauthor's home page.
• Malygin, S.N.; Postnikov, M.M. (2001) [1994],"Steenrod reduced power",Encyclopedia of Mathematics,EMS Press
• Malygin, S.N.; Postnikov, M.M. (2001) [1994],"Steenrod square",Encyclopedia of Mathematics,EMS Press
• May, J. Peter (1970),"A general algebraic approach to Steenrod operations"(PDF),The Steenrod Algebra and its Applications (Proc. Conf. to Celebrate N. E. Steenrod's Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970), Lecture Notes in Mathematics, vol. 168, Berlin, New York:Springer-Verlag, pp. 153–231,CiteSeerX 10.1.1.205.6640,doi:10.1007/BFb0058524,ISBN 978-3-540-05300-2,MR 0281196
• Milnor, John Willard (1958), "The Steenrod algebra and its dual",Annals of Mathematics, Second Series,67 (1):150–171,doi:10.2307/1969932,ISSN 0003-486X,JSTOR 1969932,MR 0099653
• Mosher, Robert E.; Tangora, Martin C. (2008) [1968],Cohomology operations and applications in homotopy theory, Mineola, NY:Dover Publications,ISBN 978-0-486-46664-4,MR 0226634,OCLC 212909028
• Ravenel, Douglas C. (1986).Complex cobordism and stable homotopy groups of spheres. Orlando:Academic Press.ISBN 978-0-08-087440-1.OCLC 316566772.
• Rudyak, Yuli B. (2001) [1994],"Steenrod algebra",Encyclopedia of Mathematics,EMS Press
• Serre, Jean-Pierre (1953), "Cohomologie modulo 2 des complexes d'Eilenberg–MacLane",Commentarii Mathematici Helvetici,27 (1):198–232,doi:10.1007/BF02564562,ISSN 0010-2571,MR 0060234,S2CID 122407123
• Smith, Larry (2007). "An algebraic introduction to the Steenrod algebra". In Hubbuck, John; Hu'ng, Nguyễn H. V.; Schwartz, Lionel (eds.).Proceedings of the School and Conference in Algebraic Topology. Geometry & Topology Monographs. Vol. 11. pp. 327–348.arXiv:0903.4997.doi:10.2140/gtm.2007.11.327.MR 2402812.S2CID 14167493.
• Steenrod, Norman E. (1947), "Products of cocycles and extensions of mappings",Annals of Mathematics, Second Series,48 (2):290–320,doi:10.2307/1969172,ISSN 0003-486X,JSTOR 1969172,MR 0022071
• Steenrod, Norman E. (1953a), "Homology groups of symmetric groups and reduced power operations",Proceedings of the National Academy of Sciences of the United States of America,39 (3):213–217,Bibcode:1953PNAS...39..213S,doi:10.1073/pnas.39.3.213,ISSN 0027-8424,JSTOR 88780,MR 0054964,PMC 1063756,PMID 16589250
• Steenrod, Norman E. (1953b), "Cyclic reduced powers of cohomology classes",Proceedings of the National Academy of Sciences of the United States of America,39 (3):217–223,Bibcode:1953PNAS...39..217S,doi:10.1073/pnas.39.3.217,ISSN 0027-8424,JSTOR 88781,MR 0054965,PMC 1063757,PMID 16589251
• Steenrod, Norman E.;Epstein, David B. A. (1962), Epstein, David B. A. (ed.),Cohomology operations, Annals of Mathematics Studies, vol. 50,Princeton University Press,ISBN 978-0-691-07924-0,MR 0145525
• Wu, Wen-tsün (1952),Sur les puissances de Steenrod, Colloque de Topologie de Strasbourg, vol. IX, La Bibliothèque Nationale et Universitaire de Strasbourg,MR 0051510

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