Steenrod algebra

In algebraic topology, a Steenrod algebra was defined by Henri Cartan (1955) to be the algebra of stable cohomology operations for mod ${\displaystyle p}$ cohomology.

For a given prime number ${\displaystyle p}$, the Steenrod algebra ${\displaystyle A_{p}}$ is the graded Hopf algebra over the field ${\displaystyle \mathbb {F} _{p}}$ of order ${\displaystyle p}$, consisting of all stable cohomology operations for mod ${\displaystyle p}$ cohomology. It is generated by the Steenrod squares introduced by Norman Steenrod (1947) for ${\displaystyle p=2}$, and by the Steenrod reduced ${\displaystyle p}$th powers introduced in Steenrod (1953) harvtxt error: multiple targets (2×): CITEREFSteenrod1953 (help) and the Bockstein homomorphism for ${\displaystyle p>2}$.

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

Cohomology operations

A cohomology operation is a natural transformation between cohomology functors. For example, if we take cohomology with coefficients in a ring, 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

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

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

Axiomatic characterization

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 functorial 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 Ádem 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 ${\displaystyle 2n>\deg(x)}$ 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 Ádem relations and commute with the suspension and boundary operators.

Ádem relations

The Ádem relations for ${\displaystyle p=2}$ were conjectured by Wen-tsün Wu (1952) and established by José Ádem (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 Ádem relations allow one to write an arbitrary composition of Steenrod squares as a sum of Serre–Cartan basis elements.

For odd ${\displaystyle p}$ the Ádem 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 ${\displaystyle a\leq pb}$.

Bullett–Macdonald identities

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

For ${\displaystyle p=2}$ put

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

then the Ádem 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

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

then the Ádem relations are equivalent to the statement that

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

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

Computations

Infinite Real Projective Space

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

${\displaystyle H^{*}(\mathbb {RP} ^{\infty };\mathbb {Z} /2)\cong \mathbb {Z} /2[x],}$

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

${\displaystyle {\begin{aligned}Sq^{0}(x)&=x\\Sq^{1}(x)&=x^{2}\\Sq^{k}(x)&=0&&{\text{ for any }}k>1\end{aligned}}}$

Using the operation

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

we note that the Cartan relation implies that

${\displaystyle Sq\colon H^{*}(X)\to H^{*}(X)}$

is a ring morphism. Hence

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

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

${\displaystyle Sq^{i}(x^{n})={n \choose i}x^{n+i}}$

Construction

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 (1953) harvtxt error: multiple targets (2×): CITEREFSteenrod1953 (help) 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 ${\displaystyle p}$.

The structure of the Steenrod algebra

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 Ádem 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

${\displaystyle i_{1},i_{2},\ldots ,i_{n}}$

is admissible if for each ${\displaystyle j}$, we have that ${\displaystyle i_{j}\geq 2i_{j+1}}$. Then the elements

${\displaystyle Sq^{I}=Sq^{i_{1}}\cdots Sq^{i_{n}},}$

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

${\displaystyle Sq_{p}^{I}=Sq_{p}^{i_{1}}\cdots Sq_{p}^{i_{n}},}$

such that

${\displaystyle i_{j}\geq pi_{j+1}}$

${\displaystyle i_{j}\equiv 0,1{\bmod {2}}(p-1)}$

${\displaystyle Sq_{p}^{2k(p-1)}=P^{k}}$

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

Hopf algebra structure and the Milnor basis

The Steenrod algebra has more structure than a graded ${\displaystyle \mathbf {F} _{p}}$-algebra. It is also a Hopf algebra, so that in particular there is a diagonal or comultiplication map

${\displaystyle \psi \colon A\to A\otimes A.}$

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

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

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

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

These formulas imply that the Steenrod algebra is co-commutative.

The linear dual of ${\displaystyle \psi }$ makes the (graded) linear dual ${\displaystyle A_{*}}$ of A into an algebra. John Milnor (1958) proved, for ${\displaystyle p=2}$, that ${\displaystyle A_{*}}$ is a polynomial algebra, with one generator ${\displaystyle \xi _{k}}$ of degree ${\displaystyle 2^{k}-1}$, for every k, and for ${\displaystyle p>2}$ the dual Steenrod algebra ${\displaystyle A_{*}}$ is the tensor product of the polynomial algebra in generators ${\displaystyle \xi _{k}}$ of degree ${\displaystyle 2p^{k}-2}$ ${\displaystyle (k\geq 1)}$ and the exterior algebra in generators τ_k of degree ${\displaystyle 2p^{k}-1}$ ${\displaystyle (k\geq 0)}$. The monomial basis for ${\displaystyle A_{*}}$ then gives another choice of basis for A, 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 ${\displaystyle A_{*}}$ is the dual of the product on A; it is given by

${\displaystyle \psi (\xi _{n})=\sum _{i=0}^{n}\xi _{n-i}^{p^{i}}\otimes \xi _{i}.}$ where ξ₀=1, and

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

The only primitive elements of A_* for p=2 are the ${\displaystyle \xi _{1}^{2^{i}}}$, and these are dual to the ${\displaystyle Sq^{2^{i}}}$ (the only indecomposables of A).

Relation to formal groups

The dual 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 p=2 then the dual Steenrod algebra is the group scheme of automorphisms of the 1-dimensional additive formal group scheme x+y that are the identity to first order. These automorphisms are of the form

${\displaystyle x\rightarrow x+\xi _{1}x^{2}+\xi _{2}x^{4}+\xi _{3}x^{8}+\cdots }$

Algebraic construction

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 ${\displaystyle p=2}$.

Applications

The most famous early applications of the Steenrod algebra to outstanding topological problems were the solutions by J. Frank Adams of the Hopf invariant one problem and the vector fields on spheres problem. Independently Milnor and Raoul Bott, as well as Michel Kervaire, gave a second solution of the Hopf invariant one problem, using operations in K-theory; these are the Adams operations. 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.

Connection to the Adams spectral sequence and the homotopy groups of spheres

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."

See also

• Pontryagin cohomology operation

References

Pedagogical

• Malkiewich, Cary, The Steenrod Algebra (PDF), archived from the original on 2017-08-15CS1 maint: BOT: original-url status unknown (link)

References

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• Bullett, Shaun R.; Macdonald, Ian G. (1982), "On the Ádem relations", Topology. An International Journal of Mathematics, 21 (3): 329–332, doi:10.1016/0040-9383(82)90015-5, ISSN 0040-9383, MR 0649764
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