Steenrod algebra

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:

H n ( X ; R ) → H 2 n ( X ; R ) x ↦ x ⌣ 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 Y is a suspension of a space X, the cup product on the cohomology of Y is trivial.) Norman Steenrod constructed stable operations

S q i : H n ( X ; Z / 2 ) → H n + i ( X ; Z / 2 )

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

P i : H n ( X ; Z / p ) → H n + 2 i ( p − 1 ) ( X ; Z / p )

The Sqⁱ generate a connected graded algebra over Z/2, where the multiplication is given by composition of operations. This is the mod 2 Steenrod algebra. In the case p > 2, the mod p Steenrod algebra is generated by the Pⁱ and the Bockstein operation β associated to the short exact sequence

0 → Z / p → Z / p 2 → Z / p → 0.

In the case p=2, the Bockstein element is Sq¹ and the reduced p-th power Pⁱ is Sq²ⁱ.

Axiomatic characterization

Steenrod & Epstein (1962) showed that the Steenrod squares Sqⁿ:H^m→H^m+n are characterized by the following 5 axioms:

1. Naturality: Sqⁿ is an additive homomorphism from H^m(X,Z/2Z) to H^m+n(X,Z/2Z), and is natural meaning that for any map f : X → Y, f*(Sqⁿx) = Sqⁿf*(x).
2. Sq⁰ is the identity homomorphism.
3. Sqⁿ is the cup square on classes of degree n.
4. If n>deg(x) then Sqⁿ(x) = 0
5. Cartan Formula: S q n ( x ⌣ y ) = ∑ i + j = n ( S q i x ) ⌣ ( S q j y )

In addition the Steenrod squares have the following properties:

Sq¹ is the Bockstein homomorphism of the exact sequence 0 → Z / 2 → Z / 4 → Z / 2 → 0.

They satisfy the Adem relations, described below.

They commute with the suspension homomorphism and the boundary operator.

Similarly the following axioms characterize the reduced p-th powers for p > 2.

1. Naturality: Pⁿ is an additive homomorphism from H^m(X,Z/pZ) to H^m+2n(p−1)(X,Z/pZ), and is natural.
2. P⁰ is the identity homomorphism.
3. Pⁿ is the cup p-th power on classes of degree 2n.
4. If 2n>dim(X) then Pⁿ(x) = 0
5. Cartan Formula: P n ( x ⌣ y ) = ∑ i + j = n ( P i x ) ⌣ ( P j y )

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

Adem relations

The Adem relations for p=2 were conjectured by Wu (1952) and proved by José Adem (1952) and are given by

S q i S q j = ∑ k = 0 [ i / 2 ] ( j − k − 1 i − 2 k ) S q i + j − k S q k

for all i, j > 0 such that 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 p the Adem relations are

P a P b = ∑ i ( − 1 ) a + i ( ( p − 1 ) ( b − i ) − 1 a − p i ) P a + b − i P i

for a<pb and

P a β P b = ∑ i ( − 1 ) a + i ( ( p − 1 ) ( b − i ) a − p i ) β P a + b − i P i + ∑ i ( − 1 ) a + i + 1 ( ( p − 1 ) ( b − i ) − 1 a − p i − 1 ) P a + b − i β P i

for a≤pb

Bullett–Macdonald identities

Bullett & Macdonald (1982) reformulated the Adem relations as the following identities.

For p=2 put

P ( t ) = ∑ i ≥ 0 t i Sq i

then the Adem relations are equivalent to

P ( s 2 + s t ) ⋅ P ( t 2 ) = P ( t 2 + s t ) ⋅ P ( s 2 )

For p>2 put

P ( t ) = ∑ i ≥ 0 t i P i

then the Adem relations are equivalent to the statement that

( 1 + s Ad β ) P ( t p + t p − 1 s + ⋯ + t s p − 1 ) P ( s p )

is symmetric in s and t. Here β is the Bockstein operation and (Ad β)P = βP−Pβ.

Construction

Suppose that π is any degree n subgroup of the symmetric group on n points, u a cohomology class in H^q(X, B), A an abelian group acted on by π, and c a cohomology class in H_i(π, A). Steenrod (1953) showed how to construct a reduced power uⁿ/c in H^kq−i(X, (A ⊗ B ⊗ ... ⊗ B) / π) as follows.

1. Taking the external product of u with itself n times gives an equivariant cocycle on Xⁿ with coefficients in B ⊗ ... ⊗ B.
2. Choose E to be a contractible space on which π acts freely and an equivariant map from E × X to Xⁿ. Pulling back uⁿ by this map gives an equivariant cocyle on E × X and therefore a cocycle of E / π × X with coefficients in B ⊗ ... ⊗ B.
3. Taking a slant product with c in H_i(E / π, A) gives a cocycle of X with coefficients in H₀(π, A ⊗ B ⊗ ... ⊗ B)

The Steenrod squares and reduced powers are special cases of this construction where π is a cyclic group of prime order p=n acting as a cyclic permutation of n elements, and the groups A and B are cyclic of order p, so that H₀(π, A ⊗ B ⊗ ... ⊗ B) is also cyclic of order p.

The structure of the Steenrod algebra

Serre (1953) (for p=2) and Cartan (1954, 1955) (for p>2) described the structure of the Steenrod algebra of stable mod 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

i 1 , i 2 , … , i n

is admissible if for each j, i_j ≥ 2i_j+1. Then the elements

S q I = S q i 1 … S q i n ,

where 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 p > 2 consisting of the elements

S q p I = S q p i 1 … S q p i n ,

such that

i j ≥ p i j + 1 i j ≡ 0 , 1 mod 2 ( p − 1 ) S q p 2 k ( p − 1 ) = P k S q p 2 k ( p − 1 ) + 1 = β P k

Hopf algebra structure and the Milnor basis

The Steenrod algebra has more structure than a graded F_p-algebra. It is also a Hopf algebra, so that in particular there is a diagonal or comultiplication map

ψ : A → A ⊗ 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

ψ ( S q k ) = ∑ i + j = k S q i ⊗ S q j ψ ( P k ) = ∑ i + j = k P i ⊗ P j ψ ( β ) = β ⊗ 1 + 1 ⊗ β .

The linear dual of ψ makes the (graded) linear dual A_* of A into an algebra. Milnor (1958) proved, for p = 2, that A_* is a polynomial algebra, with one generator ξ_k of degree 2^k - 1, for every k, and for p>2 the dual Steenrod algebra A_* is the tensor product of the polynomial algebra in generators ξ_k of degree 2p^k - 2 (k≥1) and the exterior algebra in generators τ_k of degree 2p^k - 1 (k≥0). The monomial basis for 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 A_* is the dual of the product on A; it is given by

ψ ( ξ n ) = ∑ i = 0 n ξ n − i p i ⊗ ξ i . where ξ₀=1, and ψ ( τ n ) = τ n ⊗ 1 + ∑ i = 0 n ξ n − i p i ⊗ τ i if p>2

The only primitive elements of A_* for p=2 are the ξ 1 2 i , and these are dual to the S q 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

x → x + ξ 1 x 2 + ξ 2 x 4 + ξ 3 x 8 + ⋯

Algebraic construction

Smith (2007) gave the following algebraic construction of the Steenrod algebra over a finite field F_q of order q. If V is a vector space over F_q then write SV for the symmetric algebra of V. There is an algebra homomorphism P(x)

P ( x ) : S V [ [ x ] ] → S V [ [ x ] ]

such that

P ( x ) ( v ) = v + F ( v ) x = v + v q x

for v∈V, where F is the Frobenius endomorphism of SV. If we put

P ( x ) ( f ) = ∑ P i ( f ) x i (for p>2)

or

P ( x ) ( f ) = ∑ S q 2 i ( f ) x i (for p=2)

for f∈SV then if V is infinite dimensional the elements Pⁱ 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 Sq²ⁱ for 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 Bott, as well as 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 S^{2n - 1} → Sⁿ of Hopf invariant one, then n is a power of 2.

The proof uses the fact that each 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 E₂ term for the (p-local) Adams spectral sequence, whose abutment is the p-component of the stable homotopy groups of spheres. More specifically, the E₂ term of this spectral sequence may be identified as

E x t A s , t ( F p , 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."