Steenrod algebra

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

For a given prime number p, the Steenrod algebra Ap is the graded Hopf algebra over the field Fp of order p, consisting of all stable cohomology operations for mod p cohomology. It is generated by the Steenrod squares introduced by Steenrod (1947) for p=2, and by the Steenrod reduced pth powers introduced in Steenrod (1953) and the Bockstein homomorphism for p>2.

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

Cohomology operations[edit]

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) \to H^{2n}(X;R)
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 Y is a suspension of a space X, the cup product on the cohomology of Y is trivial.) Norman Steenrod constructed stable operations

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

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. 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 \to \mathbf{Z}/p \to \mathbf{Z}/p^2 \to \mathbf{Z}/p \to 0.

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

Axiomatic characterization[edit]

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

  1. Naturality: Sqn is an additive homomorphism from Hm(X,Z/2Z) to Hm+n(X,Z/2Z), and is natural meaning that for any map f : XY, 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: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:

  • Sq1 is the Bockstein homomorphism of the exact sequence 0 \to \mathbf{Z}/2 \to \mathbf{Z}/4 \to \mathbf{Z}/2 \to 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 Hm(X,Z/pZ) to Hm+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 \smile y) = \sum_{i+j=n} (P^i x) \smile (P^j y)

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

Adem relations[edit]

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

Sq^i Sq^j = \sum_{k=0}^{[i/2]} {j-k-1 \choose i-2k} Sq^{i+j-k} Sq^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} = \sum_i (-1)^{a+i}{(p-1)(b-i)-1 \choose a-pi} P^{a+b-i}P^i

for a<pb and

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 apb

Bullett–Macdonald identities[edit]

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

For p=2 put

P(t)=\sum_{i\geq 0}t^i\text{Sq}^i

then the Adem relations are equivalent to

P(s^2+st)\cdot P(t^2)=P(t^2+st)\cdot P(s^2)

For p>2 put

P(t)=\sum_{i\geq 0}t^i\text{P}^i

then the Adem relations are equivalent to the statement that

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

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

Construction[edit]

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

  1. Taking the external product of u with itself n times gives an equivariant cocycle on Xn with coefficients in BB⊗...⊗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 BB⊗...⊗B.
  3. Taking a slant product with c in Hi(E/π,A)gives a cocycle of X with coefficients in H0(π,ABB⊗...⊗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,ABB⊗...⊗B) is also cyclic of order p.

The structure of the Steenrod algebra[edit]

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, \ldots, i_n

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

Sq^I = Sq^{i_1} \ldots Sq^{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

Sq_p^I = Sq_p^{i_1} \ldots Sq_p^{i_n},

such that

i_j\ge pi_{j+1}
i_j\equiv 0,1\bmod 2(p-1)
Sq_p^{2k(p-1)} = P^k
Sq_p^{2k(p-1)+1} = \beta P^k

Hopf algebra structure and the Milnor basis[edit]

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

\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

\psi(Sq^k) = \sum_{i+j=k} Sq^i \otimes Sq^j
\psi(P^k) = \sum_{i+j=k} P^i \otimes P^j
\psi(\beta) = \beta\otimes1+1\otimes\beta.

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 ξk of degree 2k - 1, for every k, and for p>2 the dual Steenrod algebra A* is the tensor product of the polynomial algebra in generators ξk of degree 2pk - 2 (k≥1) and the exterior algebra in generators τk of degree 2pk - 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

\psi(\xi_n) = \sum_{i=0}^n \xi_{n-i}^{p^i} \otimes \xi_i. where ξ0=1, and
\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 \xi_1^{2^i}, and these are dual to the Sq^{2^i} (the only indecomposables of A).

Relation to formal groups[edit]

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\rightarrow x + \xi_1x^2+\xi_2x^4+\xi_3x^8+\cdots

Algebraic construction[edit]

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

P(x):SV[[x]]\rightarrow SV[[x]]

such that

\displaystyle P(x)(v) = v+F(v)x=v+v^qx

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

P(x)(f)=\sum P^i(f)x^i (for p>2)

or

P(x)(f)=\sum Sq^{2i}(f)x^i (for p=2)

for fSV 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[edit]

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 S2n - 1 → Sn of Hopf invariant one, then n is a power of 2.

The proof uses the fact that each Sqk 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[edit]

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

\mathrm{Ext}^{s,t}_{A}(\mathbf{F}_p, \mathbf{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[edit]

References[edit]