For a given prime number , the Steenrod algebra is the graded Hopf algebra over the field of order , consisting of all stable cohomology operations for mod cohomology. It is generated by the Steenrod squares introduced by Norman Steenrod (1947) for , and by the Steenrod reduced th powers introduced in Steenrod (1953) harvtxt error: multiple targets (2×): CITEREFSteenrod1953 (help) and the Bockstein homomorphism for .
The term "Steenrod algebra" is also sometimes used for the algebra of cohomology operations of a generalized cohomology theory.
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:
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 is a suspension of a space , the cup product on the cohomology of is trivial.) Steenrod constructed stable operations
for all greater than zero. The notation and their name, the Steenrod squares, comes from the fact that restricted to classes of degree is the cup square. There are analogous operations for odd primary coefficients, usually denoted and called the reduced -th power operations:
The generate a connected graded algebra over , where the multiplication is given by composition of operations. This is the mod 2 Steenrod algebra. In the case , the mod Steenrod algebra is generated by the and the Bockstein operation associated to the short exact sequence
In the case , the Bockstein element is and the reduced -th power is .
- Naturality: is an additive homomorphism and is functorial with respect to any so .
- is the identity homomorphism.
- for .
- If then
- Cartan Formula:
In addition the Steenrod squares have the following properties:
- is the Bockstein homomorphism of the exact sequence
- commutes with the connecting morphism of the long exact sequence in cohomology. In particular, it commutes with respect to suspension
- They satisfy the Ádem relations, described below
Similarly the following axioms characterize the reduced -th powers for .
- Naturality: is an additive homomorphism and natural.
- is the identity homomorphism.
- is the cup -th power on classes of degree .
- If then
- Cartan Formula:
As before, the reduced p-th powers also satisfy Ádem relations and commute with the suspension and boundary operators.
for all such that . (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 the Ádem relations are
for a<pb and
then the Ádem relations are equivalent to
then the Ádem relations are equivalent to the statement that
is symmetric in and . Here is the Bockstein operation and .
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
where For the operations on we know that
Using the operation
we note that the Cartan relation implies that
is a ring morphism. Hence
Since there is only one degree component of the previous sum, we have that
Suppose that is any degree subgroup of the symmetric group on points, a cohomology class in , an abelian group acted on by , and a cohomology class in . Steenrod (1953) harvtxt error: multiple targets (2×): CITEREFSteenrod1953 (help) showed how to construct a reduced power in , as follows.
- Taking the external product of with itself times gives an equivariant cocycle on with coefficients in .
- Choose to be a contractible space on which acts freely and an equivariant map from to Pulling back by this map gives an equivariant cocycle on and therefore a cocycle of with coefficients in .
- Taking the slant product with in gives a cocycle of with coefficients in .
The Steenrod squares and reduced powers are special cases of this construction where is a cyclic group of prime order acting as a cyclic permutation of elements, and the groups and are cyclic of order , so that is also cyclic of order .
The structure of the Steenrod algebra
Jean-Pierre Serre (1953) (for ) and Henri Cartan (1954, 1955) (for ) described the structure of the Steenrod algebra of stable mod 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
is admissible if for each , we have that . Then the elements
where 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 consisting of the elements
Hopf algebra structure and the Milnor basis
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
These formulas imply that the Steenrod algebra is co-commutative.
The linear dual of makes the (graded) linear dual of A into an algebra. John Milnor (1958) proved, for , that is a polynomial algebra, with one generator of degree , for every k, and for the dual Steenrod algebra is the tensor product of the polynomial algebra in generators of degree and the exterior algebra in generators τk of degree . The monomial basis for 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 is the dual of the product on A; it is given by
- where ξ0=1, and
- if p>2
The only primitive elements of A* for p=2 are the , and these are dual to the (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
Larry Smith (2007) gave the following algebraic construction of the Steenrod algebra over a finite field of order q. If V is a vector space over then write SV for the symmetric algebra of V. There is an algebra homomorphism
where F is the Frobenius endomorphism of SV. If we put
for then if V is infinite dimensional the elements 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 for .
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 of Hopf invariant one, then n is a power of 2.
The proof uses the fact that each 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 term for the (p-local) Adams spectral sequence, whose abutment is the p-component of the stable homotopy groups of spheres. More specifically, the term of this spectral sequence may be identified as
This is what is meant by the aphorism "the cohomology of the Steenrod algebra is an approximation to the stable homotopy groups of spheres."
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