Steenrod algebra
Encyclopedia
In algebraic topology
, a Steenrod algebra was defined by to be the algebra of stable cohomology operation
s 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 operation
s for mod p cohomology
. It is generated by the Steenrod squares introduced by for p=2, and by the Steenrod reduced pth powers introduced in 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.
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 Y is a suspension of a space X, the cup product on the cohomology of Y is trivial.) Norman Steenrod
constructed stable operations
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
In the case p=2, the Bockstein element is Sq1 and the reduced p′th power Pi is Sq2i.
In addition the Steenrod squares have the following properties:
Similarly the following axioms characterize the reduced p-th powers for p > 2.
As before, the reduced pth powers also satisfy Adem relations and commute with the suspension and boundary operators.
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
for a
for a≤pb
For p=2 put
then the Adem relations are equivalent to
For p>2 put
then the Adem relations are equivalent to the statement that
is symmetric in s and t. Here β is the Bockstein operation and (Ad β)P = βP−Pβ.
showed how to construct a reduced power un/c in Hkq−i(X,(A⊗B⊗B⊗...⊗B)/π) as follows.
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⊗...⊗B) is also cyclic of order p.
is admissible if for each j, ij ≥ 2ij+1. Then the elements
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
such that
, so that in particular there is a diagonal or comultiplication map
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
The linear dual of ψ makes the (graded) linear dual
A* of A into an algebra. 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
where ξ0=1, and
if p>2
The only primitive element
s of A* for p=2 are the
, and these are dual to the 
(the only indecomposables of A).
such that
for v∈V,
where F is the Frobenius endomorphism of SV.
If we put
(for p>2)
or
(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.
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 operation
s. 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.
, 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
This is what is meant by the aphorism "the cohomology of the Steenrod algebra is an approximation to the stable homotopy groups of spheres."
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, a Steenrod algebra was defined by to be the algebra of stable cohomology operation
Cohomology operation
In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if F is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from...
s for mod p cohomology.
For a given prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
p, the Steenrod algebra Ap is the graded Hopf algebra
Hopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally...
over the field Fp of order p, consisting of all stable cohomology operation
Cohomology operation
In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if F is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from...
s for mod p cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
. It is generated by the Steenrod squares introduced by for p=2, and by the Steenrod reduced pth powers introduced in and the Bockstein homomorphism
Bockstein homomorphism
In homological algebra, the Bockstein homomorphism, introduced by , is a connecting homomorphism associated with a short exact sequenceof abelian groups, when they are introduced as coefficients into a chain complex C, and which appears in the homology groups as a homomorphism reducing degree by...
for 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 transformationNatural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
between cohomology functors. For example, if we take cohomology with coefficients in a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
, the cup product
Cup product
In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative graded commutative product operation in cohomology, turning the cohomology of a space X into a...
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 Y is a suspension of a space X, the cup product on the cohomology of Y is trivial.) Norman Steenrod
Norman Steenrod
Norman Earl Steenrod was a preeminent mathematician most widely known for his contributions to the field of algebraic topology.-Life:...
constructed stable operations
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
In the case p=2, the Bockstein element is Sq1 and the reduced p′th power Pi is Sq2i.
Axiomatic characterization
showed that the Steenrod squares Sqn:Hm→Hm+n are characterized by the following 5 axioms:- 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 : X → Y, f*(Sqnx) = Sqnf*(x).
- Sq0 is the identity homomorphism.
- Sqn is the cup square on classes of degree n.
- If n>dim(X) then Sqn(x) = 0
- Cartan Formula:
In addition the Steenrod squares have the following properties:
- Sq1 is the Bockstein homomorphism of the exact sequence
- 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.
- Naturality: Pn is an additive homomorphism from Hm(X,Z/pZ) to Hm+2n(p−1)(X,Z/pZ), and is natural.
- P0 is the identity homomorphism.
- Pn is the cup p′th power on classes of degree 2n.
- If 2n>dim(X) then Pn(x) = 0
- Cartan Formula:
As before, the reduced pth powers also satisfy Adem relations and commute with the suspension and boundary operators.
Adem relations
The Adem relations for p=2 were conjectured by and proved by and are given byfor 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
for a
for a≤pb
Bullett–Macdonald identities
reformulated the Adem relations as the following Bullett–Macdonald identities.For p=2 put
then the Adem relations are equivalent to
For p>2 put
then the Adem relations are equivalent to the statement that
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 Hq(X,B), A an abelian group acted on by π, and c a cohomology class in Hi(π,A).showed how to construct a reduced power un/c in Hkq−i(X,(A⊗B⊗B⊗...⊗B)/π) as follows.
- Taking the external product of u with itself n times gives an equivariant cocycle on Xn with coefficients in B⊗B⊗...⊗B.
- 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⊗...⊗B.
- Taking a slant product with c in Hi(E/π,A)gives a cocycle of X with coefficients in H0(π,A⊗B⊗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⊗...⊗B) is also cyclic of order p.
The structure of the Steenrod algebra
(for p=2) and (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 sequenceis admissible if for each j, ij ≥ 2ij+1. Then the elements
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
such that
Hopf algebra structure and the Milnor basis
The Steenrod algebra has more structure than a graded Fp-algebra. It is also a Hopf algebraHopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally...
, so that in particular there is a diagonal or comultiplication map
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
The linear dual of ψ makes the (graded) linear dual
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
A* of A into an algebra. 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
where ξ0=1, and if p>2The only primitive element
Primitive element
In mathematics, the term primitive element can mean:* Primitive root modulo n, in number theory* Primitive element , an element that generates a given field extension...
s 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 formAlgebraic construction
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)such that
for v∈V,
where F is the Frobenius endomorphism of SV.
If we put
(for p>2)or
(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 spheresVector fields on spheres
In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras....
problem. Independently Milnor and Bott, as well as Kervaire, gave a second solution of the Hopf invariant one problem, using operations in K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
; these are the Adams operation
Adams operation
In mathematics, an Adams operationis a cohomology operation in topological K-theory, or any allied operation in algebraic K-theory or other types of algebraic construction, defined on a pattern introduced by Frank Adams...
s. 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
The cohomology of the Steenrod algebra is the E2 term for the (p-local) Adams spectral sequenceAdams spectral sequence
In mathematics, the Adams spectral sequence is a spectral sequence introduced by . Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory...
, 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
This is what is meant by the aphorism "the cohomology of the Steenrod algebra is an approximation to the stable homotopy groups of spheres."