Steenrod algebra

In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod formula_1 cohomology.
For a given prime number formula_1, the Steenrod algebra formula_3 is the graded Hopf algebra over the field formula_4 of order formula_1, consisting of all stable cohomology operations for mod formula_1 cohomology. It is generated by the Steenrod squares introduced by for formula_7, and by the Steenrod reduced formula_1th powers introduced in and the Bockstein homomorphism for formula_9.
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 formula_10, 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 formula_13 is a suspension of a space formula_14, the cup product on the cohomology of formula_13 is trivial.) Steenrod constructed stable operations
for all formula_17 greater than zero. The notation formula_18 and their name, the Steenrod squares, comes from the fact that formula_19 restricted to classes of degree formula_20 is the cup square. There are analogous operations for odd primary coefficients, usually denoted formula_21 and called the reduced formula_1-th power operations:
The formula_24 generate a connected graded algebra over formula_25, where the multiplication is given by composition of operations. This is the mod 2 Steenrod algebra. In the case formula_26, the mod formula_1 Steenrod algebra is generated by the formula_21 and the Bockstein operation formula_29 associated to the short exact sequence
In the case formula_7, the Bockstein element is formula_32 and the reduced formula_1-th power formula_21 is formula_35.
As a cohomology ring.
We can summarize the properties of the Steenrod operations as generators in the cohomology ring of Eilenberg–Maclane spectra
since there is an isomorphism
giving a direct sum decomposition of all possible cohomology operations with coefficients in formula_4. Note the inverse limit of cohomology groups appears because it is a computation in the "stable range" of cohomology groups of Eilenberg–Maclane spaces. This result was originally computedpg 277 by Cartanpg 7 and Serre.
Note there is a dual characterizationpg 279 using homology for the dual Steenrod algebra.
Remark about generalizing to generalized cohomology theories.
It should be observed if the Eilenberg–Maclane spectrum formula_39 is replaced by an arbitrary spectrum formula_40, then there are many challenges for studying the cohomology ring formula_41. In this case, the generalized dual Steenrod algebra formula_42 should be considered instead because it has much better properties and can be tractably studied in many cases (such as formula_43)pg 280. In fact, these ring spectra are commutative and the formula_44 bimodules formula_42 are flat. In this case, these is a canonical action of formula_42 on formula_47 for any space formula_14, such that this action behaves well with respect to the stable homotopy category, i.e., there is an isomorphism formula_49 hence we can use the ring structure formula_50 to get an action of formula_42 on formula_47.
Axiomatic characterization.
 showed that the Steenrod squares formula_53 are characterized by the following 5 axioms:
In addition the Steenrod squares have the following properties: 
Similarly the following axioms characterize the reduced formula_1-th powers for formula_26.
As before, the reduced "p"-th powers also satisfy the Ádem relations and commute with the suspension and boundary operators.
Ádem relations.
The Ádem relations for formula_7 were conjectured by and established by . They are given by
for all formula_80 such that formula_81. (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 formula_1 the Ádem relations are
for "a"<"pb" and
for formula_85.
Bullett–Macdonald identities.
 reformulated the Ádem relations as the following identities.
For formula_7 put
then the Ádem relations are equivalent to 
For formula_26 put
then the Ádem relations are equivalent to the statement that 
is symmetric in formula_92 and formula_93. Here formula_29 is the Bockstein operation and formula_95.
Geometric interpretation.
There is a nice straightforward geometric interpretation of the Steenrod squares using manifolds representing cohomology classes. Suppose formula_14 is a smooth manifold and consider a cohomology class formula_97 represented geometrically as a smooth submanifold formula_98. Cohomologically, if we let formula_99 represent the fundamental class of formula_13 then the pushforward map
gives a representation of formula_102. In addition, associated to this immersion is a real vector bundle call the normal bundle formula_103. The Steenrod squares of formula_102 can now be understood — they are the pushforward of the Stiefel–Whitney classes of the normal bundle
which gives a geometric reason for why the Steenrod products eventually vanish. Note that because the Steenrod maps are group homomorphisms, if we have a class formula_29 which can be represented as a sum
where the formula_108 are represented as manifolds, we can interpret the squares of the classes as sums of the pushforwards of the normal bundles of their underlying smooth manifolds, i.e.,
Also, this equivalence is strongly related to the Wu formula.
Computations.
Complex projective spaces.
On the complex projective plane formula_110, there are only the following non-trivial cohomology groups,
as can be computed using a cellular decomposition. This implies that the only possible non-trivial Steenrod product is formula_112 on formula_113 since it gives the cup product on cohomology. As the cup product structure on formula_114 is nontrivial, this square is nontrivial. There is a similar computation on the complex projective space formula_115, where the only non-trivial squares are formula_57 and the squaring operations formula_117 on the cohomology groups formula_118 representing the cup product. In formula_119 the square
can be computed using the geometric techniques outlined above and the relation between Chern classes and Steifel–Whitney classes; note that formula_121 represents the non-zero class in formula_122. It can also be computed directly using the Cartan formula since formula_123 and
Infinite Real Projective Space.
The Steenrod operations for real projective spaces can be readily computed using the formal properties of the Steenrod squares. Recall that
where formula_126 For the operations on formula_127 we know that
The Cartan relation implies that the total square
is a ring homomorphism
Hence
Since there is only one degree formula_132 component of the previous sum, we have that
Construction.
Suppose that formula_134 is any degree formula_20 subgroup of the symmetric group on formula_20 points, formula_137 a cohomology class in formula_138, formula_139 an abelian group acted on by formula_134, and formula_141 a cohomology class in formula_142. showed how to construct a reduced power formula_143 in formula_144, as follows.
The Steenrod squares and reduced powers are special cases of this construction where formula_134 is a cyclic group of prime order formula_162 acting as a cyclic permutation of formula_20 elements, and the groups formula_139 and formula_165 are cyclic of order formula_1, so that formula_160 is also cyclic of order formula_1.
Properties of the Steenrod algebra.
In addition to the axiomatic structure the Steenrod algebra satisfies, it has a number of additional useful properties.
Basis for the Steenrod algebra.
 (for formula_7) and (for formula_9) described the structure of the Steenrod algebra of stable mod formula_1 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 formula_173, we have that formula_174. Then the elements
where formula_176 is an admissible sequence, form a basis (the Serre–Cartan basis) for the mod 2 Steenrod algebra, called the admissible basis. There is a similar basis for the case formula_9 consisting of the elements
such that
Hopf algebra structure and the Milnor basis.
The Steenrod algebra has more structure than a graded formula_183-algebra. It is also a Hopf algebra, 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. This map 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 formula_188 makes the (graded) linear dual formula_189 of "A" into an algebra. proved, for formula_190, that formula_189 is a polynomial algebra, with one generator formula_192 of degree formula_193, for every "k", and for formula_26 the dual Steenrod algebra formula_189 is the tensor product of the polynomial algebra in generators formula_192 of degree formula_197 formula_198 and the exterior algebra in generators τk of degree formula_199 formula_200. The monomial basis for formula_189 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 formula_189 is the dual of the product on "A"; it is given by
The only primitive elements of formula_189 for formula_7 are the elements of the form formula_209, and these are dual to the formula_210 (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 formula_7 then the dual Steenrod algebra is the group scheme of automorphisms of the 1-dimensional additive formal group scheme formula_212 that are the identity to first order. These automorphisms are of the form 
Finite sub-Hopf algebras.
The formula_190 Steenrod algebra admits a filtration by finite sub-Hopf algebras. As formula_215 is generated by the elements pg 47
we can form subalgebras formula_217 generated by the Steenrod squares
giving the filtration
These algebras are significant because they can be used to simplify many Adams spectral sequence computations, such as for formula_220, and formula_221pg 63-67.
Algebraic construction.
 gave the following algebraic construction of the Steenrod algebra over a finite field formula_222 of order "q". If "V" is a vector space over formula_222 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
or
for formula_227 then if "V" is infinite dimensional the elements formula_228 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 formula_117 for formula_190.
Applications.
Early applications of the Steenrod algebra were calculations by Jean-Pierre Serre of some homotopy groups of spheres, using the compatibility of transgressive differentials in the Serre spectral sequence with the Steenrod operations, and the classification by René Thom of smooth manifolds up to cobordism, through the identification of the graded ring of bordism classes with the homotopy groups of Thom complexes, in a stable range. The latter was refined to the case of oriented manifolds by C. T. C. Wall. A famous application of the Steenrod operations, involving factorizations through secondary cohomology operations associated to appropriate Adem relations, was the solution by J. Frank Adams of the Hopf invariant one problem. One application of the mod 2 Steenrod algebra that is fairly elementary is the following theorem.
Theorem. If there is a map formula_231 of Hopf invariant one, then "n" is a power of 2.
The proof uses the fact that each formula_232 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 formula_233 term for the ("p"-local) Adams spectral sequence, whose abutment is the "p"-component of the stable homotopy groups of spheres. More specifically, the formula_233 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."