There are analogous operations for odd primary coefficients, usually denoted
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
, such that this action behaves well with respect to the stable homotopy category, i.e., there is an isomorphism
As before, the reduced p-th powers also satisfy the Adem relations and commute with the suspension and boundary operators.
were conjectured by Wen-tsün Wu (1952) and established by José Adem (1952).
The Adem relations allow one to write an arbitrary composition of Steenrod squares as a sum of Serre–Cartan basis elements.
Shaun R. Bullett and Ian G. Macdonald (1982) reformulated the Adem relations as the following identities.
put then the Adem relations are equivalent to the statement that is symmetric in
There is a nice straightforward geometric interpretation of the Steenrod squares using manifolds representing cohomology classes.
can now be understood — they are the pushforward of the Stiefel–Whitney class 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
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.
, there are only the following non-trivial cohomology groups, as can be computed using a cellular decomposition.
we know that The Cartan relation implies that the total square is a ring homomorphism Hence Since there is only one degree
The Steenrod squares and reduced powers are special cases of this construction where
) described the structure of the Steenrod algebra of stable mod
This basis relies on a certain notion of admissibility for integer sequences.
consisting of the elements such that The Steenrod algebra has more structure than a graded
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.
The dual to the Steenrod algebra is often more convenient to work with, because the multiplication is (super) commutative.
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
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
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.
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.
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.
The singular cochain functor with coefficients in the algebraic closure of
induces a contravariant equivalence from the homotopy category of connected
term for the (p-local) Adams spectral sequence, whose abutment is the p-component of the stable homotopy groups of spheres.
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."