In algebraic topology, a branch of mathematics, the Steenrod algebra is a structure occurring in the theory of cohomology operations. It is an object of great importance, most especially to homotopy theorists. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... Mathematics is the study of quantity, structure, space and change. ... 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 F to itself. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...

More precisely, for a given prime number p, it is a graded algebra over the field Z/p, the integers modulo p. Briefly, it is the algebra of all stable cohomology operations for mod p singular cohomology. It is generated by the Steenrod reduced pth powers, or Steenrod squares if p=2. The requirements of calculations of homotopy groups mean that homological algebra over the Steenrod algebra must be considered. In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ... 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 F to itself. ... In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ... In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ...

Cohomology operations

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: In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... 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. ...

Note that cohomology operations need not be group homomorphisms.

The utility of these operations is limited, because they 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 In topology, the suspension SX of a topological space X is the quotient space: of the product of X with the unit interval I = [0, 1]. Intuitively we make X into a cylinder and collapse both ends to two points. ... Norman Earl Steenrod (April 22, 1910–October 14, 1971) was a leading mathematician, working in the field of topology. ...

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 mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...

(The Bockstein occurs also in the mod 2 case, as Sq¹.)

Axiomatic characterization

The Steenrod squares Sqⁿ satisfy the following axioms:

1. Naturality: For any map f : X → Y, f*(Sqⁿx) = Sqⁿf*(x).
2. Additivity: Sqⁿ(x + y) = Sqⁿ(x) + Sqⁿ(y).
3. Cartan Formula:
4. Stability: The squares commute with the suspension isomorphism (if we are careful).
5. Sqⁿ is the cup square on classes of degree n.
6. Sq⁰ is the identity homomorphism.
7. Sq¹ is the Bockstein homomorphism of the exact sequence

Together with the Adem relations, defined below, these axioms characterize the Steenrod squares uniquely. Similar axioms apply to the reduced p-th powers for p > 2.

Adem relations and the Serre-Cartan basis

One of the first questions about the Steenrod algebra is, when is a composition of operations nonzero? The ring structure of the Steenrod algebra is exceedingly intricate. Indeed, as described below, its cohomology may be viewed as an approximation to the stable homotopy groups of spheres, objects of modern mathematics famous for being hard to identify. Jean-Pierre Serre and Henri Cartan found a good basis for the Steenrod algebra by examining the Adem relations, named for their discoverer José Adem. This basis relies on a certain notion of admissibility for integer sequences. We say a sequence In mathematics, the homotopy groups of spheres are the groups πk(Sn) in algebraic topology, more specifically homotopy theory, where πk(.) for k ≥ 1 denotes the homotopy group and Sn the n-sphere. ... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... Henri Cartan (born July 8, 1904) is a son of Elie Cartan, and is, as his father was, a distinguished and influential mathematician. ...

is admissible if for each j, i_j ≥ 2i_j+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. The notion of admissibility comes from the Adem relations, which are

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.

Finally, the Adem relations allow us to express all possible compositions of Steenrod squares as elements of an infinite-dimensional algebra over the field F_p with p elements. This is what is meant by the term Steenrod algebra: in the case p = 2, it can be constructed in the following way. Let M_i be the free F₂-module on the symbol Sqⁱ, and let M be the graded F₂-module with homogeneous degree i part equal to M_i. Then the Steenrod algebra is the quotient of the tensor algebra of the module M by the ideal generated by the Adem relations. With this definition, the mod p cohomology of any space becomes a graded module over the mod p Steenrod algebra. In mathematics, a free module is a module having a free basis. ... In mathematics, the tensor algebra is an abstract algebra construction of a unital associative algebra T(V) from a vector space V. In a sense, T(V) is the most general algebra containing V. If we take basis vectors for V, those become non-commuting variables in T(V), subject...

Hopf algebra structure and the Milnor basis

The Steenrod algebra has more structure than as a graded F_p-algebra. It is also a Hopf algebra, so that in particular there is a diagonal or comultiplication map In abstract algebra, a Hopf algebra is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. ... In mathematics, coalgebras are structures that are in a certain sense dual to the unital associative algebras. ...

.

It is actually much easier to describe than the product map:

The linear dual of ψ makes the (graded) linear dual A* of A into an algebra. John Milnor proved, for p = 2, that A* is actually a polynomial algebra, with one generator ξ_k of degree 2^k - 1, for every k. 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 commutative. In the case p > 2, the dual is the tensor product of a polynomial algebra with an exterior algebra. Of course, the coproduct for A* is the dual of the product on A; it is given in the case p = 2 by In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... John Willard Milnor (b. ... In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ...

Here ξ₀ is interpreted as 1. The only primitive elements of A* are the , and these are dual to the (the only indecomposables of A). In mathematics, the term primitive element can mean: in number theory, a primitive root modulo n in field theory, an element that generates a given field extension, see primitive element (field theory). ...

First applications

The most famous early applications of the Steenrod algebra to outstanding topological problems were the solutions by Frank Adams of the Hopf invariant one problem and the vector fields on spheres problem. Adams later 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. Frank Adams may also refer to Frank Dawson Adams a Canadian geologist. ... The topic of K-theory spans the subjects of algebraic topology, abstract algebra and some areas of application like operator algebras and algebraic geometry. ... In mathematics, an Adams operation ψk is a cohomology operation in K-theory, or any allied operation in algebraic K-theory or other types of algebraic construction, defined on a pattern introduced by Frank Adams. ...

Theorem. If there is a map S^{2n - 1} → Sⁿ of Hopf invariant one, then n is a power of 2.

The proof uses the fact that each Sq^k is indecomposable for k = 2ⁿ, that is, such an element is not the 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 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

This is what is meant by the aphorism "the cohomology of the Steenrod algebra is an approximation to the stable homotopy groups of spheres."

References

• R. Mosher and M. Tangora, Cohomology Operations and Applications in Homotopy Theory. Harper and Row, 1968.
• Allen Hatcher, Algebraic Topology. Cambridge University Press, 2002. Available free online from the author's home page.