Steenrod algebra

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.

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 "p"th 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.

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:

:$H^n(X;R)\; o\; H^\{2n\}(X;R)$:$x\; mapsto\; x\; cup\; x$

Note that cohomology operations need not be homomorphisms of graded rings, see the Cartan formula below.

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

:$Sq^i\; colon\; H^n(X;mathbf\{Z\}/2)\; o\; H^\{n+i\}(X;mathbf\{Z\}/2)$:$P^i\; colon\; H^n(X;mathbf\{Z\}/p)\; o\; H^\{n+2i(p-1)\}(X;mathbf\{Z\}/p)$

for all "i" greater than zero. The notation "Sq" and their name, the Steenrod squares, comes from the fact that "Sq"^"n" restricted to classes of degree "n" is the cup square. There are analogous operations for odd primary coefficients, usually denoted "P"^"i" and called the reduced "p"-th power operations. The "Sq"^"i" 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 "P"^"i" and the Bockstein operation β associated to the short exact sequence

:$0\; o\; mathbf\{Z\}/p\; o\; mathbf\{Z\}/p^2\; o\; mathbf\{Z\}/p\; o\; 0.$

(The Bockstein occurs also in the mod 2 case, as "Sq"¹.)

Axiomatic characterization

The Steenrod squares "Sq"^"n" satisfy the following axioms:

#Naturality: For any map "f" : "X" → "Y", "f"*("Sq"^"n""x") = "Sq"^"n""f"*(x).
#Additivity: "Sq"^"n"("x" + "y") = "Sq"^"n"("x") + "Sq"^"n"("y").
#Cartan Formula:$Sq^n(x\; cup\; y)\; =\; sum\_\{i+j=n\}\; (Sq^i\; x)\; cup\; (Sq^j\; y)$
#Stability: The squares commute with the suspension isomorphism (if we are careful).
#"Sq"^"n" is the cup square on classes of degree "n".
#"Sq"⁰ is the identity homomorphism.
#"Sq"¹ is the Bockstein homomorphism of the exact sequence $0\; o\; mathbf\{Z\}/2\; o\; mathbf\{Z\}/4\; o\; mathbf\{Z\}/2\; o\; 0.$

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

:$i\_1,\; i\_2,\; ldots,\; i\_n$

is admissible if for each "j", "i"_"j" ≥ 2"i"_"j"+1. Then the elements

:$Sq^I\; =\; Sq^\{i\_1\}\; ldots\; Sq^\{i\_n\},$

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

:$Sq^i\; Sq^j\; =\; sum\_\{k=0\}^\{\; [i/2]\; \}\; \{j-k-1\; choose\; i-2k\}\; Sq^\{i+j-k\}\; Sq^k$

for all "i", "j" > 0 such that "i" < 2"j". (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 "Sq"ⁱ, 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.

Hopf algebra structure and the Milnor basis

The Steenrod algebra has more structure than a graded F_p-algebra. It is also a Hopf algebra, so that in particular there is a diagonal or comultiplication map

:$psi\; colon\; A\; o\; A\; otimes\; A.$

It is actually much easier to describe than the product

:$psi(Sq^k)\; =\; sum\_\{i+j=k\}\; Sq^i\; otimes\; Sq^j.$

The linear dual of &psi; 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 &xi;_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 comultiplication for "A"* is the dual of the product on "A"; it is given in the case "p" = 2 by

:$psi(xi\_n)\; =\; sum\_\{i=0\}^n\; xi\_\{n-i\}^\{2^i\}\; otimes\; xi\_i.$

Here &xi;₀ is interpreted as 1. The only primitive elements of "A"* are the $xi\_1^\{2^i\}$, and these are dual to the $Sq^\{2^i\}$ (the only indecomposables of "A").

First 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 spheres 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 operations. One application of the mod 2 Steenrod algebra that is fairly elementary is the following theorem.

Theorem. If there is a map S^{2"n" - 1} &rarr; S^"n" of Hopf invariant one, then "n" is a power of 2.

The proof uses the fact that each "Sq"^"k" 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 "E"₂ term for the ("p"-local) Adams spectral sequence, whose abutment is the "p"-component of the stable homotopy groups of spheres. More specifically, the "E"₂ term of this spectral sequence may be identified as

:$mathrm\{Ext\}^\{s,t\}\_\{A\}(mathbf\{F\}\_p,\; mathbf\{F\}\_p).$

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 [http://www.math.cornell.edu/~hatcher/AT/ATpage.html author's home page] .