I’ve been reading Milnor’s paper “The Steenrod algebra and its dual,” and want to talk a little about it today. The starting point of this story is the theory of cohomology operations. Given a cohomology theory {h^*} on spaces (or just CW complexes; one can always Kan extend to all spaces), one can consider cohomology operations on {h^*}. Most interesting for our purposes are the stable cohomology operations.

A stable cohomology operation of degree {k} will be a collection of homomorphisms {h^m(X) \rightarrow h^{m+k}(X)} for each {m}, which are natural in the space {X}, and which commute with the suspension isomorphisms. If we think of {h^*} as represented by a spectrum {E}, so that {h^*(X) = [X, E]} is a representable functor (in the stable homotopy category), then a stable cohomology operation comes from a homotopy class of maps {E \rightarrow E} of degree {k}.

A stable cohomology operation is additive, because it comes from a spectrum map, and the stable homotopy category is additive. Moreover, the set of all stable cohomology operations becomes a graded ring under composition. It is equivalently the graded ring {[E, E]}.

The case where {E} is an Eilenberg-MacLane spectrum, and {h^*} ordinary cohomology, is itself pretty interesting. First off, one has to work in finite characteristic—in characteristic zero, there are no nontrivial stable cohomology operations. In fact, the only (possibly unstable) natural transformations {H^*(\cdot, \mathbb{Q}) \rightarrow H^*(\cdot, \mathbb{Q})} come from taking iterated cup products because {H^*(K(\mathbb{Q}, n))} can be computed, via the spectral sequence, to be a free graded-commutative algebra over {\mathbb{Q}} generated by the universal element. These aren’t stable, so the only stable one has to be zero. So we will work with coefficients {\mathbb{Z}/p} for {p} a prime.

Here the algebra of stable cohomology operations is known and has been known since the 1950’s; it’s called the Steenrod algebra {\mathcal{A}^*}. In fact, all unstablecohomology operations are themselves known. Let me state the result for {p=2}.

Steenrod had constructed squaring operations

\displaystyle \mathrm{Sq}^i: H^*(\cdot, \mathbb{Z}/2) \rightarrow H^{* +i}(\mathbb{Z}/2) .

These are natural transformations, which have the following properties:

  1. {\mathrm{Sq}^0} is the identity operation.
  2. {\mathrm{Sq}^i} on a cohomology class {x} of dimension {n} vanishes for {i > n}. For {i = n}, {\mathrm{Sq}^i} acts by the cup square on {x}.
  3. The Steenrod squares behave well with respect to the cohomology cross (and thus cup) product: {\mathrm{Sq}^i(a \times b) = \sum_{j + k = i} \mathrm{Sq}^j a \times \mathrm{Sq}^k b}.
  4. {\mathrm{Sq}^1} is the Bockstein connecting homomorphism associated to the short exact sequence {0 \rightarrow \mathbb{Z}/2 \rightarrow \mathbb{Z}/4 \rightarrow \mathbb{Z}/2 \rightarrow 0}.
  5. {\mathrm{Sq}^i} commutes with suspension (and thus is a homomorphism). (more…)