Quillen’s Elementary Proofs

Recall 1.1. The construction of the Steenrod squares and powers proceeds in the following steps 1. For a space X, the p-th cartesian power Xp is acted on cyclically by Z/p and it fits into a fibration Xp → EZ/p×Z/p Xp → BZ/p 2. If x ∈ H∗(X, Z/p), one defines u × · · · × u ∈ H∗(Xp, Z/p) and extends this class to ũ ∈ H(EZ/p×Z/p Xp; Z/p) by a construction on the chain level. 3. ũ is then pulled back along EZ/p×Z/p (∆) : EZ/p×Z/p X → EZ/p×Z/p Xp, where X is acted on trivially. 4. This class lives in H∗(BZ/p×X; Z/p), so we apply the Kunneth formula, and taking the classes in H∗(X; Z/p), one defines Pi, the i-th power operation (Sqi when p = 2). Idea 1.2. It is not clear how to extend this idea to the context of generalized cohomology theories E because one no longer has the description of E∗(X) as the cohomology of a cochain complex. But when E = MU, and X is a smooth manifold, MU∗(X) has a description as a particular set of smooth maps f : Z → X. Hence we may define P( f ) to be the pullback of the map EZ/p×Z/p ( f×p) along EZ/p×Z/p (∆). Since any finite CW complex has the homotopy type of a smooth manifold, we don’t lose much generality in proceeding in this way.