The Steenrod squares are discussed at some length in Chapter 5 of the book (so we’re getting a bit out of sequence here). But they are also, of course, closely related to characteristic classes: Milnor and Stasheff’s book defines the Stiefel-Whitney classes of a real vector bundle in terms of the Thom isomorphism and the Steenrod squares applied to the total space. Nowadays, I would guess that a reader might be more familiar with characteristic classes than with cohomology operations, so this order of presentation might not make so much sense as an introduction. But it has some important corollaries, including the fact that it shows that Stiefel-Whitney classes can be defined for spherical fibrations: the linear structure of a vector bundle does not play a role here (as it does for the Pontrjagin classes).
The basic point about Steenrod squares is to quantify the commutativity of the cup-product operation on a space \(X\). What does this mean? We know that, given cocycles \(a,b\) for \(X\), one has \([ a \smile b] = [b\smile a] \in H^*(X;{\mathbb Z}_2)\). But this doesn’t tell us that the cup-product is commutative “on the nose”: only up to some kind of homotopy. In the chapter as written, I carry this idea forward algebraically, using chain maps and diagonal approximations and so on; as well as defining the Steenrod squares this yields the symmetric and quadratic constructions that are so important in Ranicki’s algebraic surgery theory. Here though let me sketch how the idea can be implemented geometrically, following some lecture notes of Mike Hopkins (see also the very last section of Hatcher’s algebraic topology book).
Let \(X\) be a space and \(x \in H^n(X) \) be a cohomology class (let’s agree that all cohomology groups have mod 2 coefficients). The external square \(x \times x \in H^{2n}(X\times X) \) is represented by a map
\[ X\times X \to K({\mathbb Z}_2,2n) \]
to an Eilenberg-MacLane space. The commutativity of the cup-product gives us a homotopy between this map and its composite with the “flip” automorphism of \(X\times X\). The homotopy can be encoded as a map
\[ (X\times X)\times_{{\mathbb Z}_2} S^1 \to K({\mathbb Z}_2,2n). \]
There are also “higher homotopies”, and taking all of those into account one can promote the above to a map
\[ (X\times X)\times_{{\mathbb Z}_2} S^\infty \to K({\mathbb Z}_2,2n). \]
Notice that \(S^\infty = E{\mathbb Z}_2\), so the left side here is the Borel construction or homotopy quotient \( X \times_{h{\mathbb Z}_2} X \) of \(X\times X\) by the involution. The construction shows that the original external square map \( X\times X \to K({\mathbb Z}_2,2n) \) factors through this homotopy quotient.
Okay, so now let’s consider the diagonal map \(\Delta \colon X\to X\times X\). This induces a diagonal map from \(X \times B{\mathbb Z}_2\) to the homotopy quotient. We end up with a diagram of maps like this
\[ \begin{array}{ccccc} H^n(X) & \to & H^{2n}(X\times_{h{\mathbb Z}_2} X) & \to &H^{2n}(X\times X) \\&&\downarrow&&\downarrow\\ && H^{2n}(X \times B{\mathbb Z}_2)&\to & H^{2n}(X) \end{array} \]
where the vertical maps are induced by \(\Delta\). By the Kunneth formula, \( H^*(X \times B{\mathbb Z}_2) \cong H^*(X)[\alpha] \), where \(\alpha\) is the 1-dimensional generator of the cohomology of \( B{\mathbb Z}_2 = {\mathbb{RP}}^\infty \). Thus the induced map (from the diagram above) \( P\colon H^n(X) \to H^{2n}(X \times B{\mathbb Z}_2) \) can be written in components as
\[ P(x) = \sum_{j=0}^n {Sq}^{n-j}(x) \alpha^j, \quad {Sq}^k(x) \in H^{n+k}(X). \]
By construction, \({Sq}^n(x) = x^2. \) The other components of the sum, which measure the extent to which the homotopy commutativity of the cup-product departs from a genuine commutativity, are the Steenrod squares.
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