Detailed exposition of construction of Steenrod squares from Haynes Miller's book

Question

$\DeclareMathOperator\Sq{Sq}$Chapter 75 of Haynes Miller's book on algebraic topology contains a beautiful construction of the Steenrod squares $\Sq^i$.

Roughly speaking, it goes as follows. All cohomology groups here have $\mathbb{F}_2$-coefficients. Consider a based space $X$. The cyclic group $C_2$ of order $2$ acts on the product $X^2$ by switching the factors. We can then look at the Borel equivariant cohomology $H^{\bullet}_{C_2}(X^2,X \vee X)$. This is the relative cohomology of the Borel construction $X^2 \times_{C_2} EC_2$, which is a fiber bundle over $B C_2$ with fiber $X^2$. The fiber inclusion gives a map of pairs $\iota\colon (X^2,X \vee X) \hookrightarrow (X^2 \times_{C_2} EC_2,(X \vee X) \times_{C_2} EC_2)$, and thus a map $$\iota^{\ast}\colon H^{\bullet}_{C_2}(X^2,X \vee X) \rightarrow H^{\bullet}(X^2,X \vee X).$$ What Miller does is construction a natural transformation $P\colon \tilde{H}^n(X) \rightarrow H^{2n}_{C_2}(X^2,X \vee X)$ such that $$\iota^{\ast}(P(x)) = x \otimes x \quad \text{for all $x \in \tilde{H}^n(X^2)$}.$$ He then pulls $P(x)$ back along the diagonal map $\Delta\colon X \rightarrow X \times X$ to get $$P(x) \in H^{2n}_{C_2}(X) = H^{2n}(X \times BC_2) = (H^{\bullet}(X) \otimes \mathbb{F}_2[d])_{2n}.$$ The coefficients are the Steenrod squares of $x$.

He verifies some parts of the usual properties of the squares, but he omits a lot of important verifications:

1. The fact that $\Sq^1$ is nonzero.
2. The Cartan formula.
3. The Adem relations

My question: where can I find a construction of the Steenrod squares using this kind of definition that verifies these properties? I like the fact that this works at the space level rather than the chain level. I also prefer to not get bogged down in signs since I'm still learning the Steenrod operations, so I would prefer to just do the squares.

Answer 1

Hatcher's book also proceeds by first showing that there is the power operation $P(x)$ as you say, and he proves all the properties.

Hatcher makes quite a bit of use of the fact that cohomology is represented by Eilenberg MacLane spaces, but, with care, one can avoid this, though one needs a bit of homotopy theory to show that to define $P(x)$ for all $x \in H^n(X)$ and all spaces $X$, it suffices to define it assuming that $X$ satisfies $H^m(X) = 0$ for $m<n$. I have unpublished Latexed notes of my own that I have given out to Virginia's algebraic topology students for many years that take this approach, and avoids Haynes' use of the Serre Spectral Sequence.

A couple of things to note:

(a) $P$ is not linear: $P(x+y) - P(x)-P(y) = tr(x \otimes y)$, where $tr: H^*(X^2) \rightarrow H_{C_2}^*(X^2)$ is the transfer associated to the double covering $EC_2 \times X^2 \rightarrow EC_2 \times_{C_2}X^2$. But this `error term' can be seen to map to zero under the pullback to $BC_2 \times X \rightarrow EC_2 \times_{C_2}X^2$, using standard properties of the transfer.

(b) The property that is arguably the most subtle to prove is that $Sq^0$ acts as the identity on a one dimensional class. (Looking at Haynes' notes, I see that this is a property he proves carefully.)