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

$$\DeclareMathOperator\Sq{Sq}$$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.

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.

• Sorry for the probably dumb question, but which book by Haynes Miller do you refer to?
– M.G.
Commented Jul 20 at 1:24
• @M.G.: I think he is referring specifically to this one: amazon.com/Lectures-Algebraic-Topology-Haynes-Miller/dp/… Commented Jul 20 at 1:25
• @AndyPutman: Thanks!
– M.G.
Commented Jul 20 at 1:29

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. 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.)

• I'll take a look at Hatcher. Somehow I assumed that like the rest of the book he would focus on chain-level computations. I would love to get a copy of your notes as well, if that was possible.
– Gene
Commented Jul 19 at 19:57