$Sq^1$ cohomology of spaces

Question

For any space $X$, the first Steenrod square cohomology operation $$Sq^1\colon H^\ast(X;\mathbb{Z}_2)\to H^{\ast +1}(X;\mathbb{Z}_2)$$ is a derivation, meaning that $Sq^1\circ Sq^1 = 0$ and $Sq^1(a\cup b) = Sq^1(a)\cup b + a\cup Sq^1(b)$ (there are no signs since we are working in characteristic two).

Hence we may form the $Sq^1$-cohomology of the space, $$H\left(H^\ast(X;\mathbb{Z}_2),Sq^1\right)$$ which will be a graded algebra over $\mathbb{Z}_2$.

I am looking for references on this object. From McCleary's "User's guide to spectral sequences", I know that this is related to the Bockstein spectral sequence. More specifically, I would like to know:

1. What is the precise relationship between the $Sq^1$-cohomology of a space $X$ and $2$-torsion of higher order in $H^\ast(X;\mathbb{Z})$?
2. Is there a reference with specific calculations of the $Sq^1$-cohomology of the Eilenberg-Mac Lane spaces $K(\mathbb{Z}_2,n)$?
3. Are there any canonical references I should know about (besides McCleary and Mosher-Tangora)?

Answer 1

I think the easiest way to understand the Bockstein spectral sequence is through the exact couple coming from the long exact sequence of cohomology associated to $0\to\mathbb Z\to\mathbb Z\to \mathbb Z/2\to0$. This shows first that indeed the first differential is $Sq^1$ and tells you that the next page is the direct sum of the cokernel and kernel (shifted one step) of multiplication by $2$ on $2H^\ast(X,\mathbb Z)$. Hence it is like what you would get from applying the universal coefficient formula to $2H^\ast(X,\mathbb Z)$ (instead of $H^\ast(X,\mathbb Z)$). When each cohomology group $H^\ast(X,\mathbb Z)$ is finitely generated this means concretely that you "keep" each $\mathbb Z$-factor (as well as odd torsion) and downgrade each $\mathbb Z/2^n$ to $\mathbb Z/2^{n-1}$.

In particular the difference between the dimension of $H^n(X,\mathbb Z/2)$ and that of the $Sq^1$-cohomology is equal to the number of $\mathbb Z/2$-factors in $H^n(X,\mathbb Z)$ and $H^{n+1}(X,\mathbb Z)$.

I found a reference to Q2. In Madsen, Milgram: The classifying spaces for surgery and cobordism of manifolds, Ann of Math Studies 92 where they refer to Browder: Torsion in H-spaces, Ann of Math 74 for the Bockstein s.s. of $K(\mathbb Z_{(2)},n)$ and $K(\mathbb Z/2,n)$. The Madsen-Milgram book also contains other examples of computations with the Bss.

Answer 2

I remember that when I wrote my thesis I was unable to find references for some quite basic facts about this that everyone knew. It is quite possible that there were references that I did not succeed in finding (there was no Google then!) or that someone has written a good exposition in the intervening time. For what it's worth, my thesis is at http://neil-strickland.staff.shef.ac.uk/research/thesis.pdf and the relevant material is in Section 5.1.

Answer 3

Several people have addressed question 1 (Torsten Ekedahl and Neil Strickland). Question 2 is interesting, but I don't have a good answer for it. For question 3, as Sean Tilson points out, this is a special case of "Margolis homology", a.k.a. $P^s_t$-homology. Try

• Adams and Margolis, "Modules over the Steenrod algebra", Topology 10 (1971)
• Anderson and Davis, "A vanishing theorem in homological algebra", Comment. Math. Helv. 48 (1973)
• Margolis, Spectra and the Steenrod algebra (1983)

I also wonder if there is anything helpful in

• Adams and Priddy, "Uniqueness of BSO".

You might also search for the phrase "Bockstein acyclic", since $\textrm{Sq}^1$ is the mod 2 Bockstein.

Answer 4

Bill Singer and I have looked at the "dual" question... Given a right $\mathcal{A}$-module $M$ (such as the homology of a space), $Sq^1$ acts (on the right) as a differential. For any $s \geq 1$ and $M = \tilde{H}_*((\mathbb{R}P^{\infty})^{\wedge s}, \mathbb{Z}/2)$, $Sq^1$-homology is trivial, indicating that the kernel of $Sq^1$ is the same as the image of $Sq^1$ there. There are interesting things to say even for higher squares, though they won't be differentials in general.

The following references may be useful (both available on arXiv):

Ault, Singer. On the Homology of Elementary Abelian Groups as Modules over the Steenrod Algebra.

Ault. Relations among the kernels and images of Steenrod squares acting on right $\mathcal{A}$-modules.