TITLE: $Sq^1$ cohomology of spaces
QUESTION [12 upvotes]: 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:


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

REPLY [9 votes]: 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.