$Sq^1$ cohomology of spaces - MathOverflow

$Sq^1$ cohomology of spaces

2011-08-28T09:16:13Z

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)?

Answer by Neil Strickland for $Sq^1$ cohomology of spaces

2011-08-28T09:41:39Z

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 by Torsten Ekedahl for $Sq^1$ cohomology of spaces

2011-08-28T09:43:43Z

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 by Shaun Ault for $Sq^1$ cohomology of spaces

2011-08-28T21:31:16Z

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.

Answer by John Palmieri for $Sq^1$ cohomology of spaces

2011-08-29T00:31:15Z

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.