Let $\cal A$ denote the mod 2 Steenrod algebra. Can the $\mathcal{A}(2)$-module structure on $\mathcal{A}(2)//\mathcal{A}(1)$ be enriched to an $\cal A$-module structure? If so, is there a finite spectrum $X$ such that $H^*(X) = \mathcal{A}(2)//\mathcal{A}(1)$?


2 Answers 2


The $A(2)$-module structure on $A(2)//A(1)$ does not extend to an $A$-module structure. In particular, there is no spectrum $X$ with $H^*(X; F_2) = A(2)//A(1)$ as an $A(2)$-module.

Additively, $A(2)//A(1)$ is generated by classes $g_i$ in degree $i$ for $i = 0, 4, 6, 7, 10, 11, 13$ and $17$. The Adem relation $Sq^4 Sq^6 = Sq^{10} + Sq^8 Sq^2$ implies $Sq^{10}(g_0) = g_{10}$. The Adem relation $Sq^2 Sq^8 = Sq^{10} + Sq^9 Sq^1$ implies $Sq^{10}(g_0) = 0$. This contradicts the existence of any $A$-module structure extending the given $A(2)$-module structure.

PS: Bruner's ext code (http://www.math.wayne.edu/~rrb/papers/) contains a script (newconsistency) that lets you verify that a purported presentation really defines an $A(2)$-module, and tells you what is needed to extend the presentation to an $A$-module structure. It could be handy if you want to realize other $A(2)$-modules by spectra.

  • $\begingroup$ Bruner's ext code is very useful, thanks very much for telling me about it! $\endgroup$
    – skd
    Mar 29, 2018 at 5:31

As pointed out by John Rognes, this answer is not correct (it misses the $Sq^4$ from the bottom class to the class in dimension $4$). Sorry for the confusion.

=========== previous answer ================================

The compact real Lie group $G_2$ realizes the desuspension of $A(2)/\!/A(1)$. Recall that $$H^\ast(G_2;{\mathbb F}_2) = {\mathbb F}_2[x_3,x_5]/(x_3^4,x_5^2)$$ with $Sq^2x_3=x_5$, $Sq^1x_5 = x_3^2$. If I'm not mistaken you get an isomorphism to $A(2)/\!/A(1)$ via $$x_3\leftrightarrow Sq(4),\quad x_5\leftrightarrow Sq(0,2),\quad x_3^2\leftrightarrow Sq(0,0,1).$$

  • $\begingroup$ Any suggestion whether I should just delete this answer? I feel a bit bad about gaining reputation points for posting Sunday afternoon fallacies... $\endgroup$ Jan 8, 2018 at 10:52
  • 1
    $\begingroup$ I think you could make it community wiki and thereby give up the points if you wanted to. $\endgroup$
    – Dan Ramras
    Jan 8, 2018 at 15:28
  • $\begingroup$ I would suggest not deleting it and at most doing what Dan suggests. $\endgroup$ Jan 8, 2018 at 17:23
  • $\begingroup$ Thanks for the feedback. I have made it community wiki now! $\endgroup$ Jan 8, 2018 at 17:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.