specific modules over the Steenrod algebra with one generator

I'd be happy to clarify the following. Consider the module which is a quotient of the Steenrod algebra mod $$2$$ by the left ideal generated by $$\operatorname{Sq}^1, \operatorname{Sq}^2, \operatorname{Sq}^3, \dots, \operatorname{Sq}^k$$. Is it true that there is a space (or spectrum) with such a cohomology? I found in the book by Margolis a statement that these modules (or may be at least one) are related to the connective K-theory.

This realization problem is well-known in algebraic topology. Let $$A$$ denote the mod $$2$$ Steenrod algebra. For $$k=0$$ the module $$A$$ is the mod $$2$$ cohomology of $$H\mathbb{Z}/2$$, and for $$k=1$$ the module $$A/ASq^1$$ is the mod $$2$$ cohomology of $$H\mathbb{Z}$$. For $$k=2$$ and $$k=3$$, the module $$A/A(Sq^1,Sq^2) = A/A(Sq^1, Sq^2, Sq^3)$$ is the mod $$2$$ cohomology of $$ko$$ (connective, real $$K$$-theory). This follows from calculations of Stong. For $$4 \le k < 8$$ the module $$A/A(Sq^1, \dots, Sq^k) = A/A(Sq^1, Sq^2, Sq^4)$$ is the mod $$2$$ cohomology of $$tmf$$, the connective topological modular forms spectrum. This construction of $$tmf$$ is due to Hopkins, Mahowald and Miller, and explained by Behrens in the Talbot proceedings book on "Topological Modular Forms". The claim that its cohomology is $$A/A(Sq^1, Sq^2, Sq^4)$$ is proved in a paper by Mathew. For $$8 \le k < \infty$$ there is no spectrum with cohomology $$A/A(Sq^1, \dots, Sq^k)$$, due to Adams' factorization of $$Sq^{2^i}$$ for $$i\ge4$$ as a composition of secondary cohomology operations, as part of his proof that $$Sq^{2^i}$$ does not detect a class in $$\pi_{2^i-1}(S)$$. All of the modules discussed so far are not unstable, hence cannot be the cohomology of a space. If you allow $$k=\infty$$, then $$A/A(Sq^1, \dots) = \mathbb{Z}/2$$ is the cohomology of a point, or the reduced cohomology of two points.
• What is the reference to the same question $mod p$ about quotients $A_p/A_p(\beta,P^1,P^p,\ldots,P^{p^k})$ and $A_p/A_p(P^1,P^p,\ldots,P^{p^k})$? – Dr.Martens May 3 '19 at 8:23
• @Dr.Martens Arunas Liulevicius proved (in his 1960 PhD thesis, published in 1962 as an AMS Memoir), that $P^{p^i}$ factors as a secondary cohomology operation for $i\ge1$ when $p$ is odd. These factorizations lead to nonzero $d_2$-differentials in the Adams spectral sequence for the sphere spectrum, so reading e.g. the introduction in Ravenel's "Complex cobordism and stable homotopy groups of spheres" will put these results in a better context. – John Rognes May 3 '19 at 19:22