On a Geometric Realization of $\mathcal A(2)$

Let JL be the mod p Steenrod algebra and M be a bounded below left JLmodule of finite type. M is said to be realizable if there exists some spectrum whose mod£ cohomology is isomorphic as Jl-module to M. For example, Jl itself is realized as JL^H*(HZ/p ; Z/p). It is a general problem whether or not given M is realizable, but there is no standard method to solve this problem. So we have to try case by case. For many interesting cases, this problem was solved. J. F. Adams [1] showed that there is no spectrum which realizes M = Z/2-x+Z/2'Sqx. E. H. Brown and S. Gitler [2] constructed certain spectra B(fe) such that U*E(k)^tA/JL{7L(Sq )\i>k}. H. Toda [8] stated that certain algebraic properties of M assure its readability. In this paper we shall prove that some more conditions give us useful information about the number of the homotopy types of spectra which realize M. (Theorem 1.1) JL(n) is a sub-Hopf algebra of Jl generated by |8, £, ••• , &~\ with 5>*= Sq for p=2. S. A. Mitchell [6] proved every JL(n) admits certain left Jl module structure extended from its own algebra multiplication and also constructed finite spectra whose cohomologies are Jl(n) free. Hence we should ask whether each JL(n) itself is realizable or not, because there exists a non-realizable JL -module which is a direct summand of a realizable module. Independently of Mitchell's work, D. M. Davis and M. Mahowald [3] gave four different module structures on JL(l) (p=2) and proved the uniqueness of the homotopy type of spectra which realize each Jl(l). For the case of JL(2) (p=2), W.H. Lin [4] showed 1600 different JL -module structures. Theorem 2.2 gives an affirmative answer to the realization problem for JL(2) (p=2} with any possible Jl-module structure and Theorem 2.4 shows the uniqueness of the homotopy type of spectra which realize JL(2) with the specific ^-module structure indicated by Mitchell [6].