Nonrealizability of subalgebras of $\mathfrak {A}^*$

Abstract:

At the prime two, the dual of the Steenrod algebra is a polynomial algebra in generators ${\xi _n},n \geq 1$. The Eilenberg-Mac Lane spectrum $K({Z_2})$ has homology ${Z_2}[{\xi _n}|n \geq 1]$, the Brown-Peterson spectrum BP has homology ${Z_2}[\xi _n^2|n \geq 1]$, and the symplectic Thom spectrum MSp has homology ${Z_2}[\xi _n^4|n \geq 1] \otimes \mathfrak {S}$. In this paper, we show that there is no spectrum ${B_k}$ with ${H_*}{B_k} = {Z_2}[\xi _n^{{2^k}}|n \geq 1]$ for $k \geq 2$.