A phenomenon of reciprocity in the universal Steenrod algebra

Abstract:

In this paper we compute the cohomology algebra of certain subalgebras ${L_r}$ and certain quotients ${K_s}$ of the $\bmod 2$ universal Steenrod algebra $Q$, the algebra of cohomology operations for ${H_\infty }$-ring spectra (see $[\text {M}]$). We prove that \[ \operatorname {Ext}_{{L_r}}({F_2},{F_2}) \cong {K_{ - k + 1}}, \qquad \operatorname {Ext}_{{K_s}}({F_2},{F_2}) \cong {L_{ - s + 1}}\] with $r$, $s$ integers and $r \leq 1$, $s \geq 0$. We also observe that some of the algebras ${L_r}$, ${K_s}$ are well known objects in stable homotopy theory and in fact our computation generalizes the fact that ${H^{\ast } }({A_L}) \cong \Lambda ^{{\text {opp}}}$ and ${H^{\ast } }({\Lambda ^{{\text {opp}}}}) \cong {A_L}$ (see, for instance, $[\text {P}]$). Here ${A_L}$ is the Steenrod algebra for simplicial restricted Lie algebras and $\Lambda$ is the ${E_1}$-term of the Adams spectral sequence discovered in $[\text {B-S}]$.