The Hurewicz image of the η_i family, a polynomial subalgebra of H_*Ω₀^{2^(i+1)-8+k}S^(2^i-2)

We consider the problem of calculating the Hurewicz image of Mahowald's family $\eta_i\in{_2\pi_{2^i}^S}$. This allows us to identify specific spherical classes in $H_*\Omega_0^{2^{i+1}-8+k}S^{2^i-2}$ for $0\leqslant k\leqslant 6$. We then identify the type of the subalgebras that these classes give rise to, and calculate the $A$-module and $R$-module structure of these subalgebras. We shall the discuss the relation of these calculations to the Curtis conjecture on spherical classes in $H_*Q_0S^0$, and relations with spherical classes in $H_*Q_0S^{-n}$.