The Hurewicz image of the ${\eta }_i$ family, a polynomial subalgebra of $H_{\ast }{\mathrm{\Omega }}_0^{2^{i+1}-8+k}{S}^{2^i-2}$

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 $H_{\ast }{\mathrm{\Omega }}_0^{2^{i+1}-8+k}{S}^{2^i-2}$ 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 $H_{\ast }{Q}_0{S}^0$, and relations with spherical classes in $H_{\ast }{Q}_0{S}^{-n}$.