Phan Hoang Chon

It is well-known that, as a ring object in the category of coalgebras (also known as a coalgebraic ring or a Hopf ring) the mod p homology of $\{QS^k\}_{k\geq0}$ is generated by the image of the mod p homology of $B\Sigma _p$ and $S^1$. In this talk, we use modular invariant theory to establish a complete set of relations for odd prime p, the case $p=2$ having been previously treated by Paul Turner.

We also describe the action of the mod p Dyer–Lashof algebra as well as the mod p Steenrod algebra on the coalgebraic ring.

One can derive the mod p homology of $\Sigma _n$'s as well as the maps induced in mod p homology by the maps $\Sigma _n\times \Sigma _m\rightarrow \Sigma _{n+m}$ and $\Sigma_n\times \Sigma _m\rightarrow \Sigma _{nm}$ from the $k=0$ case.