The transfer in mod-p group cohomology between Σ_p \int Σ_{p^{n-1}}, Σ_{p^{n-1}} \int Σ_p and Σ_{p^n}

Abstract

In this work we compute the induced transfer map: $$\barτ^\ast: \func{Im}(res^\ast:H^\ast(G) \to H^\ast(V)) \to \func{Im}(res^\ast: H^\ast (Σ_{p^n}) \to H^\ast(V))$$ in $\func{mod}p$-cohomology. Here $Σ_{p^{n}}$ is the symmetric group acting on an $n$-dimensional $\mathbb F_p$ vector space $V$, $G=Σ_{p^{n},p}$ a $p$-Sylow subgroup, $Σ_{p^{n-1}}\int Σ_{p}$, or $Σ_{p}\int Σ_{p^{n-1}}$. Some answers are given by natural invariants which are related to certain parabolic subgroups. We also compute a free module basis for certain rings of invariants over the classical Dickson algebra. This provides a computation of the image of the appropriate restriction map. Finally, if $ ξ:\func{Im}(res^\ast:H^\ast(G) \to H^\ast(V)) \to \func{Im}(res^\ast}: H^\ast(Σ_{p^n}) \to H^\ast(V)) $ is the natural epimorphism, then we prove that $\barτ^\ast=ξ$ in the ideal generated by the top Dickson algebra generator.