Equivariant generating hypotheses for finite groups

Abstract

We disprove Bohmann's equivariant generating hypothesis for every nontrivial finite group $G$, even when all $\RO(H)$-graded homotopy groups at every subgroup $H$ are tested. For each fixed $G$ and prime $p$ dividing $|G|$, we construct ghosts on finite $G$-spectra with arbitrarily long nonzero composition powers. We also prove that the homotopy-module functors are nonfull and construct non-equivalent finite $G$-spectra with isomorphic full homotopy modules. Our constructions use circle power maps and cyclic permutations of products of projective spaces, and are motivated by Ma--Xu's categorical method in the motivic setting and the projective-space power maps.