Equivariant generating hypotheses for finite groups

Abstract

We prove that no nontrivial finite group $G$ satisfies the equivariant generating hypothesis of Bohmann, even when all $RO(H)$-graded homotopy groups are tested for every subgroup $H\le G$. For each $G$, 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. A central ingredient of our proof is the natural ghost operation $\Psi =(p-[C_p/e])(1-g)$ on $C_p$-spectra. Motivated by Ma--Xu's examples in the motivic settings, we apply this operation to both a two-cell complex and products of complex projective spaces with cyclic permutation action, which produces arbitrarily long nonzero ghost powers. We also construct a ghost on an explicit four-cell complex that is detected in the derived category over the group ring.

arXiv commentRevised the introduction and added an example illustrating Proposition 1.8. Corrected minor typos. Comments are welcome