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Equivariant generating hypotheses for finite groups

  • Sihao Ma
  • XiaoLin Danny Shi
  • Zhouli Xu
  • Shangjie Zhang
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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≤ 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 Ψ=(p−[Cp/e])(1−g) on Cp-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.

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Cite

BibTeXRIS
BibTeX
@misc{ma2026equivariant,
  title = {{Equivariant generating hypotheses for finite groups}},
  author = {Sihao Ma and XiaoLin Danny Shi and Zhouli Xu and Shangjie Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.24390},
  eprint = {2609.24390},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24390},
}
RIS
TY  - UNPB
TI  - Equivariant generating hypotheses for finite groups
AU  - Sihao Ma
AU  - XiaoLin Danny Shi
AU  - Zhouli Xu
AU  - Shangjie Zhang
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.24390
UR  - https://arxiv.org/abs/2609.24390
AB  - 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≤ 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 Ψ=(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.
ER  -