{
  "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 $H\\leq G$. 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24390",
  "author_total": 4,
  "authors": [
    {
      "name": "Sihao Ma"
    },
    {
      "name": "XiaoLin Danny Shi"
    },
    {
      "name": "Zhouli Xu"
    },
    {
      "name": "Shangjie Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d1zenf24fyarh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24390",
  "fulltext_url": null,
  "id": "d1zenf24fyarh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24390",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AT"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7214007455",
  "pdf_url": "https://arxiv.org/pdf/2609.24390",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d1zenf24fyarh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2609.24390",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24390"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Equivariant generating hypotheses for finite groups",
  "title_text": "Equivariant generating hypotheses for finite groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/d1zenf24fyarh",
  "venue": "arXiv",
  "versions": []
}