{
  "abstract": "We study the subalgebra generated by degree-one mod-$2$ cohomology for metacyclic $2$-groups $G=\\langle a,b\\mid a^{2^α}=1,\\ b^{2^m}=a^c,\\ b^{-1}ab=a^r\\rangle$ with unique normal forms, where $α,m\\ge2$ and $r\\equiv-1\\pmod4$. Presentation compatibility forces $c=0$ or $2^{α-1}$ and $G^{\\mathrm{ab}}\\cong C_2\\times C_{2^m}$. If $c\\equiv0\\pmod4$, the degree-one-generated algebra is $\\mathbb{F}_2 [x,y]/(y^2,x^2+xy)$, a four-dimensional algebra shared by the four nonmodular types of Aaboun--Zekhnini. The unique exceptional case $α=2$, $c=2$ is modular and has infinite-dimensional degree-one algebra $\\mathbb{F}_2[x,y]/(y^2,x^2y)$. The regular algebra is PBW, Koszul, universally Koszul, and strongly Koszul, whereas the modular algebra is not Koszul. We realize the regular algebra by inflation from the common quotient $C_4\\rtimes_{-1}C_4$ and identify its quadratic dual with the characteristic-two Jordan plane. For the inversion tower $C_4\\rtimes_{-1}C_{2^m}$, we give uniform presentations of the full mod-$2$ cohomology rings and determine the kernels and images of inflation in every degree. Their direct limit yields the continuous cohomology ring of $C_4\\rtimes_{-1}\\mathbb Z_2$. We also record an arithmetic consequence for Galois groups arising over cyclotomic $\\mathbb Z_2$-extensions of real quadratic fields.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36401",
  "author_total": 1,
  "authors": [
    {
      "name": "Marina Palaisti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5gb4taznd8ccc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36401",
  "fulltext_url": null,
  "id": "5gb4taznd8ccc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36401",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7224107814",
  "pdf_url": "https://arxiv.org/pdf/2610.36401",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5gb4taznd8ccc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36401",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36401"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Degree-one cohomology of metacyclic $2$-groups: a low-bit dichotomy",
  "title_text": "Degree-one cohomology of metacyclic $2$-groups: a low-bit dichotomy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/5gb4taznd8ccc",
  "venue": "arXiv",
  "versions": []
}