{
  "abstract": "In this paper, we study the rational realization problem for the classifying space $B\\mathrm{aut}_1(X)$. We prove that if $S^{2n}$ is realized as $B\\mathrm{aut}_1(X)_{\\mathbb{Q}}$ for a simply-connected space $X$, then $X$ is $π$-infinite and has vanishing rational Gottlieb elements above degree $2n-1$. In particular, even-dimensional spheres cannot be realized as $B\\mathrm{aut}_1(X)_{\\mathbb{Q}}$ for any simply-connected $π$-finite space $X$. We also prove that, for all $n\\geq 2$ and $s,t\\geq 1$, the product of Eilenberg--MacLane spaces $K(\\mathbb{Q}^s,n)\\times K(\\mathbb{Q}^t,n+1)$ cannot be realized as $B\\mathrm{aut}_1(X)_{\\mathbb{Q}}$ for any simply-connected $π$-finite space $X$. Moreover, we prove that if $r\\geq 2$, $n\\geq 3$ and $K(\\mathbb{Q}^r,n)$ is realized as $B\\mathrm{aut}_1(X)_{\\mathbb{Q}}$ for a simply-connected $π$-finite space $X$, then $X$ has the rational homotopy type of $K(\\mathbb{Q}^r,n-1)$. The proofs are based on two structural results for Gottlieb elements in the derivation Lie algebra of a Sullivan minimal model, which provide a uniform method for these realization problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04285",
  "author_total": 3,
  "authors": [
    {
      "name": "Yang Bai"
    },
    {
      "name": "Xiugui Liu"
    },
    {
      "name": "Jiaxi Zha"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2q8betfqdxhez.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04285",
  "fulltext_url": null,
  "id": "2q8betfqdxhez",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04285",
  "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.AT"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7211897968",
  "pdf_url": "https://arxiv.org/pdf/2609.04285",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2q8betfqdxhez.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2609.04285",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04285"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On The Rational Realization of Even-dimensional Spheres and Products of Eilenberg--MacLane Spaces as Classifying Spaces",
  "title_text": "On The Rational Realization of Even-dimensional Spheres and Products of Eilenberg--MacLane Spaces as Classifying Spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/2q8betfqdxhez",
  "venue": "arXiv",
  "versions": []
}