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PreprintCC BY 4.0arXiv

On The Rational Realization of Even-dimensional Spheres and Products of Eilenberg--MacLane Spaces as Classifying Spaces

  • Yang Bai
  • Xiugui Liu
  • Jiaxi Zha
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Abstract

In this paper, we study the rational realization problem for the classifying space Baut1(X). We prove that if S2n is realized as Baut1(X)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 Baut1(X)ℚ for any simply-connected π-finite space X. We also prove that, for all n≥ 2 and s,t≥ 1, the product of Eilenberg--MacLane spaces K(ℚs,n)× K(ℚt,n+1) cannot be realized as Baut1(X)ℚ for any simply-connected π-finite space X. Moreover, we prove that if r≥ 2, n≥ 3 and K(ℚr,n) is realized as Baut1(X)ℚ for a simply-connected π-finite space X, then X has the rational homotopy type of K(ℚr,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.

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Cite

BibTeXRIS
BibTeX
@misc{bai2026rational,
  title = {{On The Rational Realization of Even-dimensional Spheres and Products of Eilenberg--MacLane Spaces as Classifying Spaces}},
  author = {Yang Bai and Xiugui Liu and Jiaxi Zha},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.04285},
  eprint = {2609.04285},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04285},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On The Rational Realization of Even-dimensional Spheres and Products of Eilenberg--MacLane Spaces as Classifying Spaces
AU  - Yang Bai
AU  - Xiugui Liu
AU  - Jiaxi Zha
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.04285
UR  - https://arxiv.org/abs/2609.04285
AB  - In this paper, we study the rational realization problem for the classifying space Baut_1(X). We prove that if S^2n is realized as Baut_1(X)_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 Baut_1(X)_ℚ for any simply-connected π-finite space X. We also prove that, for all n≥ 2 and s,t≥ 1, the product of Eilenberg--MacLane spaces K(ℚ^s,n)× K(ℚ^t,n+1) cannot be realized as Baut_1(X)_ℚ for any simply-connected π-finite space X. Moreover, we prove that if r≥ 2, n≥ 3 and K(ℚ^r,n) is realized as Baut_1(X)_ℚ for a simply-connected π-finite space X, then X has the rational homotopy type of K(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.
ER  -