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PreprintCC BY 4.0arXiv

A complete classification of the rational realization of gap-two 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 of the gap-two product K(ℚ,n)× K(ℚ,n+2) as the classifying space Baut1(X) for a simply-connected π-finite space X with rational homology of finite type. We prove that K(ℚ,n)× K(ℚ,n+2) can be realized as Baut1(X) up to rational homotopy equivalence if and only if n=3,4 or 4m+2 for m≥ 1. Moreover, we determine all their minimal Sullivan models up to isomorphism. The rational homotopy type of such X is unique for n=3 or 4m+2 for m≥ 1; there exist infinitely many rational homotopy types of such X for n=4.

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Cite

BibTeXRIS
BibTeX
@misc{bai2026complete,
  title = {{A complete classification of the rational realization of gap-two 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.2610.11671},
  eprint = {2610.11671},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11671},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A complete classification of the rational realization of gap-two 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.2610.11671
UR  - https://arxiv.org/abs/2610.11671
AB  - In this paper, we study the rational realization of the gap-two product K(ℚ,n)× K(ℚ,n+2) as the classifying space Baut_1(X) for a simply-connected π-finite space X with rational homology of finite type. We prove that K(ℚ,n)× K(ℚ,n+2) can be realized as Baut_1(X) up to rational homotopy equivalence if and only if n=3,4 or 4m+2 for m≥ 1. Moreover, we determine all their minimal Sullivan models up to isomorphism. The rational homotopy type of such X is unique for n=3 or 4m+2 for m≥ 1; there exist infinitely many rational homotopy types of such X for n=4.
ER  -