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PreprintCC BY-NC-ND 4.0arXiv

The disjoint disks property for Busemann G-spaces

  • Tadashi Fujioka
  • Shijie Gu
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Abstract

We prove that every finite-dimensional Busemann G-space of dimension at least five has the disjoint disks property (DDP). For a sufficiently small metric sphere L=S(c,r), we show that every embedded arc contained in an exact distance level is a homotopical Z2-set in L. It follows that L has the disjoint arc-disk property and the disjoint homotopies property. Daverman's product theorem then gives DDP for L×ℝ, and a local avoidance argument at the center yields DDP for the ambient G-space. Since finite-dimensional Busemann G-spaces are generalized manifolds, in dimensions at least five the remaining obstruction to the Busemann conjecture is the resolution problem.

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Cite

BibTeXRIS
BibTeX
@misc{fujioka2026disjoint,
  title = {{The disjoint disks property for Busemann $G$-spaces}},
  author = {Tadashi Fujioka and Shijie Gu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21357},
  eprint = {2609.21357},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21357},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - The disjoint disks property for Busemann G-spaces
AU  - Tadashi Fujioka
AU  - Shijie Gu
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21357
UR  - https://arxiv.org/abs/2609.21357
AB  - We prove that every finite-dimensional Busemann G-space of dimension at least five has the disjoint disks property (DDP). For a sufficiently small metric sphere L=S(c,r), we show that every embedded arc contained in an exact distance level is a homotopical Z_2-set in L. It follows that L has the disjoint arc-disk property and the disjoint homotopies property. Daverman's product theorem then gives DDP for L×ℝ, and a local avoidance argument at the center yields DDP for the ambient G-space. Since finite-dimensional Busemann G-spaces are generalized manifolds, in dimensions at least five the remaining obstruction to the Busemann conjecture is the resolution problem.
ER  -