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PreprintCC BY-NC-ND 4.0arXiv

Homotopical Nilpotency and Homology Nilpotency: A Dimension--Connectivity Bound

  • Paul-Eugene Parent
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Abstract

Let X be a (q−1)-connected rational space of finite type, where q≥2, with homotopical nilpotency nilh(X)=n≥1. If H>N(X;Q)=0 and N≤ q(n+3)−3, we prove that Hnil(X)=n. More precisely, the sufficient bound is N≤ an(MX)+2q−3, where an(MX) is the first nonzero internal cohomology degree of (MX+)n+1 in the minimal model. A family of examples proves that the constant −3 in this refined bound is optimal; optimality of the uniform bound is not asserted. The proof uses two stages of adjoining primitives to an ideal of the fixed minimal model.

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Cite

BibTeXRIS
BibTeX
@misc{parent2026homotopical,
  title = {{Homotopical Nilpotency and Homology Nilpotency: A Dimension--Connectivity Bound}},
  author = {Paul-Eugene Parent},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06518},
  eprint = {2609.06518},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06518},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Homotopical Nilpotency and Homology Nilpotency: A Dimension--Connectivity Bound
AU  - Paul-Eugene Parent
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06518
UR  - https://arxiv.org/abs/2609.06518
AB  - Let X be a (q−1)-connected rational space of finite type, where q≥2, with homotopical nilpotency nil_h(X)=n≥1. If H^>N(X;Q)=0 and N≤ q(n+3)−3, we prove that Hnil(X)=n. More precisely, the sufficient bound is N≤ a_n(M_X)+2q−3, where a_n(M_X) is the first nonzero internal cohomology degree of (M_X^+)^n+1 in the minimal model. A family of examples proves that the constant −3 in this refined bound is optimal; optimality of the uniform bound is not asserted. The proof uses two stages of adjoining primitives to an ideal of the fixed minimal model.
ER  -