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PreprintCC BY-NC-ND 4.0arXiv

Homology Nilpotency in Rational Homotopy Theory: Cell Attachments, Retractive Towers, Divergence, and Stabilization

  • Paul-Eugène Parent
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Abstract

Homology nilpotency of a simply connected minimal Sullivan algebra M is the least integer n for which (M+)n+1 is contained in an acyclic differential ideal. We develop lower- and upper-bound criteria, including methods for rational cell attachments, and construct an explicit finite-type minimal Sullivan algebra M satisfying nilh(M)=3<Hnil(M)=4. Complete strictly coordinated retractive towers identify the obstruction: they characterize homotopical nil-length while exposing the additional quotient defect measured by homology nilpotency. We then show that this rigid defect disappears after stabilization by degreewise finite wedges of simply connected rational spheres: Hnils(X)=cato(X) for every simply connected rational space X of finite type.

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Cite

BibTeXRIS
BibTeX
@misc{parent2026homology,
  title = {{Homology Nilpotency in Rational Homotopy Theory: Cell Attachments, Retractive Towers, Divergence, and Stabilization}},
  author = {Paul-Eugène Parent},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02980},
  eprint = {2609.02980},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02980},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Homology Nilpotency in Rational Homotopy Theory: Cell Attachments, Retractive Towers, Divergence, and Stabilization
AU  - Paul-Eugène Parent
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.02980
UR  - https://arxiv.org/abs/2609.02980
AB  - Homology nilpotency of a simply connected minimal Sullivan algebra M is the least integer n for which (M^+)^n+1 is contained in an acyclic differential ideal. We develop lower- and upper-bound criteria, including methods for rational cell attachments, and construct an explicit finite-type minimal Sullivan algebra M satisfying nil_h(M)=3<Hnil(M)=4. Complete strictly coordinated retractive towers identify the obstruction: they characterize homotopical nil-length while exposing the additional quotient defect measured by homology nilpotency. We then show that this rigid defect disappears after stabilization by degreewise finite wedges of simply connected rational spheres: Hnil_s(X)=cat_o(X) for every simply connected rational space X of finite type.
ER  -