PreprintCC BY-NC-ND 4.0arXiv
A sphere-product counterexample to the additivity of homology nilpotency
Abstract
Homology nilpotency need not increase under a product with a sphere. We construct a simply connected, closed smooth manifold M of dimension $53$, rationally elliptic with dimℚ(π*(M)⊗ℚ)=7, such that cat0(M)=10 and Hnil(M× S2)=Hnil(M)=11. The inequality Hnil(X× Y)≤ Hnil(X)+Hnil(Y) can therefore be strict even for a product of a closed manifold with S2 as the second factor.
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BibTeX
@misc{parent2026sphereproduct,
title = {{A sphere-product counterexample to the additivity of homology nilpotency}},
author = {Paul-Eugène Parent},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01567},
eprint = {2611.01567},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01567},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A sphere-product counterexample to the additivity of homology nilpotency
AU - Paul-Eugène Parent
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01567
UR - https://arxiv.org/abs/2611.01567
AB - Homology nilpotency need not increase under a product with a sphere. We construct a simply connected, closed smooth manifold M of dimension $53$, rationally elliptic with dim_ℚ(π_*(M)⊗ℚ)=7, such that cat_0(M)=10 and Hnil(M× S^2)=Hnil(M)=11. The inequality Hnil(X× Y)≤ Hnil(X)+Hnil(Y) can therefore be strict even for a product of a closed manifold with S^2 as the second factor.
ER -