有理同伦理论中的同调幂零性：胞腔附着、收缩塔、发散性与稳定化
Homology Nilpotency in Rational Homotopy Theory: Cell Attachments, Retractive Towers, Divergence, and Stabilization
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中文总结 AI 辅助
该研究在有理同伦理论中构造了nil_h(M)=3<Hnil(M)=4的极小Sullivan代数，揭示同调幂零性的商缺陷可通过逐次有限单连通有理球楔积稳定化消除，建立了稳定化后同调幂零性与同伦范畴的关系。
中文摘要 AI 辅助
单连通极小Sullivan代数M的同调幂零性nil_h(M)是使(M⁺)ⁿ⁺¹包含于一个无圈微分理想的最小整数n。我们建立了上下界准则，包括有理胞腔附着的方法，并构造了一个显式有限型极小Sullivan代数M，满足nil_h(M)=3<Hnil(M)=4。完全严格协调的收缩塔确定了该障碍：它们刻画了同伦幂零长度，同时揭示了由同调幂零性度量的额外商缺陷。随后我们证明，该刚性缺陷在经逐次有限楔积的单连通有理球稳定化后消失，即对所有有限型单连通有理空间X，有Hnil_s(X)=cat_o(X)。
英文摘要
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.
发表机构
- Department of Mathematics and Statistics, University of Ottawa（渥太华大学数学与统计系）
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