GLMY路径同调的一个尖锐曲率阈值
A Sharp Curvature Threshold for GLMY Path Homology
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中文总结 AI 辅助
该研究证明了GLMY路径同调的一阶尖锐曲率阈值为1/2，还得出连通图曲率为正相关的基本群有限性结论，且高阶同调不受曲率严格正性强制消失的影响。
中文摘要 AI 辅助
设G为至少含一条边的有限简单图，我们证明了尖锐消失定理：κ_min^LLY(G)>1/2 ⇒ PathH₁(G;ℝ)=0。等价地，非零一阶GLMY路径同调意味着Lin-Lu-Yau曲率至多为1/2的边，该阈值1/2是尖锐的，且由C₅达到。证明结合了一阶GLMY路径同调的循环空间描述与Lin-Lu-Yau曲率的无极限拉普拉斯特征刻画。作为保曲率通用覆盖方法的次要结论，我们证明若G是连通的且κ_min^LLY(G)>0，则π₁(X₅(G),o)是有限的，其中X₅(G)通过填充长度至多为5的每条简单循环得到，等价于基于简单5-循环环生成的正规子群在π₁^GLMY(G,o)中具有有限指数。在高阶情形中情况不同：对每个整数r≥1，笛卡尔积T_r=C₅^□r在每条边上的曲率为1/(2r)，且对每个域𝔽，PathH_p(T_r;𝔽)≅𝔽^(binom(r,p))（0≤p≤r），故Lin-Lu-Yau曲率的严格正性并不强制高阶GLMY路径同调消失。
英文摘要
Let $G$ be a finite simple graph with at least one edge. We prove the sharp vanishing theorem \[ κ_{\min}^{\mathrm{LLY}}(G)>\frac12 \quad\Longrightarrow\quad \PathH_1(G;\R)=0. \] Equivalently, nonzero first GLMY path homology forces an edge of Lin--Lu--Yau curvature at most $1/2$. The threshold $1/2$ is sharp and is attained by $C_5$. The proof combines the cycle-space description of first GLMY path homology with the limit-free Laplacian characterization of Lin--Lu--Yau curvature. As a secondary consequence of the curvature-preserving universal-cover method, we prove that if $G$ is connected and $κ_{\min}^{\mathrm{LLY}}(G)>0$, then $π_1(\Xshort{5}(G),o)$ is finite, where $\Xshort{5}(G)$ is obtained by filling every simple cycle of length at most five. Equivalently, the normal subgroup generated by based simple $5$-cycle loops has finite index in $π_1^{\mathrm{GLMY}}(G,o)$. In higher degrees the situation is different: for each integer $r\geq1$, the Cartesian product $T_r=C_5^{\square r}$ has curvature $1/(2r)$ on every edge and, for every field $\F$, \[ \PathH_p(T_r;\F)\cong\F^{\binom rp}\qquad(0\leq p\leq r), \] so strict positivity of Lin--Lu--Yau curvature does not force higher-dimensional GLMY path homology to vanish.
