AI 中文总结
研究骨架同调\(K_{n}^{\varepsilon}(X)\)，证明任意度量空间的同构关系，引入超钻石度量等新方法，推广相关定理，定义同态及实同调、幻同调，表明实同调更能代表尺度下同调，幻同调可预测黎曼流形坍缩时维度变化。
AI 中文摘要
“骨架同调”\(K_{n}^{\varepsilon}(X)\)指由骨架\((n,\varepsilon)\)-单形生成的链复形\(S_{n}^{\varepsilon}(X)\)的同调群，即从标准单形的\(0\)-骨架到度量空间\(X\)且像直径小于\(\varepsilon>0\)的函数。此前Goldfarb对有限度量空间定义了此同调群，并证明其与VR复形的单纯同调\(H_{n}^{\Delta}(VR_{\varepsilon}(X))\)同构。本文证明了任意度量空间的同构关系，引入理解尺度下同调的新方法。定义了超钻石度量，证明“接近的闭链是同调的”，修改奇异同调方法推广了Hausmann定理和Latchev定理。定义同态\(\rho_{\varepsilon}:H_{n}(X)\rightarrow K_{n}^{\varepsilon}(X)\)，其像为尺度下的“实同调”\(H_{n}^{\varepsilon}(X)\)，认为它比\(H_{n}^{\Delta }(VR_{\varepsilon}(X))\)更能代表尺度下的真实同调。定义“幻同调”\(P_{n}^{\varepsilon}(X)=K_{n}^{\varepsilon}(X)/H_{n}^{\varepsilon}(X)\)，利用\(K_{n}^{\varepsilon}(X)\)的稳定性表明在黎曼流形坍缩时幻同调能预测极限中维度的突然下降。
英文摘要
"Skeletal homology" $K_{n}^{\varepsilon}(X)$ refers to the homology of the chain complex $S_{n}^{\varepsilon}(X)$ generated by skeletal $(n,\varepsilon)$-simplices, i.e. functions from the $0$-skeleton of the standard simplex into a metric space $X$, with image diameter less than $\varepsilon>0$. This homology was previously defined by Goldfarb, who showed that for finite metric spaces, it is isomorphic to the simplicial homology $H_{n}^Δ(VR_{\varepsilon}(X))$ of the VR complex. We prove an isomorphism for arbitrary metric spaces, and introduce new methods to understand homology at scale. We define an invariant metric on $S_{n}^{\varepsilon}(X)$, called the ultradiamond metric, that extends the uniform metric on skeletal simplices. With this metric we prove that "close cycles are homologous", which quickly leads to a host of stability results. We modify methods from singular homology to prove a strong generalization of Hausmann's Theorem, one of the two main justifications to use $H_{n}^Δ(VR_{\varepsilon}(X))$ as a proxy for homology in discrete metric spaces. The second justification is Latchev's Theorem, for which we also prove a strong generalization. We define a homomorphism $ρ_{\varepsilon}:H_{n}(X)\rightarrow K_{n}^{\varepsilon}(X)$ induced by repeated barycentric subdivision and restriction, the image of which we call "real homology" $H_{n}^{\varepsilon}(X)$ at scale. We argue that $H_{n}^{\varepsilon}(X)$ better represents bona fide homology at scale than $H_{n}^{Δ}(VR_{\varepsilon}(X))$. To distinguish them, we define "phantom homology" to be $P_{n}^{\varepsilon}(X)=K_{n}^{\varepsilon}(X)/H_{n}^{\varepsilon}(X)$, and use the stability of $K_{n}^{\varepsilon}(X)$ to show that in collapse of Riemannian manifolds (e.g. the Berger Spheres), phantom homology can anticipate the abrupt drop in dimension that occurs in the limit.
CommentsMinor correction to Remark 10
