论文标题
在加权的简单同源性上
On Weighted Simplicial Homology
论文作者
论文摘要
我们开发了一个框架，用于计算具有离散评估环中系数的加权简单复合物的同源性。 Dawson [Cah]引入的加权简单复合物，$（x，v）$。 Topol。 géom。不同。 catég。 31（1990），第229--243页]是一种简单的复合体，$ x $，以及整数值函数，$ v $，将权重分配给简单，因此任何面部的重量都单调增加。此外，加权同源性，$ h_n^v（x）$，具有新的边界运算符，$ \ partial_n^v $。在与道森（Dawson）的不同之处，我们的方法以自然同构$θ$的加权链络合物为中心。关键对象是$ h^v_ {n}（x/θ）$，这是由$θ$引起的链络合物商的加权同源性，以较长的精确序列链接了具有不同权重的加权同源物。我们将为加权边界图的内核和图像构建基础，并将$ n $ simplices识别为$κ_n$ - 或$μ_n$ -vertices。加权同源组和碱基的长期精确序列使我们能够证明具有正式功率序列$ r = \ mathbb {f} [[π] $的加权简单同源性的结构定理，其中$ \ mathbb {f} $是一个字段。相对于简单同源性，新的扭转出现，我们将表明扭转模块连接到划分的$κ_n$和$μ__{n+1} $简单之间的配对。
We develop a framework for computing the homology of weighted simplicial complexes with coefficients in a discrete valuation ring. A weighted simplicial complex, $(X,v)$, introduced by Dawson [Cah. Topol. Géom. Différ. Catég. 31 (1990), pp. 229--243], is a simplicial complex, $X$, together with an integer-valued function, $v$, assigning weights to simplices, such that the weight of any of faces are monotonously increasing. In addition, weighted homology, $H_n^v(X)$, features a new boundary operator, $\partial_n^v$. In difference to Dawson, our approach is centered at a natural homomorphism $θ$ of weighted chain complexes. The key object is $H^v_{n}(X/θ)$, the weighted homology of a quotient of chain complexes induced by $θ$, appearing in a long exact sequence linking weighted homologies with different weights. We shall construct bases for the kernel and image of the weighted boundary map, identifying $n$-simplices as either $κ_n$- or $μ_n$-vertices. Long exact sequences of weighted homology groups and the bases, allow us to prove a structure theorem for the weighted simplicial homology with coefficients in a ring of formal power series $R=\mathbb{F}[[π]]$, where $\mathbb{F}$ is a field. Relative to simplicial homology new torsion arises and we shall show that the torsion modules are connected to a pairing between distinguished $κ_n$ and $μ_{n+1}$ simplices.
