Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
别名：；-chain group
词性：名词
已登记的原文定义
Nakahara · second-edition-local · §3.3.2 · PDF 第121页
Definition 3.2. The $r$ -chain group $C_r(K)$ of a simplicial complex $K$ is a free Abelian group generated by the oriented $r$ -simplexes of $K$ . If $r > \dim K$ , $C_r(K)$ is defined to be 0. An element of $C_r(K)$ is called an $r$ -chain.Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第121页
Let there be $I_r$ $r$ -simplexes in $K$ . We denote each of them by $\sigma_{r,i}$ ( $1 \leq i \leq I_r$ ). Then $c \in C_r(K)$ is expressed asNakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第121页
$$
c = \sum_ {i = 1} ^ {I _ {r}} c _ {i} \sigma_ {r, i} \quad c _ {i} \in \mathbb {Z}. \tag {3.15}
$$
Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第122页
The integers $c_{i}$ are called the coefficients of $c$ . The group structure is given as follows. The addition of two $r$ -chains, $c = \sum_{i} c_{i} \sigma_{r,i}$ and $c' = \sum_{i} c_{i}' \sigma_{r,i}$ , isNakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第122页
$$
c + c ^ {\prime} = \sum_ {i} \left(c _ {i} + c _ {i} ^ {\prime}\right) \sigma_ {r, i}. \tag {3.16}
$$
Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第122页
The unit element is $0 = \sum_{i} 0 \cdot \sigma_{r,i}$ , while the inverse element of $c$ is $-c = \sum_{i} (-c_i) \sigma_{r,i}$ . [Remark: An oppositely oriented $r$ -simplex $-\sigma_r$ is identified with $(-1) \sigma_r \in C_r(K)$ .] Thus, $C_r(K)$ is a free Abelian group of rank $I_r$ ,Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第122页
$$
C _ {r} (K) \cong \underbrace {\mathbb {Z} \oplus \mathbb {Z} \oplus \cdots \oplus \mathbb {Z}} _ {I _ {r}}. \tag {3.17}
$$
Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第122页
Before we define the cycle group and the boundary group, we need to introduce the boundary operator. Let us denote the boundary of an $r$ -simplex $\sigma_r$ by $\partial_r\sigma_r$ . $\partial_r$ should be understood as an operator acting on $\sigma_r$ to produce its boundary. This point of view will be elaborated later. Let us look at the boundaries of lower-dimensional simplexes. Since a 0-simplex has no boundary, we defineNakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第122页
$$
\partial_ {0} p _ {0} = 0. \tag {3.18}
$$
Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第122页
For a 1-simplex $(p_0p_1)$ , we defineNakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第122页
$$
\partial_ {1} \left(p _ {0} p _ {1}\right) = p _ {1} - p _ {0}. \tag {3.19}
$$
Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第122页
The reader might wonder about the appearance of a minus sign in front of $p_0$ . This is again related to the orientation. The following examples will clarify this point. In figure 3.7(a), an oriented 1-simplex $(p_0p_2)$ is divided into two, $(p_0p_1)$ and $(p_1p_2)$ . We agree that the boundary of $(p_0p_2)$ is $\{p_0\} \cup \{p_2\}$ and so should be that of $(p_0p_1) + (p_1p_2)$ . If $\partial_1(p_0p_2)$ were defined to be $p_0 + p_2$ , we would have $\partial_1(p_0p_1) + \partial_1(p_1p_2) = p_0 + p_1 + p_1 + p_2$ . This is not desirable since $p_1$ is a fictitious boundary. If, instead, we take $\partial_1(p_0p_2) = p_2 - p_0$ , we will have $\partial_1(p_0p_1) + \partial_1(p_1p_2) = p_1 - p_0 + p_2 - p_1 = p_2 - p_0$ as expected. The next example is the triangle of figure 3.7(b). It is the sum of three oriented 1-simplexes,Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第123页
$(p_0p_1) + (p_1p_2) + (p_2p_0)$ . We agree that it has no boundary. If we insisted on the rule $\partial_{1}(p_{0}p_{1}) = p_{0} + p_{1}$ , we would haveNakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第123页
$$
\partial_ {1} \left(p _ {0} p _ {1}\right) + \partial_ {1} \left(p _ {1} p _ {2}\right) + \partial_ {1} \left(p _ {2} p _ {0}\right) = p _ {0} + p _ {1} + p _ {1} + p _ {2} + p _ {2} + p _ {0}
$$
Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第123页
which contradicts our intuition. If, on the other hand, we take $\partial_1(p_0p_1) = p_1 - p_0$ , we haveNakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第123页
$$
\partial_ {1} \left(p _ {0} p _ {1}\right) + \partial_ {1} \left(p _ {1} p _ {2}\right) + \partial_ {1} \left(p _ {2} p _ {0}\right) = p _ {1} - p _ {0} + p _ {2} - p _ {1} + p _ {0} - p _ {2} = 0
$$
Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第123页
as expected. Hence, we put a plus sign if the first vertex is omitted and a minus sign if the second is omitted. We employ this fact to define the boundary of a general $r$ -simplex.Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第123页
Let $\sigma_r(p_0\ldots p_r)$ ( $r > 0$ ) be an oriented $r$ -simplex. The boundary $\partial_r\sigma_r$ of $\sigma_r$ is an $(r - 1)$ -chain defined byNakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第123页
$$
\partial_ {r} \sigma_ {r} \equiv \sum_ {i = 0} ^ {r} (- 1) ^ {i} \left(p _ {0} p _ {1} \dots \hat {p} _ {i} \dots p _ {r}\right) \tag {3.20}
$$
Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第123页
where the point $p_i$ under $\hat{\mathbf{\mu}}$ is omitted. For example,Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第123页
$$
\begin{array}{l} \partial_ {2} \left(p _ {0} p _ {1} p _ {2}\right) = \left(p _ {1} p _ {2}\right) - \left(p _ {0} p _ {2}\right) + \left(p _ {0} p _ {1}\right) \\ \partial_ {3} \left(p _ {0} p _ {1} p _ {2} p _ {3}\right) = \left(p _ {1} p _ {2} p _ {3}\right) - \left(p _ {0} p _ {2} p _ {3}\right) + \left(p _ {0} p _ {1} p _ {3}\right) - \left(p _ {0} p _ {1} p _ {2}\right). \\ \end{array}
$$
Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第123页
We formally define $\partial_0\sigma_0 = 0$ for $r = 0$ .Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第123页
The operator $\partial_r$ acts linearly on an element $c = \sum_{i} c_i \sigma_{r,i}$ of $C_r(K)$ ,Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第123页
$$
\partial_ {r} c = \sum_ {i} c _ {i} \partial_ {r} \sigma_ {r, i}. \tag {3.21}
$$
Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第123页
The RHS of (3.21) is an element of $C_{r-1}(K)$ . Accordingly, $\partial_r$ defines a mapNakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第123页
$$
\partial_ {r}: C _ {r} (K) \rightarrow C _ {r - 1} (K). \tag {3.22}
$$
Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第123页
$\partial_r$ is called the boundary operator. It is easy to see that the boundary operator is a homomorphism.Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第123页
Let $K$ be an $n$ -dimensional simplicial complex. There exists a sequence of free Abelian groups and homomorphisms,Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第123页
$$
0 \xrightarrow {i} C _ {n} (K) \xrightarrow {\partial_ {n}} C _ {n - 1} (K) \xrightarrow {\partial_ {n - 1}} \dots \xrightarrow {\partial_ {2}} C _ {1} (K) \xrightarrow {\partial_ {1}} C _ {0} (K) \xrightarrow {\partial_ {0}} 0 \tag {3.23}
$$
Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §3.3.2 · PDF 第123页
where $i: 0 \hookrightarrow C_n(K)$ is an inclusion map (0 is regarded as the unit element of $C_n(K)$ ). This sequence is called the chain complex associated with $K$ and is denoted by $C(K)$ . It is interesting to study the image and kernel of the homomorphisms $\partial_r$ .Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
按作者与版本查阅
以下按已登记位置排列。出现顺序、原文引用与已登记依赖分别展示；最早位置不代表全书首次定义。
尚无可定位的当前书页出处；未解析登记见下方。
机器入库的作者陈述（未经人工审阅）
机器入库 · 未经人工审阅
以下由语义核糖体自动读取原著生成并入库。引文逐字取自 OCR，概念归属、角色与依赖均为模型判断；可按运行整体撤销。