同调边缘：p-链 C 若存在 (p+1)-链 C′ 使 C=∂C′，则 C 为 p-boundary，即 C∈im ∂_{p+1}。来源 Szekeres §17.4；不是集合的 closure-minus-interior 边界，也不是边缘算子或边缘群本身。证据 reports/coordination-20261020/boundary-sense-review/evidence.json#szekeres-homological-boundary。
别名：p-boundary；r-boundary；-边界；-边缘；-边界；-边缘；边界
已登记的原文定义
Szekeres · source-snapshot-bc3404156f6e · §17.4
and the boundary of a $p$-simplex $\Delta$ is defined as the $( p - 1 )$ -chain
$$
\partial \Delta = \sum_ {i = 0} ^ {p} \Delta_ {i} = \sum_ {i = 0} ^ {p} (- 1) ^ {i} \langle \mathbf {x} _ {0}, \ldots , \widehat {\mathbf {x}} _ {i}, \ldots , \mathbf {x} _ {p} \rangle ,
$$
\
*Figure 17.5 Standard low dimensional $p$-simplexes*
and extends to all $p$-chains by linearity.链群语境的边缘；各构造按原书保留，不与点集边界合并。
Szekeres · source-snapshot-bc3404156f6e · §17.4
A $p$-chain $C$ is said to be a <u class="concept-term">cycle</u> if it has no <u class="concept-term">boundary</u>, $\partial C = 0 .$ . It is said to be a boundary if there exists a $(p+1)$-chain $C^{\prime}$ such that $C = \partial C ^ { \prime }$ . Clearly every boundary is a $p$-cycle since $\partial C = \partial ^ { 2 } C ^ { \prime } = 0$ , but the converse need not be true.链群语境的边缘；各构造按原书保留，不与点集边界合并。
Szekeres · source-snapshot-bc3404156f6e · §17.4
A $p$-boundary $B$ is a singular $p$-cycle on $M$ that is the boundary of a $(p+1)$-chain, $B = \partial _ { p + 1 } C . \mathrm { ~ A ~ }$ $p$-cycle $C$ is a singular $p$-chain on $M$ whose boundary vanishes, $\partial _ { p } C = 0$ Since $\partial ^ { 2 } = 0$ it is clear that every $p$-boundary is a $p$-cycle.链群语境的边缘；各构造按原书保留，不与点集边界合并。
Nakahara · second-edition-local · §3.3.2 · PDF 第124页
Definition 3.4. Let $K$ be an $n$ -dimensional simplicial complex and let $c \in C_r(K)$ . If there exists an element $d \in C_{r + 1}(K)$ such thatNakahara Definition 3.4：边界。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124页
$$
c = \partial_ {r + 1} d \tag {3.25}
$$
Nakahara Definition 3.4：边界。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124页
then $c$ is called an $r$ -boundary. The set of $r$ -boundaries $B_r(K)$ is a subgroup of $C_r(K)$ and is called the $r$ -boundary group. Note that $B_r(K) = \operatorname{im} \partial_{r+1}$ . [Remark: $B_n(K)$ is defined to be 0.]Nakahara Definition 3.4：边界。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124页
From lemma 3.1, it follows that $Z_{r}(K)$ and $B_{r}(K)$ are subgroups of $C_{r}(K)$ . We now prove an important relation between $Z_{r}(K)$ and $B_{r}(K)$ , which is crucial in the definition of homology groups.Nakahara Definition 3.4：边界。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。
Nakahara · second-edition-local
已登记出处中的最早位置：§3.3 · zh-CN。
来源短摘录与定位
§3.3 · zh-CN · machine_draft · unreviewed
正文请在原位置阅读；长段落、数学边界或未定位引用不截断嵌入，精确范围与指纹保存在结构化索引。
§4 · zh-CN · machine_draft · unreviewed
正文请在原位置阅读；长段落、数学边界或未定位引用不截断嵌入，精确范围与指纹保存在结构化索引。
§6.1 · zh-CN · machine_draft · unreviewed
正文请在原位置阅读；长段落、数学边界或未定位引用不截断嵌入，精确范围与指纹保存在结构化索引。
§6.2 · zh-CN · machine_draft · unreviewed
正文请在原位置阅读；长段落、数学边界或未定位引用不截断嵌入，精确范围与指纹保存在结构化索引。
Szekeres · source-snapshot-bc3404156f6e
已登记出处中的最早位置：§17.4 · zh-CN。
来源短摘录与定位
§17.4 · zh-CN · translation · unreviewed
一个-链若没有边界(boundary)，即，则称其为循环(cycle)。若存在一个-链使得，则称其为边界。显然，每个边界都是-循环，因为，但反之未必成立。
§17.4 · en · original_transcription · unreviewed
A -chain is said to be a cycle if it has no boundary, . It is said to be a boundary if there exists a -chain such that . Clearly every boundary is a -cycle since , but the converse need not be true.
原文中已定位的引用（不自动视为依赖）
- href=“03-section-3-3.md#bound-definition-1c3bccaa36fa6f6567f60365”
- href=”../chapter-03/03-section-3-3.md#bound-definition-1ccd5413f50b8ab807558a49”
- href=”../chapter-03/03-section-3-3.md#bound-definition-b6c4b4cd58ce97d21892883c”
- href=“05-section-4-5.md#bound-definition-aa4fa93d3f466df668189f39”
- href=”../chapter-02/03-section-2-3.md#bound-definition-bea63120260758e422dc7e27”
- href=“01-section-4-1.md#bound-definition-4a2489d02e04924c4017048c”
- href=”../chapter-02/01-section-2-1.md#bound-definition-c624ec2b869cd03c185835db”
- href=”../chapter-05/index.md#bound-definition-2d833ac0bb979a1a384519d5”
- href=”../chapter-03/03-section-3-3.md#bound-definition-b6c4b4cd58ce97d21892883c”
- href=”../chapter-03/02-section-3-2.md#section-3-2”
- href=”../chapter-03/03-section-3-3.md#bound-definition-b6c4b4cd58ce97d21892883c”
- [练习 6.4](02-section-6-2.md#exercise-6-4)