{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:02f7a498-5c17-4366-9479-8666d5f9bb2e", "labels": { "zh-CN": "同调群", "en": "homology group" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "6696b8d9f72d6880380686de94fe07274822c8cc9579b77d9f3ccc5ff6dbd407", "occurrences": [ { "concept_id": "urn:uuid:02f7a498-5c17-4366-9479-8666d5f9bb2e", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-17/04-homology-and-cohomology", "anchor": "first-10254fb45e81b1ade885-zh", "chapter": "17", "section": "17.4", "offset": 10494, "range": [ 14791, 15029 ], "source_revision": "962da89618680522177106de9b1899c86eb3dedc9484f12c33659d432388a117", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "同调群(homology group)", "context": { "range": [ 14780, 15627 ], "quote": "通常这被称为第$p$个同调群(homology group)，其中只有阿贝尔群性质是相关的。若两个循环$C _ { 1 }$与$C _ { 2 }$属于同一个同调类，即存在一个链$C$使得$C _ { 1 } - C _ { 2 } = \\partial C$，则称它们是同调的(homologous)的。第$p$个同调空间的维数称为第$p$个贝蒂数(Betti number)，", "quote_sha256": "8de502e60460eae59becddbfdfb4feafece5152c148f3f7d22c008e6fa4e610f", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-17/04-homology-and-cohomology", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-e3ee5f5ad0e58e3ddf3c77df" }, { "concept_id": "urn:uuid:02f7a498-5c17-4366-9479-8666d5f9bb2e", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-17/04-homology-and-cohomology", "anchor": "first-10254fb45e81b1ade885-en", "chapter": "17", "section": "17.4", "offset": 13018, "range": [ 16842, 17075 ], "source_revision": "bcdaffd9c93fa5d4dbb16ac427fef782512ff58c5d5472ea282c7769a2173a3d", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "homology group", "context": { "range": [ 16808, 17836 ], "quote": "Commonly this is called the $p$th homology group, only the abelian group property being relevant. Two cycles $C _ { 1 }$ and $C _ { 2 }$ are said to be homologous if they belong to the same homology class – that is, if there exists a chain $C$ such that $C _ { 1 } - C _ { 2 } = \\partial C$ . The dimension of the $p$th homology space is known as the $p$ th Betti number,", "quote_sha256": "1371f5c4e5c39df1f8cd84a5e4b90593dea95626a22fccfeaad025afe55b7a95", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-17/04-homology-and-cohomology", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-a0de0bcc0f43067eec03d112" } ], "registered_positions": [ { "route": "en/chapter-17/04-homology-and-cohomology", "lang": "en", "section": "17.4", "signal": "term-mark", "label": "homology group", "english": "homology group", "offset": 11299 }, { "route": "zh/chapter-17/04-homology-and-cohomology", "lang": "zh-CN", "section": "17.4", "signal": "term-mark", "label": "同调群", "english": "homology group", "offset": 8880 } ], "statements": [ { "id": "statement-27db493e412102426ffd26d7", "scope": { "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "source_id": "nakahara-mineru-bd76b709" }, "chapter": "nakahara-3", "chapter_number": 3, "author_label": "Nakahara 第3章", "label": "边界同态：式(3.20)–(3.22)", "role": "definition", "conditions": "本章整数系数单纯链；链群为自由阿贝尔群。该路径以群、子群和有向单形知识为已具备背景。", "review_status": "codex-source-reviewed; 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pending-independent-human-review", "mathematical_status": "author-stated-unverified", "source_kind": "ribosome-atoms", "route": "author-statements/nakahara-3", "anchor": "statement-45381cbdd77285069b293927" }, { "id": "statement-68ebd9974e3ebfba6471dbb4", "scope": { "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "source_id": "nakahara-mineru-bd76b709" }, "chapter": "nakahara-4-pilot", "chapter_number": 4, "author_label": "Nakahara §4.4.3 · 新单元候选", "label": "例 4.11 Klein 瓶", "role": "use", "conditions": "Klein 瓶基本群的表示已知。", "review_status": "codex-source-reviewed; pending-independent-human-review", "mathematical_status": "author-stated-unverified", "source_kind": "ribosome-atoms", "route": "author-statements/nakahara-4-pilot", "anchor": "statement-68ebd9974e3ebfba6471dbb4" }, { "id": "statement-8ae86ef7b666cb37cb3076ea", "scope": { "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "source_id": "nakahara-mineru-bd76b709" }, "chapter": "nakahara-3", "chapter_number": 3, "author_label": "Nakahara 第3章", "label": "定义3.5 同调群", "role": "definition", "conditions": "本章主要采用整数系数；$0\\le r\\le n$。不能与实系数同调空间按定义完全等同。", "review_status": "codex-source-reviewed; pending-independent-human-review", "mathematical_status": "author-stated-unverified", "source_kind": "ribosome-atoms", "route": "author-statements/nakahara-3", "anchor": "statement-8ae86ef7b666cb37cb3076ea" }, { "id": "statement-8d8d2177e831615076629fce", "scope": { "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "source_id": "nakahara-mineru-bd76b709" }, "chapter": "nakahara-4-pilot", "chapter_number": 4, "author_label": "Nakahara §4.4.3 · 新单元候选", "label": "定理 4.9", "role": "theorem", "conditions": "K 为连通单纯复形；F 为 π_1(|K|) 的换位子群。", "review_status": "codex-source-reviewed; pending-independent-human-review", "mathematical_status": "author-stated-unverified", "source_kind": "ribosome-atoms", "route": "author-statements/nakahara-4-pilot", "anchor": "statement-8d8d2177e831615076629fce" }, { "id": "statement-a9109552ffd06f4f8a886bfa", "scope": { "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "source_id": "nakahara-mineru-bd76b709" }, "chapter": "nakahara-3", "chapter_number": 3, "author_label": "Nakahara 第3章", "label": "定义3.2：整数系数链群", "role": "definition", "conditions": "本章整数系数单纯链；链群为自由阿贝尔群。该路径以群、子群和有向单形知识为已具备背景。", "review_status": "codex-source-reviewed; pending-independent-human-review", "mathematical_status": "author-stated-unverified", "source_kind": "ribosome-atoms", "route": "author-statements/nakahara-3", "anchor": "statement-a9109552ffd06f4f8a886bfa" }, { "id": "statement-ac7699973242d9783a9bb1ca", "scope": { "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "source_id": "nakahara-mineru-bd76b709" }, "chapter": "nakahara-3", "chapter_number": 3, "author_label": "Nakahara 第3章", "label": "定义3.3：循环群是边界同态的核", "role": "definition", "conditions": "本章整数系数单纯链；链群为自由阿贝尔群。该路径以群、子群和有向单形知识为已具备背景。", "review_status": "codex-source-reviewed; pending-independent-human-review", "mathematical_status": "author-stated-unverified", "source_kind": "ribosome-atoms", "route": "author-statements/nakahara-3", "anchor": "statement-ac7699973242d9783a9bb1ca" }, { "id": "statement-ae16e6e7da0a84f365502acf", "scope": { "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "source_id": "nakahara-mineru-bd76b709" }, "chapter": "nakahara-3", "chapter_number": 3, "author_label": "Nakahara 第3章", "label": "定义3.5：同调群及作者的良定义说明", "role": "definition", "conditions": "本章整数系数单纯链；链群为自由阿贝尔群。该路径以群、子群和有向单形知识为已具备背景。 可另行采用实数或模二系数；不将不同系数的定义自动等同。", "review_status": "codex-source-reviewed; pending-independent-human-review", "mathematical_status": "author-stated-unverified", "source_kind": "ribosome-atoms", "route": "author-statements/nakahara-3", "anchor": "statement-ae16e6e7da0a84f365502acf" }, { "id": "statement-b9954f1c50ffc2105adda9f2", "scope": { "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "source_id": "nakahara-mineru-bd76b709" }, "chapter": "nakahara-4-pilot", "chapter_number": 4, "author_label": "Nakahara §4.4.3 · 新单元候选", "label": "推论 4.3", "role": "theorem", "conditions": "X 与 Y 具有相同的同伦型。", "review_status": "codex-source-reviewed; pending-independent-human-review", "mathematical_status": "author-stated-unverified", "source_kind": "ribosome-atoms", "route": "author-statements/nakahara-4-pilot", "anchor": "statement-b9954f1c50ffc2105adda9f2" }, { "id": "statement-cb6bb26d0da659fa31ee1333", "scope": { "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "source_id": "szekeres-chapter-17" }, "chapter": "library-szekeres-17", "chapter_number": 17, "author_label": "Szekeres §17.4", "label": "实系数同调空间与同调群称呼", "role": "definition", "conditions": "Szekeres 用光滑奇异单形的实系数链；整数系数的挠信息不能由该入口抹去。", "review_status": "codex-source-reviewed; pending-independent-human-review", "mathematical_status": "author-stated-unverified", "source_kind": "library-excerpt", "route": "author-statements/library-szekeres-17", "anchor": "statement-cb6bb26d0da659fa31ee1333" }, { "id": "statement-cf70f6e882eed0df163c05a1", "scope": { "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "source_id": "nakahara-mineru-bd76b709" }, "chapter": "nakahara-3", "chapter_number": 3, "author_label": "Nakahara 第3章", "label": "定理3.3 边界群包含于循环群", "role": "proof", "conditions": "这一步为同调商群提供子群包含关系。", "review_status": "codex-source-reviewed; pending-independent-human-review", "mathematical_status": "author-stated-unverified", "source_kind": "ribosome-atoms", "route": "author-statements/nakahara-3", "anchor": "statement-cf70f6e882eed0df163c05a1" }, { "id": "statement-dd7721e0c9adf5e7b891bfcd", "scope": { "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "source_id": "nakahara-mineru-bd76b709" }, "chapter": "nakahara-3", "chapter_number": 3, "author_label": "Nakahara 第3章", "label": "引理3.3 连续两次取边界为零", "role": "theorem", "conditions": "r 链与边界算子按本书有向单形约定。", "review_status": "codex-source-reviewed; pending-independent-human-review", "mathematical_status": "author-stated-unverified", "source_kind": "ribosome-atoms", "route": "author-statements/nakahara-3", "anchor": "statement-dd7721e0c9adf5e7b891bfcd" }, { "id": "statement-ddc45b084c0fae9befd3525f", "scope": { "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "source_id": "nakahara-mineru-bd76b709" }, "chapter": "nakahara-3", "chapter_number": 3, "author_label": "Nakahara 第3章", "label": "§3.1.1 商群 (3.2)–(3.3)", "role": "definition", "conditions": "Nakahara 本章限于阿贝尔群，运算记 +，单位元记 0；不把本章约定推广到任意群。 本章每个子群均正规，不能省去一般群商构造中的正规性条件。", "review_status": "codex-source-reviewed; pending-independent-human-review", "mathematical_status": "author-stated-unverified", "source_kind": "ribosome-atoms", "route": "author-statements/nakahara-3", "anchor": "statement-ddc45b084c0fae9befd3525f" }, { "id": "statement-e9cdbaabe0ae1a11cb9cc9b3", "scope": { "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "source_id": "nakahara-mineru-bd76b709" }, "chapter": "nakahara-3", "chapter_number": 3, "author_label": "Nakahara 第3章", "label": "定义3.4：边界群是前一边界同态的像", "role": "definition", "conditions": "本章整数系数单纯链；链群为自由阿贝尔群。该路径以群、子群和有向单形知识为已具备背景。", "review_status": "codex-source-reviewed; pending-independent-human-review", "mathematical_status": "author-stated-unverified", "source_kind": "ribosome-atoms", "route": "author-statements/nakahara-3", "anchor": "statement-e9cdbaabe0ae1a11cb9cc9b3" }, { "id": "statement-f5c8dc7bcb0740921fe8819b", "scope": { "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "source_id": "nakahara-mineru-bd76b709" }, "chapter": "nakahara-3", "chapter_number": 3, "author_label": "Nakahara 第3章", "label": "引理3.1 核与像为子群", "role": "theorem", "conditions": "Nakahara 本章限于阿贝尔群，运算记 +，单位元记 0；不把本章约定推广到任意群。", "review_status": "codex-source-reviewed; pending-independent-human-review", "mathematical_status": "author-stated-unverified", "source_kind": "ribosome-atoms", "route": "author-statements/nakahara-3", "anchor": "statement-f5c8dc7bcb0740921fe8819b" }, { "id": "statement-247c80347353b6759a829348", "scope": { "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "source_id": "nakahara-mineru-bd76b709" }, "chapter": "nakahara-3", "chapter_number": 3, "author_label": "Nakahara 第3章", "label": "定理3.1 的证明", "role": "proof", "conditions": "Nakahara 本章限于阿贝尔群，运算记 +，单位元记 0；不把本章约定推广到任意群。", "review_status": "codex-source-reviewed; pending-independent-human-review", "mathematical_status": "author-stated-unverified", "source_kind": "ribosome-atoms", "route": "author-statements/nakahara-3", "anchor": "statement-247c80347353b6759a829348" }, { "id": "statement-e147ac6a63afbf8aa27b13b9", "scope": { "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "source_id": "nakahara-mineru-bd76b709" }, "chapter": "nakahara-4-pilot", "chapter_number": 4, "author_label": "Nakahara §4.4.3 · 新单元候选", "label": "推论 4.2", "role": "theorem", "conditions": "X 为连通拓扑空间。", "review_status": "codex-source-reviewed; pending-independent-human-review", "mathematical_status": "author-stated-unverified", "source_kind": "ribosome-atoms", "route": "author-statements/nakahara-4-pilot", "anchor": "statement-e147ac6a63afbf8aa27b13b9" } ], "resolved_references": [], "reading_order": [ "occ-e3ee5f5ad0e58e3ddf3c77df", "occ-a0de0bcc0f43067eec03d112" ], "recorded_dependencies": [ { "id": "dependency-0ec8fbcd4724a3ea6d6a737a", "relation": "uses-author-statement", "basis": "explicit-author-reference", "from_statement": "statement-247c80347353b6759a829348", "to_statement": "statement-f5c8dc7bcb0740921fe8819b", "quote": "Proof. Both sides are groups according to lemma 3.1. Define a map $\\varphi : G_1 / \\ker f \\to \\operatorname{im} f$ by $\\varphi([x]) = f(x)$ . This map is well defined since for $x' \\in [x]$ , there exists $h \\in \\ker f$ such that $x' = x + h$ and $f(x') = f(x + h) = f(x) + f(h) = f(x)$ . Now we show that $\\varphi$ is an isomorphism. First, $\\varphi$ is a homomorphism,", "citation": "according to lemma 3.1", "target_marker": "Lemma 3.1.", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-581d033a4d3b64de8b484b60", "relation": "uses-author-statement", "basis": "explicit-author-reference", "from_statement": "statement-45381cbdd77285069b293927", "to_statement": "statement-f5c8dc7bcb0740921fe8819b", "quote": "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.", "citation": "From lemma 3.1", "target_marker": "Lemma 3.1.", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-f504b265a21f6d412d13d8bb", "relation": "uses-author-statement", "basis": "explicit-author-reference", "from_statement": "statement-cf70f6e882eed0df163c05a1", "to_statement": "statement-dd7721e0c9adf5e7b891bfcd", "quote": "Proof. This is obvious from lemma 3.3. Any element $c$ of $B_r(K)$ is written as $c = \\partial_{r+1}d$ for some $d \\in C_{r+1}(K)$ . Then we find $\\partial_r c = \\partial_r(\\partial_{r+1}d) = 0$ , that is, $c \\in Z_r(K)$ . This implies $Z_r(K) \\supset B_r(K)$ .", "citation": "from lemma 3.3", "target_marker": "Lemma 3.3.", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-d882a2d2a9998b9472515b5f", "relation": "uses-author-statement", "from_statement": "statement-ae16e6e7da0a84f365502acf", "to_statement": "statement-cf70f6e882eed0df163c05a1", "basis": "author-transition", "quote": "Since $B_r(K)$ is a subgroup of $Z_r(K)$ , $H_r(K)$ is well defined.", "citation": null, "target_marker": "Theorem 3.3. Let $Z_{r}(K)$ and $B_{r}(K)$ be the ", "reason": "作者的 Since 承接使用了定理3.3的包含关系，保证分母是分子中的子群；阿贝尔条件另由链群定义提供。", "evidence_status": "codex-source-reviewed-candidate", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-32cccb33b8b8fee29a6214e5", "relation": "uses-author-statement", "from_statement": "statement-ae16e6e7da0a84f365502acf", "to_statement": "statement-ddc45b084c0fae9befd3525f", "basis": "construction-use", "quote": "$$\nH _ {r} (K) \\equiv Z _ {r} (K) / B _ {r} (K). \\tag {3.28}\n$$", "citation": null, "target_marker": "Let $H$ be a subgroup of $G$ . We say $x, y \\in G$ ", "reason": "这里实际使用商群构造；本章阿贝尔条件保证无需另加一般群的正规性前提。此边是来源构造使用候选，不是编号回指。", "evidence_status": "codex-source-reviewed-candidate", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-1ee9cf68f1b8bc922b9b17c5", "relation": "uses-author-statement", "from_statement": "statement-ae16e6e7da0a84f365502acf", "to_statement": "statement-a9109552ffd06f4f8a886bfa", "basis": "construction-use", "quote": "Since $B_r(K)$ is a subgroup of $Z_r(K)$ , $H_r(K)$ is well defined.", "citation": null, "target_marker": "Definition 3.2. The $r$ -chain group $C_r(K)$ of a s", "reason": "链群的阿贝尔结构经子群继承，说明本处的商群条件；这不是把一般群的子群都当成正规子群。", "evidence_status": "codex-source-reviewed-candidate", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-350d5feb5ff2f69c0982e91a", "relation": "uses-author-statement", "from_statement": "statement-ae16e6e7da0a84f365502acf", "to_statement": "statement-ac7699973242d9783a9bb1ca", "basis": "construction-use", "quote": "$$\nH _ {r} (K) \\equiv Z _ {r} (K) / B _ {r} (K). \\tag {3.28}\n$$", "citation": null, "target_marker": "Definition 3.3. If $c \\in C_r(K)$ satisfies", "reason": "商群分子 Zr 必须按作者的核定义理解，而非把所有链都当作循环。", "evidence_status": "codex-source-reviewed-candidate", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-4d9ea95fb7a4a526094a73db", "relation": "uses-author-statement", "from_statement": "statement-ae16e6e7da0a84f365502acf", "to_statement": "statement-e9cdbaabe0ae1a11cb9cc9b3", "basis": "construction-use", "quote": "$$\nH _ {r} (K) \\equiv Z _ {r} (K) / B _ {r} (K). \\tag {3.28}\n$$", "citation": null, "target_marker": "Definition 3.4. Let $K$ be an $n$ -dimensional simpl", "reason": "商群分母 Br 必须按作者的像定义理解，商掉的是边界循环。", "evidence_status": "codex-source-reviewed-candidate", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-5ab4c184178868d2f11eed23", "relation": "uses-author-statement", "from_statement": "statement-ae16e6e7da0a84f365502acf", "to_statement": "statement-45381cbdd77285069b293927", "basis": "construction-use", "quote": "Since $B_r(K)$ is a subgroup of $Z_r(K)$ , $H_r(K)$ is well defined.", "citation": null, "target_marker": "From lemma 3.1, it follows that $Z_{r}(K)$ and $B_{r", "reason": "先确认 Zr、Br 的群资格，再结合定理3.3的包含关系组成商群。该前置结论单独不证明 Br 包含于 Zr。", "evidence_status": "codex-source-reviewed-candidate", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-fd0436467a9a3a8bd6545786", "relation": "uses-author-statement", "from_statement": "statement-45381cbdd77285069b293927", "to_statement": "statement-ac7699973242d9783a9bb1ca", "basis": "construction-use", "quote": "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.", "citation": null, "target_marker": "Definition 3.3. If $c \\in C_r(K)$ satisfies", "reason": "引理3.1应用到核时，核正是定义3.3的循环群。", "evidence_status": "codex-source-reviewed-candidate", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-514dec634fe89684d29346b0", "relation": "uses-author-statement", "from_statement": "statement-45381cbdd77285069b293927", "to_statement": "statement-e9cdbaabe0ae1a11cb9cc9b3", "basis": "construction-use", "quote": "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.", "citation": null, "target_marker": "Definition 3.4. Let $K$ be an $n$ -dimensional simpl", "reason": "引理3.1应用到像时，像正是定义3.4的边界群。", "evidence_status": "codex-source-reviewed-candidate", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-897207a3e13dbddb26b2a837", "relation": "uses-author-statement", "from_statement": "statement-ac7699973242d9783a9bb1ca", "to_statement": "statement-27db493e412102426ffd26d7", "basis": "construction-use", "quote": "$c$ is called an $r$ -cycle. The set of $r$ -cycles $Z_{r}(K)$ is a subgroup of $C_{r}(K)$ and is called the $r$ -cycle group. Note that $Z_{r}(K) = \\ker \\partial_{r}$ . [Remark: If $r = 0$ , $\\partial_0 c$ vanishes identically and $Z_{0}(K) = C_{0}(K)$ , see (3.23).]", "citation": null, "target_marker": "Let $\\sigma_r(p_0\\ldots p_r)$ ( $r > 0$ ) be an orien", "reason": "循环条件和核需要此前定义的边界同态。", "evidence_status": "codex-source-reviewed-candidate", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-cb72658754eea40f65bd416a", "relation": "uses-author-statement", "from_statement": "statement-e9cdbaabe0ae1a11cb9cc9b3", "to_statement": "statement-27db493e412102426ffd26d7", "basis": "construction-use", "quote": "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.]", "citation": null, "target_marker": "Let $\\sigma_r(p_0\\ldots p_r)$ ( $r > 0$ ) be an orien", "reason": "边界及像需要此前从 r+1 链到 r 链的边界同态。", "evidence_status": "codex-source-reviewed-candidate", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-e997bfe83e609355c7530f3b", "relation": "uses-author-statement", "from_statement": "statement-27db493e412102426ffd26d7", "to_statement": "statement-a9109552ffd06f4f8a886bfa", "basis": "author-transition", "quote": "The RHS of (3.21) is an element of $C_{r-1}(K)$ . Accordingly, $\\partial_r$ defines a map", "citation": null, "target_marker": "Definition 3.2. The $r$ -chain group $C_r(K)$ of a s", "reason": "作者将边界线性延拓到链，因此映射的定义域、值域取自定义3.2的链群。", "evidence_status": "codex-source-reviewed-candidate", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-dc78aa117f41159a20c5afd3", "relation": "uses-author-statement", "from_statement": "statement-dd7721e0c9adf5e7b891bfcd", "to_statement": "statement-27db493e412102426ffd26d7", "basis": "construction-use", "quote": "Lemma 3.3. The composite map $\\partial_r\\circ \\partial_{r + 1}:C_{r + 1}(K)\\to C_{r - 1}(K)$ is a zero map; that is, $\\partial_r(\\partial_{r + 1}c) = 0$ for any $c\\in C_{r + 1}(K)$ .", "citation": null, "target_marker": "Let $\\sigma_r(p_0\\ldots p_r)$ ( $r > 0$ ) be an orien", "reason": "连续取边界的零映射结论使用此前的边界算子；这条边不替代引理3.3的证明。", "evidence_status": "codex-source-reviewed-candidate", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-79d536f23a188cd3c9478bc0", "relation": "uses-author-statement", "from_statement": "statement-68ebd9974e3ebfba6471dbb4", "to_statement": "statement-8d8d2177e831615076629fce", "basis": "construction-use", "quote": "Example 4.11. From $\\pi_1(\\text{Klein bottle}) \\cong (x, y; xyxy^{-1})$ , we have", "target_marker": "Theorem 4.9. Let $K$ be a connected simplicial complex. Then $H_{1}(K)$ is isomorphic to $\\pi_1(|K|)/F$", "citation": null, "reason": "例 4.11 的计算过程正是定理 4.9 所述构造的具体实施：取基本群模去换位子群。", "review_status": "codex-source-reviewed; pending-independent-human-review", "evidence_status": "codex-source-reviewed-candidate" }, { "id": "dependency-368799583a78ce7e483543d8", "relation": "uses-author-statement", "from_statement": "statement-e147ac6a63afbf8aa27b13b9", "to_statement": "statement-8d8d2177e831615076629fce", "basis": "explicit-author-reference", "quote": "[Use theorem 4.9.]", "target_marker": "Theorem 4.9. Let $K$ be a connected simplicial complex.", "citation": "theorem 4.9", "reason": "作者明确指示使用定理 4.9 来推导此推论。", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-faa1666301a7c6eb4731bb1f", "relation": "uses-author-statement", "from_statement": "statement-b9954f1c50ffc2105adda9f2", "to_statement": "statement-8d8d2177e831615076629fce", "basis": "explicit-author-reference", "quote": "[Use theorems 4.9 and 4.3.]", "target_marker": "Theorem 4.9. Let $K$ be a connected simplicial complex.", "citation": "theorems 4.9", "reason": "作者明确指示使用定理 4.9 作为推导依据之一。", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-d6aa06fe2b4be3c8aa50649d", "relation": "uses-author-statement", "from_statement": "statement-445d27e174a7c2fa11b79bb7", "to_statement": "statement-8d8d2177e831615076629fce", "basis": "author-transition", "quote": "The theorem states that if $\\pi_1(|K|) = (x_i : r_m)$ , then $H_1(K) \\cong (x_i : r_m, x_i x_j x_i^{-1} x_j^{-1})$ .", "target_marker": "Theorem 4.9. Let $K$ be a connected simplicial complex.", "citation": null, "reason": "作者以 'The theorem states' 回指定理 4.9 的结论来推导 bouquet 的同调群。", "review_status": "codex-source-reviewed; pending-independent-human-review", "evidence_status": "codex-source-reviewed-candidate" } ], "reading_paths": [ { "id": "nakahara-homology-definition", "concept_id": "urn:uuid:02f7a498-5c17-4366-9479-8666d5f9bb2e", "title": "读到定义3.5：为什么能取这个商群", "goal_statement": "statement-ae16e6e7da0a84f365502acf", "statements": [ "statement-ddc45b084c0fae9befd3525f", "statement-f5c8dc7bcb0740921fe8819b", "statement-a9109552ffd06f4f8a886bfa", "statement-27db493e412102426ffd26d7", "statement-ac7699973242d9783a9bb1ca", "statement-e9cdbaabe0ae1a11cb9cc9b3", "statement-45381cbdd77285069b293927", "statement-dd7721e0c9adf5e7b891bfcd", "statement-cf70f6e882eed0df163c05a1", "statement-ae16e6e7da0a84f365502acf" ], "step_roles": { "statement-ddc45b084c0fae9befd3525f": "提供等价类及代表元无关的商群构造。", "statement-f5c8dc7bcb0740921fe8819b": "保证同态的核与像是子群。", "statement-a9109552ffd06f4f8a886bfa": "固定整数系数与阿贝尔链群。", "statement-27db493e412102426ffd26d7": "建立链群之间的边界同态。", "statement-ac7699973242d9783a9bb1ca": "用核刻画没有边界的循环。", "statement-e9cdbaabe0ae1a11cb9cc9b3": "用像刻画来自更高一维链的边界。", "statement-45381cbdd77285069b293927": "将一般的核／像子群结论用于 Zr、Br。", "statement-dd7721e0c9adf5e7b891bfcd": "说明连续两次取边界为零。", "statement-cf70f6e882eed0df163c05a1": "据零映射证明 Br 包含于 Zr。", "statement-ae16e6e7da0a84f365502acf": "据群资格、阿贝尔条件和包含关系构造同调商群，并读作者的良定义说明。" }, "scope_note": "这是从群／子群及有向单形背景到本章同调群定义的有限作者路径，不声称追溯全部数学基础。无编号的边分别标作作者承接或构造使用候选，均待独立人工复核。原始定义范围与扩展说明同时保留。" } ], "coverage": { "registered_positions": 2, "located_occurrences": 2, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "e3d76681023040a14d51ff7d5b09e40a291c54f2cfe81c51465cdb323c60b64d", "optional_enrichments": [ { "id": "archived-02f7a498-5c17-4366-9479-8666d5f9bb2e", "kind": "archived-composition", "actor": { 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"hom-author-view-nakahara", "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "summary": "第3章先给出群的阿贝尔加法约定及有限生成背景；定义3.2以有向 $r$-单形生成自由阿贝尔链群，系数取整数，再以循环群模边界群定义 $H_r(K)$。", "conditions": "引言中的有限生成说法仅作为 Nakahara 第3章的作者背景陈述记录；具体定义条件为 $K$ 是 $n$ 维单纯复形、$0\\le r\\le n$，且此处使用整数系数。不能仅由 $n$ 维推出有限生成，也不能把该章背景强加到其他作者的定义；书中另提可选实数或 $\\mathbb{Z}_2$ 系数。", "notation": "$C_r(K)$ 为自由阿贝尔群，$H_r(K)=Z_r(K)/B_r(K)$；本章用加法记号。", "evidence_ids": [ "e-d0b8aaaef85c87c196def54f", "statement-a9109552ffd06f4f8a886bfa", "statement-ae16e6e7da0a84f365502acf", "e-6f80385868a9be86bd60fc9e", "e-b9d6b7ae4a73dc305c6c21ae", "e-69d50dad276500a72a608c6d" ] }, { "id": "hom-author-view-szekeres", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "summary": "在光滑流形上，Szekeres 以光滑奇异单形组成实系数链空间，并以循环空间模边界空间定义同调空间；他说明此处称“同调群”只需阿贝尔群性质。", "conditions": "对象是 Szekeres 所述流形上的光滑奇异单形及其实系数链空间；保留“同调空间”与“同调群”这一作者用语说明，不套用 Nakahara 第3章引言的有限生成背景。", "notation": "$H_p(M)=Z_p(M)/B_p(M)$；链系数为实数。", 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