{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:bed044e3-0bd6-48d8-8aab-253bd7096faa", "labels": { "zh-CN": "基本群", "en": "fundamental group" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "2a5c57f6740fa228c88addd66ab73077bfde321052340b2211a7e837f7e9f4de", "occurrences": [ { "concept_id": "urn:uuid:bed044e3-0bd6-48d8-8aab-253bd7096faa", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions", "anchor": "concept-6-fund", "chapter": "6", "section": "6.2", "offset": 1353, "range": [ 9644, 9964 ], "source_revision": "d5776d5bdae66f74cdbd2aaeda0ef4ffb4148f08eeb21a72c16dd61bab582129", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "基本群(fundamental group)", "context": { "range": [ 9588, 10928 ], "quote": "\n> [!theorem] 拓展\n> *其拓扑原因是：当 $n>2$ 时，$\\mathrm{SO}(n)$ 的[基本群(fundamental group)](../../../../../concepts/topology/fundamental-group)非平凡，且 $\\pi_1(\\mathrm{SO}(n))\\cong\\mathbb Z_2$。特别地，$\\mathrm{SO}(n)$ 中存在一条不可缩的环路，沿它绕行两圈后便可缩。自旋群在拓扑上是 $\\mathrm{SO}(n)$ 的单连通[双重覆盖(double cover)](../../../../../concepts/topology/double-cover)，而[覆盖映射(covering map)](../../../../../concepts/topology/covering-map) $\\Phi\\colon\\operatorname{Spin}(n)\\to\\mathrm{SO}(n)$ 可以选为群同态。*\n", "quote_sha256": "dea483579b808c5e6807bb9c15a43714bb0cfbaf6305b3ac6d8b953b69aed79b", "environment": "theorem", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-16ec397f02cc92b787cda31f" }, { "concept_id": "urn:uuid:bed044e3-0bd6-48d8-8aab-253bd7096faa", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions", "anchor": "concept-6-fund", "chapter": "6", "section": "6.2", "offset": 1910, "range": [ 9359, 9674 ], "source_revision": "525b7bc4b6973b05449bbbbcbdef1c067e7411218e20b20cb388b95687d1c4d8", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "fundamental group", "context": { "range": [ 9283, 10795 ], "quote": "\n> [!theorem] Digression\n> *The topological reason is that, for $n>2$, the [fundamental group](../../../../../concepts/topology/fundamental-group) of $\\mathrm{SO}(n)$ is nontrivial: $\\pi_1(\\mathrm{SO}(n))\\cong\\mathbb Z_2$. In particular, $\\mathrm{SO}(n)$ contains a noncontractible loop that becomes contractible when traversed twice. The spin group is the simply connected [double cover](../../../../../concepts/topology/double-cover) of $\\mathrm{SO}(n)$, and the [covering map](../../../../../concepts/topology/covering-map) $\\Phi\\colon\\operatorname{Spin}(n)\\to\\mathrm{SO}(n)$ can be chosen to be a group homomorphism.*\n", "quote_sha256": "ce87774c3310d1aaa59f1b40bf63bbe946c9381253db1be646d265bb3a49a506", "environment": "theorem", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-8eaa0c544a48b0d46393a16d" } ], "registered_positions": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions", "lang": "en", "section": "6.2", "signal": "term-mark", "label": "fundamental group", "english": "fundamental group", "offset": 1442 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions", "lang": "zh-CN", "section": "6.2", "signal": "term-mark", "label": "基本群", "english": "fundamental group", "offset": 1041 } ], "statements": [ { "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" }, { "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" } ], "resolved_references": [], "reading_order": [ "occ-16ec397f02cc92b787cda31f", "occ-8eaa0c544a48b0d46393a16d" ], "recorded_dependencies": [ { "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" } ], "reading_paths": [], "coverage": { "registered_positions": 2, "located_occurrences": 2, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "f4949b96976ddc27026d7eba41e7903c5eed9520f4c651aee9e00b34bb868d58", "optional_enrichments": [ { "id": "archived-bed044e3-0bd6-48d8-8aab-253bd7096faa", "kind": "archived-composition", "actor": { "model": "gpt-6-luna", "reasoning_effort": "high" }, "source_status": "current", "stale_reasons": [], "review_status": { "generation": "complete", "structure": "validated", "semantic_review": "independent-luna-reviewed; human-review-not-performed", "translation_review": "not-performed; inherited-source-status" }, "result": { "overview_claims": [ { "id": "overview-fundamental-homology-relations", "text": "对连通单纯复形 $K$，Nakahara 定理称第一同调群同构于基本群 $\\pi_1(|K|)$ 除以换位子群 $F$；对连通拓扑空间 $X$，推论称基本群与第一同调群同构当且仅当基本群交换，若基本群由单个生成元生成亦同构。 这表达了基本群交换化后与第一同调群的关系。", "evidence_ids": [ "statement-8d8d2177e831615076629fce", "statement-e147ac6a63afbf8aa27b13b9" ] }, { "id": "overview-fundamental-so-example", "text": "Woit §6.2 以 $SO(n)$ 为例：当 $n>2$ 时其基本群是二阶循环群，环路绕两圈后可缩，并将此与自旋群的双覆盖联系起来。这是具体例子；本次材料未给出基本群的一般定义。 这里是该群族中的一个拓扑例子。", "evidence_ids": [ "e-2bfdec6ad5c23143e18b857c" ] } ], "author_views": [ { "id": "view-nakahara-fundamental-group", "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "summary": "定理 4.9 对连通单纯复形给出第一同调群与基本群交换化的同构；推论 4.2 对连通拓扑空间说明，基本群与第一同调群同构恰当于基本群交换，并指出单生成元情形。", "conditions": "定理中的 $K$ 必须是连通单纯复形，$F$ 是 $\\pi_1(|K|)$ 的换位子群；推论中的 $X$ 为连通拓扑空间。引用保留作者原文的 $K$ 与 $X$ 两种范围。", "notation": "沿用 Nakahara 的 $K$、$X$、$\\pi_1(|K|)$、$H_{1}(K)$ 和换位子群 $F$。", "evidence_ids": [ "statement-8d8d2177e831615076629fce", "statement-e147ac6a63afbf8aa27b13b9" ] }, { "id": "view-woit-fundamental-group", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "summary": "§6.2 在解释自旋群覆盖时，以 $SO(n)$ 为例，在 $n>2$ 时指出其基本群非平凡，并给出 $\\pi _ { 1 } ( S O ( n ) ) = { \\bf Z } _ { 2 }$；作者还说绕不可缩环路两圈后可缩。", "conditions": "这是关于 $SO(n)$ 的特定例子，条件为 $n>2$；本节英文阅读稿与中文译稿仍是未逐页核验的机器稿，不据此替代基本群的一般定义。", "notation": "沿用 Woit 的 $SO(n)$、$\\pi_1(SO(n))$ 与 $\\mathbb{Z}_2$ 记号。", "evidence_ids": [ "e-2bfdec6ad5c23143e18b857c", "e-4adb4fdec29745927008b82e" ] } ], "reading_suggestions": [ { "id": "read-fundamental-theorem", "text": "先读定理 4.9，注意连通单纯复形这一条件，以及商群中的换位子群。", "target_ids": [ "statement-8d8d2177e831615076629fce" ], "evidence_ids": [ "statement-8d8d2177e831615076629fce" ] }, { "id": "read-fundamental-corollary", "text": "再读推论 4.2，比较连通拓扑空间上的一般交换条件与单生成元这一充分情形。", "target_ids": [ "statement-e147ac6a63afbf8aa27b13b9" ], "evidence_ids": [ "statement-e147ac6a63afbf8aa27b13b9" ] }, { "id": "read-fundamental-example", "text": "最后看 Woit §6.2 的 $SO(n)$ 例子，留意 $n>2$ 限定和双覆盖背景。", "target_ids": [ "e-2bfdec6ad5c23143e18b857c" ], "evidence_ids": [ "e-2bfdec6ad5c23143e18b857c" ] } ], "assessment": { "composition": "complete", "key_material": "missing", "critical_gaps": [ "基本群的一般定义与构造材料缺失，当前内容只能说明与第一同调群的关系及一个 SO(n) 例子。" ], "whole_book_coverage": "not-exhausted" }, "gaps": [ "本次材料没有提供基本群的一般定义，也没有在这里展开其基点及同伦类构造；页面因此限于已给出的同调关系和 SO(n) 例子。Woit §6.2 的英文与中文材料为机器稿，资格按来源记录保留。" ] }, "evidence": { "e-ef404ab91c22aa4a567ab746": { "route": null, "anchor": "", "statement_id": "statement-8d8d2177e831615076629fce", "label": "Nakahara §4.4.3 · 新单元候选 · 定理 4.9", "quote_sha256": "bac32ab96338c90c74331328747ab3accbf7a8b26e8a0fc45661ee5f17e35e9f" }, "e-335f427797f904179fb2ebdc": { "route": null, "anchor": "", "statement_id": "statement-e147ac6a63afbf8aa27b13b9", "label": "Nakahara §4.4.3 · 新单元候选 · 推论 4.2", "quote_sha256": "6a7665b89eade69aae81b09d24eb6246000551d715ef15a21a11fc91cd20054f" }, "e-4adb4fdec29745927008b82e": { "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions", "anchor": "", "statement_id": null, "label": "Woit · §6.2", "quote_sha256": "38e804ca8bf72da0caa6a848fac738b6f955b8260e7a6c19e663a7bb54abe689" }, "e-2bfdec6ad5c23143e18b857c": { "route": "books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions", "anchor": "", "statement_id": null, "label": "Woit · §6.2", "quote_sha256": "0bce8a280ff15de23146eb82d5a78e663c125b1ab804f21620a6ed08b8a2ea4f" } }, "candidate_dependencies": [], "input_sha256": "2c762bb37a03d2ee51e95b1d1dd5bcea52f2bee65b0624bde3eadb2c96501233", "archive_sha256": "50851cf91d01f18f4f79b966fba0c4e6a6d37436f56eb89ee3ac8cec7b933e7d" } ] }