{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:d3626907-41e1-4ef2-bc68-2d2efdec0b42", "labels": { "zh-CN": "覆盖映射", "en": "covering map" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "e4b49074e183c1a48a3406b411a55120d3ecd3510539ec72f4811ceb1c2782e9", "occurrences": [ { "concept_id": "urn:uuid:d3626907-41e1-4ef2-bc68-2d2efdec0b42", "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-cover", "chapter": "6", "section": "6.2", "offset": 2213, "range": [ 10504, 10817 ], "source_revision": "d5776d5bdae66f74cdbd2aaeda0ef4ffb4148f08eeb21a72c16dd61bab582129", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "覆盖映射(covering map)", "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-7363337f797738de22903fcd" }, { "concept_id": "urn:uuid:d3626907-41e1-4ef2-bc68-2d2efdec0b42", "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-cover", "chapter": "6", "section": "6.2", "offset": 2895, "range": [ 10344, 10651 ], "source_revision": "525b7bc4b6973b05449bbbbcbdef1c067e7411218e20b20cb388b95687d1c4d8", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "covering map", "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-16e8523d01d9be1dc2a636a1" }, { "concept_id": "urn:uuid:d3626907-41e1-4ef2-bc68-2d2efdec0b42", "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-cover", "chapter": "6", "section": "6.2", "offset": 8103, "range": [ 16394, 16686 ], "source_revision": "d5776d5bdae66f74cdbd2aaeda0ef4ffb4148f08eeb21a72c16dd61bab582129", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "覆盖映射(covering map)", "context": { "range": [ 16289, 17321 ], "quote": " $u$ 与 $-u$ 对 $\\vec v$ 的作用相同，因此 $\\mathrm{SO}(3)$ 中每个元素在 $\\mathrm{Sp}(1)$ 中恰有两个原像。映射 $\\Phi$ 是满射，所以它是次数为二的[覆盖映射(covering map)](../../../../../concepts/topology/covering-map)。因此，$\\mathrm{Sp}(1)$ 二重覆盖 $\\mathrm{SO}(3)$，这个群记作 $\\operatorname{Spin}(3)$。从几何上看，$\\mathrm{SO}(3)$ 是将三球面 $S^3=\\mathrm{Sp}(1)=\\operatorname{Spin}(3)$ 上的对径点等同后得到的空间。该空间即[实射影 3-空间(real projective 3-space)](../../../../../concepts/topology/real-projective-space)，也可视为 $\\mathbb R^4$ 中过原点的直线所成的空间。", "quote_sha256": "d22411a74e75bfd70a425cc841fc36b27615337e3907930390b95b25d8021652", "environment": null, "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-c3fc2842e0ed2ffec0b3f49f" }, { "concept_id": "urn:uuid:d3626907-41e1-4ef2-bc68-2d2efdec0b42", "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-cover", "chapter": "6", "section": "6.2", "offset": 10516, "range": [ 17965, 18251 ], "source_revision": "525b7bc4b6973b05449bbbbcbdef1c067e7411218e20b20cb388b95687d1c4d8", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "covering map", "context": { "range": [ 17798, 19000 ], "quote": "Both $u$ and $-u$ act identically on $\\vec v$, so each element of $\\mathrm{SO}(3)$ has two preimages in $\\mathrm{Sp}(1)$. The map $\\Phi$ is surjective, hence it is a [covering map](../../../../../concepts/topology/covering-map) of degree two. Thus $\\mathrm{Sp}(1)$ double-covers $\\mathrm{SO}(3)$, and we denote this group by $\\operatorname{Spin}(3)$. Geometrically, $\\mathrm{SO}(3)$ is $S^3=\\mathrm{Sp}(1)=\\operatorname{Spin}(3)$ with antipodal points identified. The resulting space is [real projective 3-space](../../../../../concepts/topology/real-projective-space), also the space of lines through the origin in $\\mathbb R^4$.", "quote_sha256": "b3e03b7a579310acf6ca3d62bad5da6c1e0a6bfcda77c76b0691630c378647cb", "environment": null, "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-9fef8152689336949366654c" }, { "concept_id": "urn:uuid:d3626907-41e1-4ef2-bc68-2d2efdec0b42", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/07-a-summary", "anchor": "concept-6-3-covering-map", "chapter": "6", "section": "6.3", "offset": 4692, "range": [ 14035, 14366 ], "source_revision": "51224e8d409a684fb67417edf938c1deffb0aa708773316223994618d7a07300", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "覆盖映射(covering map)", "context": { "range": [ 13545, 14441 ], "quote": "存在一个[群同态(group homomorphism)](../../../../../concepts/group-theory/group-homomorphism) $\\Phi:\\operatorname{Spin}(3)\\to\\mathrm{SO}(3)$，它将第一个群映到第二个群，并且是一个二重[覆盖映射(covering map)](../../../../../concepts/topology/covering-map)。其导数 $\\Phi'$ 是两个群的李代数之间的同构。", "quote_sha256": "87deb2d1480d0ce2e344831fe477a8c6f776b3d850751a1920a024acc4529a18", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-06/07-a-summary", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-e029c8097a199ee258d80ab3" }, { "concept_id": "urn:uuid:d3626907-41e1-4ef2-bc68-2d2efdec0b42", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-06/07-a-summary", "anchor": "concept-6-3-covering-map", "chapter": "6", "section": "6.3", "offset": 4931, "range": [ 13440, 13765 ], "source_revision": "c6959dd1685931e5d5c3b3ad92a453f6e8e23f74fb6df74bd7839b5bfd7d43ac", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "covering map", "context": { "range": [ 12913, 13894 ], "quote": "There is a [group homomorphism](../../../../../concepts/group-theory/group-homomorphism) $\\Phi:\\operatorname{Spin}(3)\\to\\mathrm{SO}(3)$ taking the first group to the second; it is a two-fold [covering map](../../../../../concepts/topology/covering-map). Its derivative $\\Phi'$ is an isomorphism of the Lie algebras of the two groups.", "quote_sha256": "08809614d85ba392168df26b083ef8a7078671c7ccc0c6949f77eaccbc58fde8", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/07-a-summary", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-18ffc86e0ed8999ec825c90a" } ], "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": "covering map", "english": "covering map", "offset": 2115 }, { "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": "covering map", "english": "covering map", "offset": 8765 }, { "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": "覆盖映射", "english": "covering map", "offset": 1589 }, { "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": "覆盖映射", "english": "covering map", "offset": 6508 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-06/07-a-summary", "lang": "en", "section": "6.3", "signal": "term-mark", "label": "covering map", "english": "covering map", "offset": 3527 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/07-a-summary", "lang": "zh-CN", "section": "6.3", "signal": "term-mark", "label": "覆盖映射", "english": "covering map", "offset": 3288 } ], "statements": [], "resolved_references": [], "reading_order": [ "occ-7363337f797738de22903fcd", "occ-16e8523d01d9be1dc2a636a1", "occ-c3fc2842e0ed2ffec0b3f49f", "occ-9fef8152689336949366654c", "occ-e029c8097a199ee258d80ab3", "occ-18ffc86e0ed8999ec825c90a" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 6, "located_occurrences": 6, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "a6bbbc6c1e1db62594bb894e31b1198f113979a872333369dc7437548a206860", "optional_enrichments": [] }