{"id":1507,"date":"2022-02-13T16:00:15","date_gmt":"2022-02-13T08:00:15","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1507"},"modified":"2022-10-13T18:55:20","modified_gmt":"2022-10-13T10:55:20","slug":"examples_seifert_van_kampens_theorem_mark","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/02\/13\/examples_seifert_van_kampens_theorem_mark\/","title":{"rendered":"Seifert &#038; Van Kampen\u5b9a\u7406\u76f8\u5173\u4f8b\u5b50\u6ce8\u8bb0"},"content":{"rendered":"<p><script type=\"text\/javascript\" async src=\"https:\/\/www.caiqinyi.cn\/wp-content\/MathJax\/MathJax.js?config=TeX-AMS_CHTML\">\n<\/script><br \/>\n<script type=\"text\/x-mathjax-config\">\n    MathJax.Hub.Config({\n        tex2jax: {inlineMath: [['$','$']]},\n        TeX: {equationNumbers: {autoNumber: [\"AMS\"], useLabelIds: true}},\n        \"HTML-CSS\": {linebreaks: {automatic: true}},\n        SVG: {linebreaks: {automatic: true}}\n    });\n<\/script><\/p>\n<p>\u56e0\u4e3a\u676d\u5dde\u75ab\u60c5\u88ab&#8221;\u56f0&#8221;\u5728\u5bb6\u4e2d\u4e86\u2026\u2026 \u672c\u6587\u4e3b\u8981\u8865\u5145\u4e66\u4e0aSeifert &#038; Van Kampen\u5b9a\u7406\u7684\u76f8\u5173\u4f8b\u5b50\u4e2d\u4e00\u4e9b\u7591\u60d1\u70b9\u7684\u6ce8\u8bb0.<\/p>\n<p><!--more--><\/p>\n<p><strong>1. \u5355\u8fde\u901a\u7a7a\u95f4\u5e94\u7528Seifert &#038; Van Kampen\u5b9a\u7406\u7684\u76f8\u5173\u6ce8\u8bb0<\/strong><\/p>\n<p>\u7531\u5355\u8fde\u901a\u7a7a\u95f4\u5b9a\u4e49\u77e5, \u5176\u57fa\u672c\u7fa4\u4e3a\u5e73\u51e1\u7fa4(\u4ec5\u542b\u5355\u4f4d\u5143). \u53c8\u7531\u751f\u6210\u5143\u53ca\u751f\u6210\u5173\u7cfb\u7684\u5b9a\u4e49\u77e5, \u751f\u6210\u5143\u4e3a\u7fa4\u4e2d\u7684\u975e\u5355\u4f4d\u5143, \u800c\u751f\u6210\u5173\u7cfb\u4e3a\u751f\u6210\u5143\u6240\u751f\u6210\u7684\u81ea\u7531\u7fa4\u4e2d\u7684\u5143\u7d20, \u6545\u5355\u8fde\u901a\u7a7a\u95f4\u7684\u8868\u51fa\u4e3a$\\left \\langle \\ | \\ \\right \\rangle$.<br \/>\n$\\\\$ \u8fdb\u4e00\u6b65\u5730, \u53ef\u7f29\u7a7a\u95f4\u662f\u9053\u8def\u8fde\u901a\u7684(\u56e0\u4e3a\u5b83\u4e0e\u9053\u8def\u8fde\u901a\u7684\u5355\u70b9\u7a7a\u95f4\u540c\u4f26\u7b49\u4ef7), \u5e76\u4e14\u5177\u6709\u5e73\u51e1\u7684\u57fa\u672c\u7fa4, \u56e0\u800c\u662f\u5355\u8fde\u901a\u7684. \u8fdb\u800c\u7531\u4e0a\u8ff0\u8ba8\u8bba\u53ef\u77e5, \u53ef\u7f29\u7a7a\u95f4\u7684\u8868\u51fa\u4ea6\u4e3a$\\left \\langle \\ | \\ \\right \\rangle$.<\/p>\n<p><strong>2. \u73af\u9762$T^2$\u57fa\u672c\u7fa4\u8868\u51fa\u8ba1\u7b97\u8fc7\u7a0b\u4e2d\u7684\u76f8\u5173\u6ce8\u8bb0<\/strong><\/p>\n<p>\u7531\u4e8e\u8ba1\u7b97\u8fc7\u7a0b\u8f83\u957f, \u6545\u6b64\u5904\u4e0d\u5217\u51fa\u5168\u90e8\u8bc1\u660e\u8fc7\u7a0b. \u7a0d\u6709\u7591\u60d1\u7684\u70b9\u5728\u4e8e\u4e66\u4e0aP205\u7b2c\u4e8c\u6bb5\u4e2d\u7684\u8bba\u8ff0: \u4ece\u6b63\u65b9\u5f62\u53f3\u4e0a\u89d2\u51fa\u53d1\u9006\u65f6\u9488\u4e00\u5708\u7684\u9053\u8def\u6070\u597d\u662f$ab \\bar{a} \\bar{b}$, \u56e0\u6b64\u5728$Y$\u4e2d$c \\simeq \\bar{w} ab \\bar{a} \\bar{b}w$, \u5373$$\\left \\langle c \\right \\rangle_Y = w_{\\#}(\\left \\langle a \\right \\rangle_Y \\left \\langle b \\right \\rangle_Y \\left \\langle a \\right \\rangle_Y^{-1} \\left \\langle b \\right \\rangle_Y^{-1}).$$\u7531\u4e8e\u5728$Y$\u4e2d, $c$\u4e3a\u4ee5$y_0$\u4e3a\u57fa\u70b9\u7684\u95ed\u9053\u8def, $ab \\bar{a} \\bar{b}$\u4ea6\u4e3a\u4e00\u6761\u95ed\u9053\u8def, \u5176\u57fa\u70b9\u4e0d\u4e00\u5b9a\u4e3a$y_0$. \u53c8$Y$\u4e3a\u4e00\u4e2a\u9053\u8def\u8fde\u901a\u7a7a\u95f4, \u5219\u6709$c \\simeq \\bar{w} ab \\bar{a} \\bar{b} w$, \u5176\u4e2d, $w$\u4e3a\u4e00\u6761\u4ece$ab \\bar{a} \\bar{b}$\u7684\u57fa\u70b9\u5230$c$\u7684\u57fa\u70b9$y_0$\u7684\u9053\u8def.<\/p>\n<p>PS: \u53d1\u73b0\u4e24\u4e2a\u4e0d\u9519\u7684\u4e2d\u6587\u6570\u5b66\u793e\u533a.<br \/>\n1. <a href=\"https:\/\/wuli.wiki\/book\/HomT5\">\u57fa\u672c\u7fa4\u7684\u8ba1\u7b97<\/a><br \/>\n2. <a href=\"https:\/\/www.bananaspace.org\/wiki\/%E8%AE%B2%E4%B9%89:%E6%8B%93%E6%89%91%E5%AD%A6\/%E5%9F%BA%E6%9C%AC%E7%BE%A4_(I)\">7. \u57fa\u672c\u7fa4 (I)<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u56e0\u4e3a\u676d\u5dde\u75ab\u60c5\u88ab&#8221;\u56f0&#8221;\u5728\u5bb6\u4e2d\u4e86\u2026\u2026 \u672c\u6587\u4e3b\u8981\u8865\u5145\u4e66\u4e0aSeifert &#038; Van &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/02\/13\/examples_seifert_van_kampens_theorem_mark\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">Seifert &#038; Van Kampen\u5b9a\u7406\u76f8\u5173\u4f8b\u5b50\u6ce8\u8bb0<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1507"}],"collection":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/comments?post=1507"}],"version-history":[{"count":29,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1507\/revisions"}],"predecessor-version":[{"id":2498,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1507\/revisions\/2498"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1507"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1507"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1507"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}