{"id":1254,"date":"2021-11-07T20:31:59","date_gmt":"2021-11-07T12:31:59","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1254"},"modified":"2022-10-13T14:45:53","modified_gmt":"2022-10-13T06:45:53","slug":"fundamental_group_related_exercises","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2021\/11\/07\/fundamental_group_related_exercises\/","title":{"rendered":"\u57fa\u672c\u7fa4\u76f8\u5173\u4e60\u9898"},"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>\u8fd9\u5468\u5f00\u59cb\u7ec8\u4e8e\u53c8\u62e5\u6709\u5468\u672b\u4e86, \u539f\u5b9a\u4e8e11\u6708\u4efd\u7684\u52a0\u73ed\u63a8\u8fdf\u523012\u6708\u4efd\u4e86, \u4e0d\u8fc7\u73ed\u8fd8\u662f\u8981\u52a0\u7684=.= \u4e4b\u524d\u4e5f\u8bf4\u8fc7\u4e86, \u63a5\u4e0b\u6765\u7684\u5468\u672b\u4f1a\u4f18\u5148\u5b66\u4e60\u5085\u5b5d\u660e\u8001\u5e08\u7684\u6570\u5b57\u51e0\u4f55\u5904\u7406\u7f51\u8bfe, \u6240\u4ee5\u4ee3\u6570\u62d3\u6251\u7684\u535a\u5ba2\u8d28\u91cf\u8fd8\u662f\u4f1a\u53d7\u5230\u4e00\u4e9b\u5f71\u54cd. (\u54c7\u5494\u5494, \u53c8\u4e3a\u81ea\u5df1\u6c34\u6587\u627e\u5230\u4e86\u4e00\u4e2a\u6b63\u5f53\u7406\u7531\u543c\u543c(\u25cf&#8217;\u25e1&#8217;\u25cf)) \u672c\u6587\u4e3b\u8981\u662f\u7528\u4e8e\u8bb0\u5f55\u51e0\u9053\u81ea\u6211\u611f\u89c9\u503c\u5f97\u8bf4\u660e\u7684\u57fa\u672c\u7fa4\u76f8\u5173\u7684\u4e60\u9898.<\/p>\n<p><!--more--><\/p>\n<p>1. \u8bbe$X$\u662f$E^n$\u4e2d\u7684\u51f8\u5b50\u96c6, \u8bc1\u660e$X$\u5355\u8fde\u901a.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u5373\u8981\u8bc1\u4efb\u610f\u95ed\u9053\u8def$p(s), s \\in [0, 1]$\u5b9a\u7aef\u540c\u4f26\u4e8e\u70b9\u9053\u8def$a$. \u53ef\u53d6\u4f26\u79fb$$H(s, t) = ta + (1 &#8211; t)p(s), t \\in [0, 1],$$\u5176\u4e2d$H(s, 0) = p(s)$, $H(s, 1) = a$, \u4e14\u7531\u4e8e$X$\u662f$E^n$\u4e2d\u7684\u51f8\u5b50\u96c6, \u4f26\u79fb$H(s, t)$\u7684\u50cf\u59cb\u7ec8\u5728$X$\u4e2d, \u6545\u547d\u9898\u5f97\u8bc1.<\/p>\n<p>2. \u8bbe$X$\u6709\u4e00\u65cf\u5355\u8fde\u901a\u7684\u5f00\u5b50\u7a7a\u95f4$X_1, X_2, \\cdots$, \u4f7f\u5f97\u6bcf\u4e2a$X_i \\subseteq X_{i + 1}$, \u5e76\u4e14$X = $$ \\bigcup_{i = 0}^{\\infty} X_i$, \u8bc1\u660e$X$\u5355\u8fde\u901a.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u4efb\u53d6\u95ed\u9053\u8def$a: [0, 1] \\to X$, \u5219$a([0, 1])$\u7d27\u81f4, \u56e0\u6b64\u88ab\u6709\u9650\u591a\u4e2a$X_i$\u8986\u76d6, \u53c8\u7531\u6bcf\u4e2a$X_i \\subseteq X_{i + 1}$\u53ef\u77e5\u95ed\u9053\u8def$a$\u5fc5\u88ab\u5305\u542b\u4e8e\u67d0\u4e2a$X_k$\u4e2d, \u4ece\u800c\u95ed\u9053\u8def$a$\u5b9a\u7aef\u540c\u4f26\u4e8e\u4efb\u610f\u70b9\u9053\u8def(\u70b9\u9053\u8def\u4e5f\u662f\u4e00\u79cd\u7279\u6b8a\u7684\u95ed\u9053\u8def), \u6545\u547d\u9898\u5f97\u8bc1.<\/p>\n<p>3. \u8bbe$A \\subseteq E^2$\u9053\u8def\u8fde\u901a, \u5e76\u4e14$S^1 \\subseteq A, (0, 0) \\notin A$. \u8bc1\u660e$A$\u4e0d\u5355\u8fde\u901a.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> $S^1$\u662f$A$\u7684\u6536\u7f29\u6838, \u800c\u6536\u7f29\u6620\u5c04$r$\u6ee1\u8db3$r \\circ i = id_A$, \u4ece\u800c$r_\\pi \\circ i_\\pi = $$ (r \\circ i)_\\pi$\u4e0e$(id_A)_\\pi$\u4ec5\u76f8\u5dee\u4e00\u4e2a\u57fa\u70b9\u7684\u63a8\u9001\u540c\u6784, \u8fd9\u8bf4\u660e$r_\\pi \\circ i_\\pi$\u4e5f\u662f\u540c\u6784. \u6545\u6536\u7f29\u6620\u5c04$r$\u8bf1\u5bfc\u4e86\u4e00\u4e2a\u6ee1\u7684\u57fa\u672c\u7fa4\u540c\u6001$r_\\pi : \\pi_1(A, x_0) \\to \\pi_1( $$ S^1, r(x_0))$, \u800c$S^1$\u4e0d\u662f\u5355\u8fde\u901a\u7684(\u53c2\u8003<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/11\/03\/s1_multiply_connected\/\">S^1\u662f\u591a\u8fde\u901a\u7684<\/a>), \u4ece\u800c$A$\u4ea6\u975e\u5355\u8fde\u901a, \u547d\u9898\u5f97\u8bc1.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u8fd9\u5468\u5f00\u59cb\u7ec8\u4e8e\u53c8\u62e5\u6709\u5468\u672b\u4e86, \u539f\u5b9a\u4e8e11\u6708\u4efd\u7684\u52a0\u73ed\u63a8\u8fdf\u523012\u6708\u4efd\u4e86, \u4e0d\u8fc7\u73ed\u8fd8\u662f\u8981\u52a0\u7684=.= \u4e4b\u524d\u4e5f\u8bf4\u8fc7\u4e86, \u63a5 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/11\/07\/fundamental_group_related_exercises\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u57fa\u672c\u7fa4\u76f8\u5173\u4e60\u9898<\/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\/1254"}],"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=1254"}],"version-history":[{"count":7,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1254\/revisions"}],"predecessor-version":[{"id":2490,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1254\/revisions\/2490"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1254"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1254"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1254"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}