{"id":1225,"date":"2021-10-24T23:17:04","date_gmt":"2021-10-24T15:17:04","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1225"},"modified":"2022-10-13T14:10:42","modified_gmt":"2022-10-13T06:10:42","slug":"fundamental_group_mark","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2021\/10\/24\/fundamental_group_mark\/","title":{"rendered":"\u57fa\u672c\u7fa4\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>\u4eca\u5929\u8fd8\u662f\u575a\u6301\u66f4\u65b0\u6765\u4e86, \u543c\u543c~ \u4eca\u5929\u4e3b\u8981\u7684\u5b66\u4e60\u7cbe\u529b\u90fd\u653e\u5728\u5085\u5b5d\u660e\u8001\u5e08\u7684\u6570\u5b57\u51e0\u4f55\u5904\u7406\u7f51\u8bfe\u4e0a\u4e86, \u56e0\u4e3a\u5185\u5fc3\u91cc\u8fd8\u662f\u60f3\u7cfb\u7edf\u5730\u5b66\u5b8c\u6570\u5b57\u51e0\u4f55\u5904\u7406\u7684. \u6240\u4ee5\u9884\u8ba1\u63a5\u4e0b\u6765\u7684\u4ee3\u6570\u62d3\u6251\u7cfb\u5217\u7684\u6587\u7ae0\u90fd\u4f1a\u5212\u6c34\u5f97\u591a\u4e86(\u6ca1\u9519, \u5305\u62ec\u672c\u65872333)\u2026\u2026 \u57fa\u672c\u7fa4\u662f\u4e00\u4e2a\u975e\u5e38\u91cd\u8981\u7684\u4ee3\u6570\u62d3\u6251\u4e0d\u53d8\u91cf, \u540c\u80da\u7684\u62d3\u6251\u7a7a\u95f4\u5177\u6709\u540c\u6784\u7684\u57fa\u672c\u7fa4, \u56e0\u6b64\u53ef\u4ee5\u628a\u4e00\u4e9b\u62d3\u6251\u95ee\u9898\u8f6c\u5316\u4e3a\u4ee3\u6570\u95ee\u9898\u89e3\u51b3, \u867d\u7136\u57fa\u672c\u7fa4\u5e76\u4e0d\u6bd4\u540c\u8c03\u7fa4\u597d\u7b97\u5c31\u5bf9\u4e86\u2026\u2026 \u8bba\u6587An Efficient Computation of Handle and Tunnel Loops via Reeb Graphs\u91cc\u8ba1\u7b97\u7684\u4e5f\u662f\u540c\u8c03\u7fa4\u800c\u975e\u57fa\u672c\u7fa4~<\/p>\n<p><!--more--><\/p>\n<p><strong>1. \u57fa\u672c\u7fa4<\/strong><\/p>\n<p>\u57fa\u672c\u7fa4$\\pi_1(X, x_0)$\u662f1\u7ef4\u7684\u540c\u4f26\u7fa4, \u6b64\u5904\u7684\u7ef4\u6570\u6307\u7684\u662f\u6211\u4eec\u8003\u8651\u7684\u9053\u8def\u7c7b\u4e2d\u7684\u9053\u8def\u7684\u5b9a\u4e49\u57df\u7684\u7ef4\u6570, \u5982\u57fa\u672c\u7fa4\u8ba8\u8bba\u7684\u9053\u8def\u7c7b\u4e2d\u7684\u9053\u8def\u7684\u5b9a\u4e49\u57df\u662f$[0, 1]$, \u5176\u7ef4\u6570\u4e3a1. \u57fa\u672c\u7fa4\u7684\u5143\u7d20\u662f\u9053\u8def\u7c7b, \u53ef\u4ee5\u8ba4\u4e3a\u662f\u628a\u9053\u8def\u518d\u62bd\u8c61\u4e00\u6b21\u5f97\u5230\u7684\u4e00\u4e2a\u4ee3\u6570\u7ed3\u6784, \u56e0\u4e3a\u9053\u8def\u7684\u672c\u8d28\u662f\u4e00\u4e2a\u4ece$[0, 1]$\u5230\u62d3\u6251\u7a7a\u95f4$X$\u7684\u6620\u5c04, \u56e0\u6b64<strong>\u57fa\u672c\u7fa4\u672c\u8d28\u4e0a\u4fbf\u662f\u4e00\u4e2a\u5b9a\u7aef\u540c\u4f26\u7684\u6620\u5c04\u7684\u96c6\u5408\u7684\u96c6\u5408<\/strong>. \u4e0d\u5f97\u4e0d\u4f69\u670d\u62d3\u6251\u5b66\u5bb6\u4eec, \u603b\u662f\u80fd\u591f\u901a\u8fc7\u4e0d\u65ad\u62bd\u8c61\u7684\u601d\u7ef4\u770b\u5230\u66f4\u4e3a\u672c\u8d28\u7684\u5185\u5bb9.<\/p>\n<p><strong>2. \u63a8\u9001\u540c\u6784<\/strong><\/p>\n<p>\u7531\u57fa\u672c\u7fa4\u7684\u5b9a\u4e49\u53ef\u4ee5\u770b\u51fa, \u57fa\u672c\u7fa4\u662f\u4f9d\u8d56\u4e8e\u57fa\u70b9\u7684\u9009\u62e9\u7684. \u800c\u63a8\u9001\u540c\u6784\u8003\u8651\u7684\u4fbf\u662f\u5982\u679c\u57fa\u70b9\u7684\u9009\u62e9\u53d1\u751f\u6539\u53d8, \u5bf9\u57fa\u672c\u7fa4\u7684\u8ba1\u7b97\u7ed3\u679c\u4f1a\u4ea7\u751f\u4ec0\u4e48\u6837\u7684\u5f71\u54cd\u5462? \u9700\u8981\u6ce8\u610f\u7684\u662f, \u4e66\u4e0aP178 \u5b9a\u74064.4.2\u4e2d\u5e76\u6ca1\u6709\u5f3a\u8c03$x_0$\u4e0e$x_1$\u662f\u5904\u4e8e\u540c\u4e00\u4e2a\u9053\u8def\u5206\u652f\u91cc\u7684, \u800c\u8fd9\u4e2a\u6761\u4ef6\u5176\u5b9e\u662f\u5fc5\u8981\u7684, \u56e0\u4e3a\u5982\u679c$x_0$\u4e0e$x_1$\u4e0d\u5904\u4e8e\u540c\u4e00\u4e2a\u9053\u8def\u5206\u652f\u91cc, \u5219\u8ba1\u7b97\u5f97\u5230\u7684$\\pi_1(X, x_0)$\u4e0e$\\pi_1(X, x_1)$\u6ca1\u6709\u4efb\u4f55\u5173\u7cfb. \u7531\u63a8\u9001\u540c\u6784\u7684\u5b9a\u4e49\u53ef\u4ee5\u770b\u51fa, \u63a8\u9001\u540c\u6784\u5efa\u7acb\u8d77\u4e86$\\pi_1(X, x_0)$\u5230$\\pi_1(X, x_1)$\u7684\u540c\u6784\u5173\u7cfb, \u4e5f\u4ece\u4fa7\u9762\u8bf4\u660e, \u5728\u540c\u6784\u610f\u4e49\u4e0a, \u4e00\u4e2a\u9053\u8def\u8fde\u901a\u7a7a\u95f4\u7684\u57fa\u672c\u7fa4\u4e0e\u57fa\u70b9\u7684\u9009\u62e9\u65e0\u5173.<br \/>\n$\\\\$ \u6b64\u5916, \u63a8\u9001\u540c\u6784\u6d89\u53ca\u5230\u9053\u8def\u7c7b\u7684\u4e58\u79ef, \u5728\u8bc1\u660e\u65f6\u9700\u8981\u8f6c\u5316\u4e3a\u540c\u4f26\u7684\u95ee\u9898, \u8fd9\u6837\u4e00\u79cd\u8f6c\u5316\u601d\u7ef4\u662f\u7531\u9053\u8def\u7c7b\u7684\u4e58\u79ef\u5b9a\u4e49\u51b3\u5b9a\u7684, \u5373\u82e5$a, b, c$\u662f\u62d3\u6251\u7a7a\u95f4$X$\u4e0a\u7684\u9053\u8def, \u4e14\u9053\u8def$a$\u7684\u7ec8\u70b9\u662f\u9053\u8def$b$\u7684\u8d77\u70b9, \u5219$$\\left \\langle a \\right \\rangle \\left \\langle b \\right \\rangle = \\left \\langle c \\right \\rangle \\Leftrightarrow ab \\cong c,$$\u5176\u4e2d$\\left \\langle a \\right \\rangle$\u8868\u793a$a$\u6240\u5728\u7684\u9053\u8def\u7c7b.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4eca\u5929\u8fd8\u662f\u575a\u6301\u66f4\u65b0\u6765\u4e86, \u543c\u543c~ \u4eca\u5929\u4e3b\u8981\u7684\u5b66\u4e60\u7cbe\u529b\u90fd\u653e\u5728\u5085\u5b5d\u660e\u8001\u5e08\u7684\u6570\u5b57\u51e0\u4f55\u5904\u7406\u7f51\u8bfe\u4e0a\u4e86, \u56e0\u4e3a\u5185\u5fc3\u91cc\u8fd8\u662f\u60f3\u7cfb &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/10\/24\/fundamental_group_mark\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u57fa\u672c\u7fa4\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\/1225"}],"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=1225"}],"version-history":[{"count":13,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1225\/revisions"}],"predecessor-version":[{"id":2488,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1225\/revisions\/2488"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1225"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1225"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1225"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}