{"id":3284,"date":"2024-01-21T23:17:33","date_gmt":"2024-01-21T15:17:33","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=3284"},"modified":"2025-02-26T11:34:18","modified_gmt":"2025-02-26T03:34:18","slug":"fundamental_group_circle","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2024\/01\/21\/fundamental_group_circle\/","title":{"rendered":"\u5706\u5468\u7684\u57fa\u672c\u7fa4"},"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>\u6700\u8fd1\u5728\u8ddf\u7740Ph.D Pierre Albin\u8001\u5e08\u7684\u4ee3\u6570\u62d3\u6251\u89c6\u9891\u8bfe\u7a0b\u8fdb\u884c\u4ee3\u6570\u62d3\u6251\u7684\u590d\u4e60, \u6240\u7528\u7684\u6559\u6750\u4e3aHatcher\u7684Algebraic Topology. \u539f\u672c\u4ee5\u4e3aPh.D Pierre Albin\u8001\u5e08\u4f1a\u628a\u6559\u6750\u4e0a\u7684\u6bcf\u4e00\u4e2a\u77e5\u8bc6\u70b9\u90fd\u8bb2\u89e3\u5f97\u5341\u5206\u8be6\u7ec6, \u7ed3\u679c\u7b2c0\u7ae0\u5c31\u5be5\u5be5\u6570\u8bed\u5e26\u8fc7\u53bb\u4e86, \u4e5f\u662f\u4ee4\u6211\u6709\u70b9\u5c0f\u90c1\u95f7, \u56e0\u4e3a\u4e4b\u524d\u521a\u597d\u5728\u7b2c0\u7ae0\u7684\u67d0\u4e9b\u8bba\u8ff0\u4e0a\u5361\u4f4f\u4e86\u2026\u2026 \u4e0a\u5468\u521a\u770b\u5b8c\u5706\u5468\u7684\u57fa\u672c\u7fa4\u4e00\u5c0f\u8282\u7684\u89c6\u9891, \u8fd8\u662f\u6709\u4e9b\u8ff7\u60d1\u4e4b\u5904, \u8fd9\u6b21\u5c31\u518d\u8be6\u7ec6\u5730\u8bfb\u8bfb\u6559\u6750\u4e0a\u7684\u76f8\u5173\u8bc1\u660e~<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. Hatcher A .Algebraic Topology[J].second order equations with nonnegative characteristic form, 2002.DOI:10.1002\/9781118535523.ch9.<br \/>\n2. <a href=\"https:\/\/www.bilibili.com\/list\/watchlater?oid=92141490&#038;bvid=BV1P7411N7fW&#038;spm_id_from=333.1365.top_right_bar_window_view_later.content.click&#038;p=39\">S^1\u7684\u57fa\u672c\u7fa4<\/a><\/p>\n<p><strong>\u5f15\u74061<\/strong> \u7ed9\u5b9a\u4e00\u4e2a\u6620\u5c04$F: Y \\times I \\to X$, \u5176\u5728$Y \\times \\{ 0 \\}$\u4e0a\u7684\u9650\u5236$F | Y \\times $$ \\{ 0 \\}$\u7684\u63d0\u5347\u4e3a$\\widetilde{F}: Y \\times \\{ 0 \\} \\to \\widetilde{X}$, \u5219\u6620\u5c04$F$\u5b58\u5728\u4e00\u4e2a\u552f\u4e00\u7684\u63d0\u5347$\\widetilde{F}: $$ Y \\times I \\to \\widetilde{X}$, \u5176\u5728$Y \\times \\{ 0 \\}$\u4e0a\u7684\u9650\u5236\u4e3a$\\widetilde{F}$.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u9996\u5148, \u5bf9\u4e8e\u7ed9\u5b9a\u70b9$y_0 \\in Y$\u53ca\u5176\u5728$Y$\u4e2d\u7684\u90bb\u57df$N$, \u6211\u4eec\u9700\u8981\u6784\u5efa\u4e00\u4e2a\u63d0\u5347$\\widetilde{F}: $$ N \\times I \\to \\widetilde{X}$. \u7531\u4e8e$F$\u662f\u8fde\u7eed\u7684, $\\forall (y_0, t) \\in Y \\times I$\u5747\u5b58\u5728\u4e00\u4e2a\u90bb\u57df$N_t \\times (a_t, $$ b_t)$, s.t. $F(N_t \\times (a_t, b_t))$\u88ab\u5305\u542b\u4e8e$F(y_0, t)$\u7684\u4e00\u4e2a\u5747\u5300\u8986\u76d6\u7684\u90bb\u57df\u5185. \u7531$\\{ y_0 \\} $$ \\times I$\u7684\u7d27\u6027, \u5b58\u5728\u6709\u9650\u591a\u4e2a\u90bb\u57df$N_t \\times (a_t, $$ b_t)$\u8986\u76d6$\\{ y_0 \\} \\times I$. \u8fd9\u610f\u5473\u7740\u6211\u4eec\u53ef\u4ee5\u9009\u62e9$y_0$\u7684\u4e00\u4e2a\u90bb\u57df$N$\u4e0e$I$\u7684\u4e00\u4e2a\u5256\u5206$0 = t_0 < t_1 < \\cdots < t_m = 1$, s.t. \u5bf9\u4e8e$\\forall i$, $F(N \\times [t_i, t_{i + 1}])$\u88ab\u5305\u542b\u4e8e\u4e00\u4e2a\u5747\u5300\u8986\u76d6\u7684\u90bb\u57df$U_i$\u5185. \u63a5\u4e0b\u6765\u4f7f\u7528\u5f52\u7eb3\u6cd5\u9009\u62e9\u90bb\u57df$N$\u5e76\u6784\u9020\u63d0\u5347$\\widetilde{F}$, \u4e0d\u59a8\u5047\u8bbe\u7531$N \\times \\{ 0 \\}$\u4e0a\u7684$\\widetilde{F}$\u51fa\u53d1, \u53ef\u5728$N \\times [0, t_i]$\u4e0a\u6784\u9020\u5f97\u5230$\\widetilde{F}$. \u7531\u4e8e$U_i$\u662f\u5747\u5300\u8986\u76d6\u7684, \u6545\u5b58\u5728\u4e00\u4e2a\u5305\u542b\u70b9$\\widetilde{F}(y_0, t_i)$\u7684\u8fde\u901a\u5f00\u96c6$\\widetilde{U_i} \\subset \\widetilde{X}$, \u5176\u901a\u8fc7$p$\u540c\u80da\u6620\u5c04\u81f3$U_i$. \u5728\u5c06$N \\times \\{ t_i \\}$\u66ff\u6362\u4e3a\u5176\u4e0e$$(\\widetilde{F} | N \\times \\{ t_i \\})^{-1}(\\widetilde{U_i}) = p^{-1} \\circ (F | N \\times \\{ t_i \\})(\\widetilde{U_i})$$\u7684\u4ea4\u96c6\u540e, \u6211\u4eec\u53ef\u4ee5\u786e\u4fdd$F(N \\times [t_i, t_{i + 1}]) \\subset U_i$, \u4e14$\\widetilde{F}(N \\times \\{ t_i \\})$\u88ab\u5305\u542b\u4e8e$\\widetilde{U_i}$\u5185. \u5982\u6b64\u4e00\u6765, \u6211\u4eec\u4fbf\u53ef\u5728$N \\times [t_i, t_{i + 1}]$\u4e0a\u5c06$\\widetilde{F}$\u5b9a\u4e49\u4e3a$F$\u4e0e\u540c\u80da$p^{-1}: U_i \\to $$ \\widetilde{U_i}$\u7684\u590d\u5408\u51fd\u6570. \u7ecf\u8fc7\u6709\u9650\u6b65\u9aa4\u540e, \u5bf9\u4e8e$y_0$\u7684\u67d0\u4e9b\u90bb\u57df$N$, \u6211\u4eec\u53ef\u5f97\u5176\u63d0\u5347$\\widetilde{F}: N $$ \\times I \\to \\widetilde{X}$.\n$\\\\$ \u63a5\u4e0b\u6765\u6211\u4eec\u7ed9\u51fa\u5f53$Y$\u4e3a\u5355\u70b9\u96c6\u65f6\u5f15\u7406\u7684\u552f\u4e00\u6027. \u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b, \u6211\u4eec\u53ef\u5728\u8bc1\u660e\u8fc7\u7a0b\u4e2d\u7701\u7565$Y$\u7684\u7b26\u53f7. \u4e0d\u59a8\u5047\u8bbe$\\widetilde{F}$\u4e0e$\\widetilde{F}'$\u4e3a$F: I \\to X$\u7684\u4e24\u4e2a\u63d0\u5347, s.t. $\\widetilde{F}(0) = $$ \\widetilde{F}'(0)$. \u7c7b\u4f3c\u5730, \u9009\u62e9$I$\u7684\u4e00\u4e2a\u5256\u5206$0 = t_0 < t_1 < \\cdots $$ < t_m = 1$, s.t. $\\forall i$, $F([t_i, $$ t_{i + 1}])$\u88ab\u5305\u542b\u4e8e\u4e00\u4e2a\u5747\u5300\u8986\u76d6\u7684\u90bb\u57df$U_i$\u5185. \u4f7f\u7528\u5f52\u7eb3\u6cd5\u5047\u8bbe\u5728$[0, t_i]$\u4e0a\u6709$\\widetilde{F} = $$ \\widetilde{F}'$, \u7531\u4e8e$[t_i, $$ t_{i + 1}]$\u662f\u8fde\u901a\u7684, \u6545$\\widetilde{F}([ $$ t_i, t_{i + 1}])$\u4ea6\u662f\u8fde\u901a\u7684, \u4ece\u800c$\\widetilde{F}([t_i, t_{i + 1}])$\u5fc5\u88ab\u5305\u542b\u4e8e\u4e00\u4e2a\u4e0e$U_i$\u540c\u80da\u7684\u8fde\u901a\u5f00\u96c6$\\widetilde{U_i}$\u5185. \u7c7b\u4f3c\u5730, \n$\\widetilde{F}'([t_i, t_{i + 1}])$\u4ea6\u88ab\u5305\u542b\u4e8e\u8fde\u901a\u5f00\u96c6$\\widetilde{U_i}$\u5185($\\widetilde{F}(t_i) $$ = \\widetilde{F}'(t_i)$). \u7531\u4e8e$p$\u5728$\\widetilde{U_i}$\u4e0a\u4e3a\u4e00\u5355\u5c04, \u4e14$p \\widetilde{F} = $$ p \\widetilde{F}'$, \u6545\u5728$[t_i, t_{i + 1}]$\u4e0a\u6211\u4eec\u6709$\\widetilde{F} = \\widetilde{F}'$, \u4ece\u800c\u7531\u5f52\u7eb3\u6cd5\u53ef\u77e5\u5728$I$\u4e0a$\\widetilde{F} = $$ \\widetilde{F}'$.\n$\\\\$ \u6700\u540e, \u6211\u4eec\u6ce8\u610f\u5230, \u901a\u8fc7\u4e0a\u8ff0\u6b65\u9aa4\u5728\u5f62\u5f0f\u4e3a$N \\times I$\u7684\u96c6\u5408\u4e0a\u6784\u9020\u5f97\u5230\u7684\u63d0\u5347$\\widetilde{F}'$, \u5c06\u5176\u9650\u5236\u5728\u6bcf\u4e00\u6bb5$\\{ y \\} \\times I$\u4e0a\u5747\u662f\u552f\u4e00\u7684, i.e. \u5f53\u4e24\u4e2a\u5f62\u5f0f\u4e3a$N \\times I$\u7684\u96c6\u5408\u5b58\u5728\u4ea4\u96c6\u65f6, \u5176\u4ea4\u96c6\u4e0a\u6784\u9020\u5f97\u5230\u7684\u63d0\u5347$\\widetilde{F}'$\u5fc5\u662f\u552f\u4e00\u7684, \u6545\u6211\u4eec\u6700\u7ec8\u53ef\u5728\u6574\u4e2a$Y \\times I$\u4e0a\u5f97\u5230\u4e00\u4e2aWell-Defined\u7684\u63d0\u5347$\\widetilde{F}'$. \u8be5\u63d0\u5347$\\widetilde{F}'$\u662f\u8fde\u7eed\u7684, \u56e0\u4e3a\u5b83\u5728\u6bcf\u4e00\u4e2a\u5f62\u5f0f\u4e3a$N \\times I$\u7684\u96c6\u5408\u4e0a\u5747\u662f\u8fde\u7eed\u7684; \u540c\u65f6, \u8be5\u63d0\u5347$\\widetilde{F}'$\u662f\u552f\u4e00\u7684, \u56e0\u4e3a\u5b83\u5728\u6bcf\u4e00\u6bb5$\\{ y \\} \\times $$ I$\u4e0a\u5747\u662f\u552f\u4e00\u7684.\n\n$\\\\$ $\\\\$ $\\\\$ <strong>\u5f15\u74062(\u9053\u8def\u63d0\u5347\u5f15\u7406)<\/strong> \u5bf9\u4e8e\u4efb\u4e00\u4ee5\u70b9$x_0 \\in X$\u4e3a\u8d77\u70b9\u7684\u8def\u5f84$f: I \\to $$ X$\u4e0e\u4efb\u4e00$\\widetilde{x_0} \\in p^{-1}(x_0)$, \u5b58\u5728\u552f\u4e00\u4e00\u4e2a\u4ee5$\\widetilde{x_0}$\u4e3a\u8d77\u70b9\u7684\u63d0\u5347$\\widetilde{f}: I \\to \\widetilde{X}$.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u4ee4$Y$\u4e3a\u4e00\u4e2a\u5355\u70b9\u96c6, \u5e94\u7528\u5f15\u74061\u5373\u53ef\u5f97\u8bc1.<\/p>\n<p><strong>\u5f15\u74063(\u540c\u4f26\u63d0\u5347\u5f15\u7406)<\/strong> \u5bf9\u4e8e\u4efb\u4e00\u4ee5$x_0$\u4e3a\u8d77\u70b9\u7684\u4f26\u79fb$f_t: I \\to X$\u4e0e\u4efb\u4e00$\\widetilde{x_0} \\in $$ p^{-1}(x_0)$, \u5b58\u5728\u552f\u4e00\u4e00\u4e2a\u4ee5$\\widetilde{x_0}$\u4e3a\u8d77\u70b9\u7684\u4f26\u79fb\u7684\u63d0\u5347$\\widetilde{f_t}: I \\to \\widetilde{X}$.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u4ee4$Y = I$, \u5e94\u7528\u5f15\u74061\u5373\u53ef\u5f97\u8bc1.<\/p>\n<p>\u4ee4$F(s, t) = f_t(s)$, \u53ef\u7531\u540c\u4f26\u63d0\u5347\u5f15\u7406\u4e2d\u7684\u4f26\u79fb$f_t$\u5f97\u5230\u4e00\u4e2a\u6620\u5c04$F: I $$ \\times I \\to $$ X$. \u53c8\u7531\u9053\u8def\u63d0\u5347\u5f15\u7406\u53ef\u5f97\u4e00\u4e2a\u552f\u4e00\u7684\u63d0\u5347$\\widetilde{F}: I \\times \\{ 0 \\} \\to $$ \\widetilde{X}$, \u4ece\u800c\u7531\u5f15\u74061\u53ef\u5f97\u4e00\u4e2a\u552f\u4e00\u7684\u63d0\u5347$\\widetilde{F}: I \\times I \\to \\widetilde{X}$. \u5176\u4e2d, $\\widetilde{F} | \\{ 0 \\} \\times $$ I$\u4e0e$\\widetilde{F} | \\{ 1 \\} \\times I$\u5747\u4e3a\u5e38\u9053\u8def\u7684\u63d0\u5347, \u7531\u9053\u8def\u63d0\u5347\u5f15\u7406\u7684\u552f\u4e00\u6027\u90e8\u5206\u7684\u8bba\u8ff0\u53ef\u77e5$\\widetilde{F} | \\{ 0 \\} \\times I$\u4e0e$\\widetilde{F} | \\{ 1 \\} \\times I$\u4ea6\u5fc5\u4e3a\u5e38\u9053\u8def. \u6b64\u5916, \u7531\u4e8e$p \\widetilde{F} = F$, \u6545$\\widetilde{f_t}(s) = \\widetilde{F}(s, t)$\u4e3a\u4e00\u4e2a\u4f26\u79fb, $\\widetilde{f_t}$\u4e3a$f_t$\u7684\u63d0\u5347.<\/p>\n<p><strong>\u5b9a\u74061<\/strong> $\\pi_1(S^1)$\u4e3a\u4e00\u4e2a\u7531\u57fa\u70b9\u4e3a$(1, 0)$\u7684\u5708\u9053\u8def$\\omega(s) = (cos 2 \\pi s, $$ sin 2 \\pi s)$\u7684\u540c\u4f26\u7c7b\u751f\u6210\u7684\u65e0\u9650\u5faa\u73af\u7fa4.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u6ce8\u610f\u5230$[\\omega]^n = [\\omega_n]$, \u5176\u4e2d, \u5bf9\u4e8e$\\forall n \\in \\mathbb{Z}$, $\\omega_n(s) = (cos 2 \\pi ns, $$ sin 2 \\pi ns)$. \u8be5\u5b9a\u7406\u7b49\u4ef7\u4e8e: \u4efb\u4e00\u57fa\u70b9\u4e3a$(1, 0)$\u7684$S^1$\u4e0a\u7684\u5708\u9053\u8def\u5747\u540c\u4f26\u4e8e$\\omega_n$, \u5176\u4e2d, $n \\in \\mathbb{Z}$\u662f\u552f\u4e00\u7684. \u4e3a\u4e86\u8bc1\u660e\u8fd9\u4e00\u70b9, \u6211\u4eec\u53ef\u4ee5\u901a\u8fc7$p(s) = ( $$ cos 2 \\pi s, sin 2 \\pi s)$\u7ed9\u51fa\u7684\u6620\u5c04$p: \\mathbb{R} \\to S^1$\u6765\u6bd4\u8f83$S^1$\u4e2d\u7684\u8def\u5f84\u4e0e$\\mathbb{R}$\u4e2d\u7684\u8def\u5f84. \u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u87ba\u65cb\u53c2\u6570\u5316$s \\mapsto  ( $$ cos 2 \\pi s, sin 2 \\pi s, s)$\u5c06$\\mathbb{R}$\u5d4c\u5165\u81f3$\\mathbb{R}^3$\u4e2d, \u5219$p$\u4e3a$\\mathbb{R}^3$\u5230$\\mathbb{R}^2$\u7684\u6295\u5f71$(x, y, z) \\mapsto (x, $$ y)$\u5728\u87ba\u65cb\u4e0a\u7684\u9650\u5236. \u6ce8\u610f\u5230, \u5708\u9053\u8def$\\omega_n$\u4e3a\u590d\u5408\u6620\u5c04$p \\widetilde{\\omega}_n$, \u5176\u4e2d, $\\widetilde{\\omega}_n: I \\to \\mathbb{R}$\u4e3a\u4ee5\u8d77\u70b9\u4e3a0, \u4ee5\u7ec8\u70b9\u4e3a$n$\u7684\u9053\u8def, \u5b83\u7ed5\u7740\u87ba\u65cb\u4e2d\u5fc3\u8f74\u65cb\u8f6c$|n|$\u6b21, \u5f53$n > 0$\u65f6\u8868\u793a\u5411\u4e0a\u65cb\u8f6c, \u5f53$n < 0$\u65f6\u5219\u8868\u793a\u5411\u4e0b\u65cb\u8f6c. \u7531$\\omega_n = p \\widetilde{\\omega}_n$\u53ef\u77e5$\\widetilde{\\omega}_n$\u4e3a$\\omega_n$\u7684\u63d0\u5347.\n$\\\\$ \u6211\u4eec\u5c06\u901a\u8fc7\u7814\u7a76\u5982\u4f55\u5c06$S^1$\u4e2d\u7684\u9053\u8def\u63d0\u5347\u81f3$\\mathbb{R}$\u4e2d\u6765\u8bc1\u660e\u672c\u5b9a\u7406.\n$\\\\$ \u4ee4$f: I \\to S^1$\u4e3a\u4ee5$x_0 = (1, 0)$\u4e3a\u57fa\u70b9\u7684\u5708\u9053\u8def, \u8868\u793a$\\pi_1(S^1, x_0)$\u4e2d\u7684\u4e00\u4e2a\u5143\u7d20. \u7531\u9053\u8def\u63d0\u5347\u5f15\u7406\u53ef\u5f97\u4e00\u4e2a\u8d77\u70b9\u4e3a0\u7684\u63d0\u5347$\\widetilde{f}$. \u7531\u4e8e$p \\widetilde{f}(1) $$ = f(1) = x_0$, \u4e14$p^{-1}(x_0) = \\mathbb{Z} \\subset R$, \u6545\u9053\u8def$\\widetilde{f}$\u7684\u7ec8\u70b9\u4e3a\u67d0\u4e2a\u6574\u6570$n$. $\\widetilde{\\omega}_n$\u4e3a$\\mathbb{R}$\u4e2d\u53e6\u5916\u4e00\u6761\u4ece0\u5230$n$\u7684\u9053\u8def, \u4e14\u7531\u7ebf\u6027\u540c\u4f26$(1 - t) \\widetilde{f} + t \\widetilde{\\omega}_n$\u53ef\u77e5$\\widetilde{f} \\simeq \\widetilde{\\omega}_n$. \u5c06\u5176\u4e0e$p$\u8fdb\u884c\u590d\u5408\u8fd0\u7b97\u540e\u53ef\u5f97\u4e00\u4e2a\u540c\u4f26$f \\simeq \\omega_n$, \u6545$[f] = $$ [\\omega_n]$.\n$\\\\$ \u4e3a\u4e86\u8bc1\u660e$n$\u662f\u7531$[f]$\u552f\u4e00\u786e\u5b9a\u7684, \u4e0d\u59a8\u5047\u8bbe$f \\simeq \\omega_n$\u4e14$f \\simeq \\omega_m$, \u6545$\\omega_m \\simeq $$ \\omega_n$. \u4ee4$f_t$\u4e3a\u4ece$\\omega_m = f_0$\u5230$\\omega_n = f_1$\u7684\u4f26\u79fb. \u7531\u540c\u4f26\u63d0\u5347\u5f15\u7406\u53ef\u5c06\u4f26\u79fb$f_t$\u63d0\u5347\u81f3\u8d77\u70b9\u4e3a0\u7684\u4f26\u79fb$\\widetilde{f_t}$. \u53c8\u7531\u9053\u8def\u63d0\u5347\u5f15\u7406\u7684\u552f\u4e00\u6027\u90e8\u5206\u7684\u8bba\u8ff0\u53ef\u77e5$\\widetilde{f_0} = \\widetilde{\\omega}_m$\u4e14$\\widetilde{f_1} = \\widetilde{\\omega}_n$. \u7531\u4e8e$\\widetilde{f_t}$\u4e3a\u8def\u5f84\u7684\u4f26\u79fb, \u8d77\u7ec8\u70b9$\\widetilde{f_t}(1)$\u4e0e$t$\u662f\u65e0\u5173\u7684. \u5f53$t = 0$\u65f6\u5176\u7ec8\u70b9\u4e3a$m$; \u800c\u5f53$t = 1$\u65f6\u5176\u7ec8\u70b9\u4e3a$n$, \u6545$m = n$.\n$\\\\$ \u7efc\u4e0a\u6240\u8ff0, \u547d\u9898\u5f97\u8bc1.\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6700\u8fd1\u5728\u8ddf\u7740Ph.D Pierre Albin\u8001\u5e08\u7684\u4ee3\u6570\u62d3\u6251\u89c6\u9891\u8bfe\u7a0b\u8fdb\u884c\u4ee3\u6570\u62d3\u6251\u7684\u590d\u4e60, \u6240\u7528\u7684\u6559\u6750\u4e3aHatch &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2024\/01\/21\/fundamental_group_circle\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u5706\u5468\u7684\u57fa\u672c\u7fa4<\/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\/3284"}],"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=3284"}],"version-history":[{"count":38,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3284\/revisions"}],"predecessor-version":[{"id":3643,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3284\/revisions\/3643"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=3284"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=3284"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=3284"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}