{"id":1565,"date":"2022-03-05T21:30:33","date_gmt":"2022-03-05T13:30:33","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1565"},"modified":"2025-02-26T11:04:33","modified_gmt":"2025-02-26T03:04:33","slug":"seifert_van_kampen_theorem_related_exercises","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/03\/05\/seifert_van_kampen_theorem_related_exercises\/","title":{"rendered":"Seifert &#038; Van Kampen\u5b9a\u7406\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>\u6700\u8fd1\u5de5\u4f5c\u4e0a\u8fd8\u662f\u53d1\u751f\u4e86\u86ee\u591a\u4e8b\u7684, \u6700\u4e3b\u8981\u7684\u53d8\u52a8\u5c31\u662f\u81ea\u5df1\u5728\u7684\u9879\u76ee\u7ec4\u88ab&#8221;\u780d&#8221;\u4e86\u2026\u2026 \u51e1\u4e8b\u90fd\u5177\u5907\u4e24\u9762\u6027, \u8fd9\u4ef6\u4e8b\u4e5f\u4e0d\u4f8b\u5916. \u5c3d\u7ba1\u5341\u5206\u4e0d\u820d\u5f97\u76ee\u524d\u7684\u9879\u76ee\u7ec4, \u4f46\u81ea\u5df1\u4e5f\u501f\u8fd9\u4e2a\u673a\u4f1a\u6210\u529f\u8f6c\u4e3a\u5f15\u64ce\u7814\u53d1, \u7b97\u662f\u8f83\u5f7b\u5e95\u5730\u544a\u522b\u4e86GamePlay. \u5e0c\u671b\u8fd9\u662f\u4e00\u4e2a\u597d\u7684\u5951\u673a\u4e0e\u5f00\u59cb\u53ed~ \u4e0b\u5468\u4e00\u8fd8\u6709\u5929\u7f8e\u9762\u8bd5, \u5c3d\u7ba1\u73b0\u5728\u5df2\u7ecf\u5bf9\u8df3\u69fd\u53bb\u5929\u7f8e\u8fd9\u4ef6\u4e8b\u4e0d\u662f\u5f88\u4e0a\u5fc3\u4e86, \u56e0\u4e3a\u66f4\u503e\u5411\u4e8e\u5728\u7f51\u6613\u7ee7\u7eed\u5b66\u4e60\u4e0e\u63d0\u5347\u81ea\u5df1. \u597d\u4e86, \u8bdd\u4e0d\u591a\u8bf4, \u8fdb\u5165\u6b63\u9898, \u672c\u6587\u4e3b\u8981\u662fSeifert &#038; Van Kampen\u5b9a\u7406\u7684\u76f8\u5173\u4e60\u9898\u89e3\u7b54.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/math.stackexchange.com\/questions\/2281603\/understanding-how-to-compute-pi-1s1-vee-s2?noredirect=1&#038;lq=1\">Understanding how to compute $\\pi_1(S^1 \\vee S^2)$<\/a><br \/>\n2. <a href=\"https:\/\/math.stackexchange.com\/questions\/22980\/fundamental-group-of-a-torus-with-points-removed\">Fundamental group of a torus with points removed<\/a><br \/>\n3. <a href=\"https:\/\/www.math.pku.edu.cn\/teachers\/baozq\/topology\/s3\/s3.htm\">\u4e13\u9898\u8ba8\u8bba: \u4e09\u7ef4\u7403\u9762<\/a><br \/>\n4. <a href=\"https:\/\/www.math3ma.com\/blog\/the-fundamental-group-of-the-real-projective-plane\">The Fundamental Group of the Real Projective Plane<\/a><\/p>\n<p>1. \u5728\u62d3\u6251\u7a7a\u95f4$X$\u548c\u7403\u9762$S^2$\u4e0a\u5404\u53d6\u4e00\u70b9, \u7136\u540e\u628a\u8fd9\u4e24\u70b9\u62d3\u6251\u5730\u7c98\u5408, \u5f97\u5230\u4e00\u70b9\u5e76$X $$ \\vee S^2$, \u8bc1\u660e\u5176\u57fa\u672c\u7fa4\u540c\u6784\u4e8e$\\pi_1(X)$(\u6ce8\u610f, $S^2$\u4e0d\u4e00\u5b9a\u662f\u5b83\u5728$X \\vee S^2$\u4e2d\u7684\u90bb\u57df\u7684\u5f62\u53d8\u6536\u7f29\u6838, \u56e0\u6b64\u4e0d\u80fd\u76f4\u63a5\u5e94\u7528Van Kampen\u5b9a\u7406).<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u5c1d\u8bd5\u8bc1\u660e\u4e00\u4e2a\u66f4\u5177\u4e00\u822c\u6027\u7684\u547d\u9898: \u82e5\u62d3\u6251\u7a7a\u95f4$A$\u4e3a\u53ef\u7f29\u7a7a\u95f4, \u4e0e\u62d3\u6251\u7a7a\u95f4$B$\u76f8\u4ea4\u4e8e\u70b9$p$, \u5219$\\pi_1(A \\cup B) \\cong \\pi_1(B)$.<br \/>\n$\\\\$ \u8bbe$\\gamma$\u4e3a$A \\cup B$\u4e2d\u7684\u4e00\u6761\u95ed\u9053\u8def(\u975e\u70b9\u9053\u8def), $e_p^A$\u4e3a\u62d3\u6251\u7a7a\u95f4$A$\u4e2d\u4ee5$p$\u4e3a\u57fa\u70b9\u7684\u70b9\u9053\u8def, $e_p^B$\u4e3a\u62d3\u6251\u7a7a\u95f4$B$\u4e2d\u4ee5$p$\u4e3a\u57fa\u70b9\u7684\u70b9\u9053\u8def, \u5bf9\u5176\u8fdb\u884c\u5206\u7c7b\u8ba8\u8bba.<br \/>\n$\\\\$ $\\cdot$ \u95ed\u9053\u8def$\\gamma$\u7684\u57fa\u70b9$b$($\\ne p$) \u4f4d\u4e8e\u62d3\u6251\u7a7a\u95f4$A$\u5185, \u4e14$\\gamma \\cap B = \\emptyset$\u6216$\\gamma \\cap B $$ = \\{ p \\}$, \u5219\u95ed\u9053\u8def$\\gamma$\u6240\u5728\u7684\u95ed\u9053\u8def\u7c7b$\\left \\langle \\gamma \\right \\rangle = \\left \\langle e_p^A \\right \\rangle$, \u5b83\u4e0e$\\pi_1(B)$\u4e2d\u7684\u95ed\u9053\u8def\u7c7b$\\left \\langle e_p^B \\right \\rangle$\u4e00\u4e00\u5bf9\u5e94.<br \/>\n$\\\\$ $\\cdot$ \u95ed\u9053\u8def$\\gamma$\u7684\u57fa\u70b9$b$($\\ne p$) \u4f4d\u4e8e\u62d3\u6251\u7a7a\u95f4$A$\u5185, \u4e14$\\gamma \\cap B \\ne \\emptyset$, \u5219\u53ef\u5c06\u95ed\u9053\u8def$\\gamma$\u5206\u89e3\u5f97$\\gamma = \\gamma_1 \\cup \\eta_1 \\cup \\cdots \\cup \\eta_n \\cup \\gamma_n$, \u5176\u4e2d$\\gamma_i \\cap A = \\{ p \\}$, $\\eta_i \\cap B = $$ \\{ p \\}$. \u7531\u4e8e\u62d3\u6251\u7a7a\u95f4$A$\u4e3a\u4e00\u4e2a\u53ef\u7f29\u7a7a\u95f4, \u6545\u95ed\u9053\u8def$\\gamma$\u540c\u4f26\u4e8e\u4ee5$p$\u4e3a\u57fa\u70b9\u7684\u95ed\u9053\u8def$\\eta_1 \\cup \\cdots \\cup \\eta_n$, \u4ece\u800c$\\left \\langle \\gamma \\right \\rangle $$ = \\left \\langle \\eta_1 \\cup \\cdots \\cup \\eta_n \\right \\rangle$, \u4e0e$\\pi_1(B)$\u4e2d\u7684\u4e00\u4e2a\u95ed\u9053\u8def\u7c7b\u5f62\u6210\u4e00\u4e00\u5bf9\u5e94.<br \/>\n$\\\\$ $\\cdot$ \u95ed\u9053\u8def$\\gamma$\u7684\u57fa\u70b9$b = p$, \u5219$\\gamma \\cap A \\ne \\emptyset$, $\\gamma \\cap B \\ne \\emptyset$. \u6211\u4eec\u5c06\u95ed\u9053\u8def$\\gamma$\u5206\u89e3\u5f97$\\gamma =<br \/>\n $$ \\gamma_1 \\cup \\eta_1 \\cup \\cdots \\cup \\eta_n \\cup \\gamma_n$, \u5176\u4e2d$\\gamma_i \\cap A = \\{ p \\}$, $\\eta_i \\cap B $$ = \\{ p \\}$. \u7531\u4e8e\u62d3\u6251\u7a7a\u95f4$A$\u4e3a\u4e00\u4e2a\u53ef\u7f29\u7a7a\u95f4, \u6545\u95ed\u9053\u8def$\\gamma$\u540c\u4f26\u4e8e\u4ee5$p$\u4e3a\u57fa\u70b9\u7684\u95ed\u9053\u8def$\\eta_1 \\cup \\cdots \\cup \\eta_n$, \u4ece\u800c$\\left \\langle \\gamma \\right \\rangle = \\left \\langle \\eta_1 \\cup \\cdots \\cup \\eta_n \\right \\rangle$, \u4e0e$\\pi_1(B)$\u4e2d\u7684\u4e00\u4e2a\u95ed\u9053\u8def\u7c7b\u5f62\u6210\u4e00\u4e00\u5bf9\u5e94.<br \/>\n$\\\\$ $\\cdot$ \u95ed\u9053\u8def$\\gamma$\u7684\u57fa\u70b9$b$($\\ne p$) \u4f4d\u4e8e\u62d3\u6251\u7a7a\u95f4$B$\u5185, \u4e14$\\gamma \\cap A = \\emptyset$\u6216$\\gamma \\cap A $$ = \\{ p \\}$, \u5219\u95ed\u9053\u8def$\\gamma$\u6240\u5728\u7684\u95ed\u9053\u8def\u7c7b$\\left \\langle \\gamma \\right \\rangle$\u901a\u8fc7\u4e00\u4e2a\u6295\u5f71\u6620\u5c04\u4e0e$\\pi_1(B)$\u4e2d\u7684\u4e00\u4e2a\u95ed\u9053\u8def\u7c7b\u5f62\u6210\u4e00\u4e00\u5bf9\u5e94.<br \/>\n$\\\\$ $\\cdot$ \u95ed\u9053\u8def$\\gamma$\u7684\u57fa\u70b9$b$($\\ne p$) \u4f4d\u4e8e\u62d3\u6251\u7a7a\u95f4$B$\u5185, \u4e14$\\gamma \\cap A \\ne \\emptyset$, \u5219\u53ef\u5c06\u95ed\u9053\u8def$\\gamma$\u5206\u89e3\u5f97$\\gamma = \\gamma_1 \\cup \\eta_1 \\cup \\cdots \\cup \\eta_n \\cup \\gamma_n$, \u5176\u4e2d, $\\gamma_i \\cap A = \\{ $$ p \\}$, $\\eta_i \\cap B $$ = \\{ p \\}$. \u7531\u4e8e\u62d3\u6251\u7a7a\u95f4$A$\u4e3a\u4e00\u4e2a\u53ef\u7f29\u7a7a\u95f4, \u6545\u95ed\u9053\u8def$\\gamma$\u540c\u4f26\u4e8e\u4ee5$p$\u4e3a\u57fa\u70b9\u7684\u95ed\u9053\u8def$\\eta_1 \\cup \\cdots \\cup \\eta_n$, \u4ece\u800c$$\\left \\langle \\gamma \\right \\rangle = \\left \\langle \\eta_1 \\cup \\cdots \\cup \\eta_n \\right \\rangle,$$\u4e0e$\\pi_1(B)$\u4e2d\u7684\u4e00\u4e2a\u95ed\u9053\u8def\u7c7b\u5f62\u6210\u4e00\u4e00\u5bf9\u5e94.<\/p>\n<p>2. \u5199\u51fa\u6316\u53bb\u4e24\u4e2a\u70b9\u7684\u73af\u9762\u7684\u57fa\u672c\u7fa4\u7684\u4e00\u4e2a\u8868\u51fa.<br \/>\n$\\\\$ <strong>\u89e3:<\/strong> \u6211\u4eec\u5148\u8003\u8651\u6316\u53bb\u4e00\u4e2a\u70b9$p$\u7684\u73af\u9762\u7684\u57fa\u672c\u7fa4\u7684\u4e00\u4e2a\u8868\u51fa. \u4ece\u73af\u9762\u591a\u8fb9\u5f62\u8868\u793a\u7684\u89d2\u5ea6\u51fa\u53d1, \u73af\u9762$T^2 = Q \/ \\sim$, \u5176\u4e2d, $Q := [-1, 1] \\times [-1, 1]$, \u5bf9\u4e8e$x \\in [-1, 1]$\u6709$(x, -1) \\sim (x, 1)$, \u5bf9\u4e8e$y \\in [-1, 1]$, \u5982\u4e0b\u56fe\u6240\u793a.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/\u73af\u9762\u591a\u8fb9\u5f62\u8868\u793a.png\" alt=\"\" width=\"173\" height=\"174\" class=\"aligncenter size-full wp-image-1592\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/\u73af\u9762\u591a\u8fb9\u5f62\u8868\u793a.png 173w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/\u73af\u9762\u591a\u8fb9\u5f62\u8868\u793a-150x150.png 150w\" sizes=\"(max-width: 173px) 100vw, 173px\" \/><\/p>\n<p>\u6b64\u5916, \u5047\u8bbe$p = (0, 0)$\u4f4d\u4e8e\u56db\u8fb9\u5f62\u4e2d\u5fc3. \u6613\u77e5$Q \\backslash  \\{ p \\}$\u53ef\u901a\u8fc7\u5982\u4e0b\u540c\u4f26\u5f62\u53d8\u6536\u7f29\u81f3\u56db\u8fb9\u5f62\u7684\u8fb9\u754c$\\partial Q$,$$H((x, y), t) = (1 &#8211; t)(x, y) + tb(x, y),$$\u5176\u4e2d, $b(x, y) \\in \\partial Q$\u662f$\\partial Q$\u4e0e\u4ece$p$\u51fa\u53d1\u7ecf\u8fc7$(x, y)$\u7684\u5c04\u7ebf\u7684\u552f\u4e00\u7684\u4ea4\u70b9. \u6545$(Q \/ $$ \\sim) \\backslash \\{ p \\}$\u7684\u57fa\u672c\u7fa4\u4e0e$\\partial Q \/ \\sim$\u7684\u57fa\u672c\u7fa4\u662f\u540c\u6784\u7684. \u53c8$\\partial Q \/ $$ \\sim$\u540c\u80da\u4e8e\u4e24\u4e2a\u5706\u5468\u7684\u4e00\u70b9\u5e76(\u53c8\u79f02\u5706\u675f), \u6545\u6211\u4eec\u53ef\u901a\u8fc7Van Kampen\u5b9a\u7406\u5f97\u5230$$\\pi_1(T^2 \\backslash \\{ p \\}) = \\pi_1(\\partial Q \/ \\sim) = \\pi_1(S^1 \\vee S^1) \\\\ = \\pi_1(S^1) * \\pi_1(S^1) = Z * Z.$$\u540c\u7406\u53ef\u5f97, \u5f53\u4ece\u73af\u9762\u6316\u53bb$n$\u4e2a\u70b9\u540e, \u6613\u77e5$\\partial Q \/ \\sim$\u540c\u80da\u4e8e$n + 1$\u4e2a\u5706\u5468\u7684\u4e00\u70b9\u5e76(\u53c8\u79f0$n + 1$\u5706\u675f), \u4ece\u800c\u6316\u53bb$n$\u4e2a\u70b9\u7684\u73af\u9762\u7684\u57fa\u672c\u7fa4\u7684\u4e00\u4e2a\u8868\u51fa\u4e3a$\\left \\langle \\gamma_1, \\cdots, \\gamma_{n + 1} \\ | \\ \\right \\rangle$, \u5176\u4e2d\u6bcf\u4e2a$\\gamma_i$\u662f\u7ed5\u7b2c$i$\u4e2a\u5706\u5468\u4e00\u5708\u7684\u95ed\u9053\u8def\u7c7b, \u5982\u4e0b\u56fe\u6240\u793a.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/n\u5706\u675f.png\" alt=\"\" width=\"1280\" height=\"916\" class=\"aligncenter size-full wp-image-1600\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/n\u5706\u675f.png 1280w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/n\u5706\u675f-300x215.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/n\u5706\u675f-768x550.png 768w\" sizes=\"(max-width: 1280px) 100vw, 1280px\" \/><\/p>\n<p>3. \u8bbe$M$\u4e3a\u4e09\u7ef4\u6d41\u5f62, $p \\in M$, \u8bc1\u660e\u5728$M$\u4e2d\u4efb\u53d6\u57fa\u70b9$x_0 \\ne p$, $$\\pi_1(M, x_0) \\cong \\pi_1(M \\backslash \\{ p \\}, x_0).$$<strong>\u8bc1: <\/strong> \u53d6$U$\u4e3a$p$\u7684\u540c\u80da\u4e8e\u5f00\u5b9e\u5fc3\u7403\u7684\u90bb\u57df, \u8bb0$V = M \\backslash \\{ p \\}$, \u7531\u4e8e\u5f00\u5b9e\u5fc3\u7403\u4e3a$E^3$\u4e2d\u7684\u51f8\u5b50\u96c6, \u6545\u5176\u57fa\u672c\u7fa4\u4e3a\u5e73\u51e1\u7fa4(\u76f8\u5173\u8bc1\u660e\u53ef\u53c2\u8003<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/11\/07\/fundamental_group_related_exercises\/\">\u57fa\u672c\u7fa4\u76f8\u5173\u4e60\u9898<\/a>), \u4ece\u800c$U$\u7684\u57fa\u672c\u7fa4\u4ea6\u4e3a\u5e73\u51e1\u7fa4. \u53c8$U \\cap V$\u540c\u80da\u4e8e\u4e00\u4e2a\u53bb\u5fc3\u7684\u5f00\u5b9e\u5fc3\u7403, \u5b83\u53ef\u901a\u8fc7\u6620\u5c04$F: $$ [0, 1] \\times $$ (U \\cap V) \\to S^3$\u5f62\u53d8\u6536\u7f29\u81f3$S^3$, \u6620\u5c04$F$\u5b9a\u4e49\u5982\u4e0b\u6240\u793a,$$F(\\lambda, x) = (1 &#8211; \\lambda)x + \\lambda \\frac{x}{\\left \\| x \\right \\|}.$$$\\therefore \\pi_1(U \\cap V) \\cong \\pi_1(S^3) = \\{ \\left \\langle e_{S^3} \\right \\rangle \\}$, \u5176\u4e2d, $e_{S^3}$\u4e3a$S^3$\u4e2d\u7684\u70b9\u9053\u8def. \u6700\u540e\u5e94\u7528Seifert &#038; Van Kampen\u5b9a\u7406, \u547d\u9898\u5373\u53ef\u5f97\u8bc1.<\/p>\n<p>4. \u4eff\u7167\u73af\u9762\u57fa\u672c\u7fa4\u7684\u8ba1\u7b97\u8fc7\u7a0b, \u5199\u51fa\u5c04\u5f71\u5e73\u9762\u7684\u57fa\u672c\u7fa4\u7684\u4e00\u4e2a\u8ba1\u7b97\u8fc7\u7a0b\u4ee5\u53ca\u7ed3\u679c.<br \/>\n<strong>\u89e3: <\/strong> \u4ee4$RP^2 = A \\cup B$, \u5176\u4e2d$A$\u4e3a\u7ea2\u8272\u5706\u76d8, $B$\u4e3a\u84dd\u8272\u73af, $A$\u4e0e$B$\u7684\u4ea4\u96c6\u4e3a\u7d2b\u8272\u7684\u5c0f\u73af, \u5982\u4e0b\u56fe\u6240\u793a.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/RP^2-CW-Complex.jpg\" alt=\"\" width=\"316\" height=\"421\" class=\"aligncenter size-full wp-image-1619\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/RP^2-CW-Complex.jpg 1263w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/RP^2-CW-Complex-225x300.jpg 225w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/RP^2-CW-Complex-768x1023.jpg 768w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/RP^2-CW-Complex-1153x1536.jpg 1153w\" sizes=\"(max-width: 316px) 100vw, 316px\" \/><\/p>\n<p>\u7531Seifert &#038; Van Kampen\u5b9a\u7406\u53ef\u5f97$$\\pi_1(RP^2) = \\pi_1(A) *_{\\pi_1(A \\cap B)} \\pi_1(B).$$\u63a5\u4e0b\u6765\u5206\u522b\u8ba1\u7b97$\\pi_1(A), \\pi_1(B)$\u4e0e$\\pi_1(A \\cap B)$. \u7531\u4e8e$A$\u662f\u53ef\u7f29\u7684, \u6545$$\\pi_1(A) \\cong \\{ \\left \\langle e_A \\right \\rangle \\} = \\left \\langle \\ \\emptyset \\ | \\ \\emptyset \\ \\right \\rangle,$$\u5176\u4e2d, $e_A$\u4e3a$A$\u4e2d\u7684\u70b9\u9053\u8def. \u540c\u7406, \u7531\u4e8e$A \\cap B$\u4e3a\u4e00\u4e2a\u73af, \u5176\u53ef\u5f62\u53d8\u6536\u7f29\u81f3\u4e00\u4e2a\u5706\u5708, \u4e0d\u59a8\u8bbe\u5176\u751f\u6210\u5143\u4e3a$\\gamma$, \u5219$$\\pi_1(A \\cap B) \\cong Z = \\left \\langle \\gamma \\ | \\ \\emptyset \\ \\right \\rangle.$$\u800c\u5bf9\u4e8e$\\pi_1(B)$, \u6ce8\u610f\u5230$B$\u662f\u7531$RP^2$\u51cf\u53bb\u4e00\u4e2a\u5706\u76d8\u5f97\u5230\u7684, \u6545$B$\u540c\u80da\u4e8e\u4e00\u4e2a\u83ab\u6bd4\u4e4c\u65af\u5e26. \u800c\u83ab\u6bd4\u4e4c\u65af\u5e26\u53ef\u5f62\u53d8\u6536\u7f29\u81f3\u4e00\u4e2a\u5706\u5708(\u901a\u8fc7\u4e00\u4e2a\u7b80\u5355\u7684\u6295\u5f71), \u4e0d\u59a8\u8bbe\u5176\u751f\u6210\u5143\u4e3a$b$, \u5219$$\\pi_1(B) \\cong Z = \\left \\langle b \\ | \\ \\emptyset \\ \\right \\rangle.$$\u4ece\u800c\u6211\u4eec\u6709$$\\pi_1(RP^2) = \\{ \\left \\langle e_A \\right \\rangle \\} *_Z Z.$$\u7531\u81ea\u7531\u79ef\u7684\u5b9a\u4e49, \u6211\u4eec\u5217\u51fa\u6240\u6709\u76f8\u5173\u7684\u751f\u6210\u5143\u4e0e\u751f\u6210\u5173\u7cfb, \u5e76\u5c06\u4e0a\u5f0f\u6539\u5199\u4e3a\u5982\u4e0b\u5f62\u5f0f:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/presentation-of-pi_1RP^2-scaled.jpg\" alt=\"\" width=\"2560\" height=\"733\" class=\"aligncenter size-full wp-image-1620\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/presentation-of-pi_1RP^2-scaled.jpg 2560w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/presentation-of-pi_1RP^2-300x86.jpg 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/presentation-of-pi_1RP^2-768x220.jpg 768w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/presentation-of-pi_1RP^2-1536x440.jpg 1536w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/presentation-of-pi_1RP^2-2048x587.jpg 2048w\" sizes=\"(max-width: 2560px) 100vw, 2560px\" \/><\/p>\n<p>\u5176\u4e2d, $i_{A_*} : \\pi_(A \\cap B) \\to \\pi_1(A)$\u662f\u7531\u542b\u5165\u6620\u5c04$i_A : A \\cap B \\mapsto A$\u8bf1\u5bfc\u7684\u540c\u6001\u6620\u5c04, $i_{B_*} : \\pi_(A \\cap B) \\to \\pi_1(B)$\u662f\u7531\u542b\u5165\u6620\u5c04$i_B : A $$ \\cap B \\mapsto B$\u8bf1\u5bfc\u7684\u540c\u6001\u6620\u5c04.\u63a5\u4e0b\u6765\u6211\u4eec\u4ec5\u9700\u8ba1\u7b97\u6620\u5c04$i_{A_*}, i_{B_*}$\u5373\u53ef. \u800c\u6620\u5c04$i_{A_*}$\u662f\u5bb9\u6613\u8ba1\u7b97\u7684, \u56e0\u4e3a$\\pi_1(A) $$ = \\{ \\left \\langle e \\right \\rangle \\}$, \u6545$i_{A_*}$\u662f\u4e00\u4e2a\u5e73\u51e1\u6620\u5c04, \u5373$i_{A_*}(\\left \\langle \\gamma \\right \\rangle) = \\left \\langle e_A \\right \\rangle$.<br \/>\n$\\\\$ \u4f46\u5bf9\u4e8e\u751f\u6210\u5143$\\gamma$\u5728\u6620\u5c04$i_{B_*}$\u4e0b\u7684\u50cf, \u6211\u4eec\u9700\u8981\u8003\u8651\u751f\u6210\u5143$b$\u4e0e\u751f\u6210\u5143$\\gamma$\u4e4b\u95f4\u7684\u5173\u7cfb(\u4e00\u4e2a\u540c\u6001\u6620\u5c04\u53ef\u7531\u5b9a\u4e49\u57df\u4e0e\u503c\u57df\u7684\u751f\u6210\u5143\u51b3\u5b9a), \u793a\u610f\u56fe\u5982\u4e0b\u6240\u793a.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/relationship-between-generators-scaled.jpg\" alt=\"\" width=\"2560\" height=\"330\" class=\"aligncenter size-full wp-image-1623\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/relationship-between-generators-scaled.jpg 2560w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/relationship-between-generators-300x39.jpg 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/relationship-between-generators-768x99.jpg 768w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/relationship-between-generators-1536x198.jpg 1536w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/03\/relationship-between-generators-2048x264.jpg 2048w\" sizes=\"(max-width: 2560px) 100vw, 2560px\" \/><\/p>\n<p>\u5176\u4e2d, \u751f\u6210\u5143$b$\u4e3a\u83ab\u6bd4\u4e4c\u65af\u5e26\u4e2d\u4f4d\u4e8e\u4e2d\u5fc3\u5904\u7684\u5706\u5708, \u751f\u6210\u5143$\\gamma$\u4e3a\u5176\u8fb9\u754c\u5706. \u7531\u4e0a\u56fe\u53ef\u77e5, $i_{B_*}(\\left \\langle \\gamma \\right \\rangle) = b^2$.<br \/>\n$\\\\$ \u7efc\u4e0a\u6240\u8ff0, \u6211\u4eec\u53ef\u4ee5\u5f97\u5230$\\pi_1(RP^2)$\u7684\u8868\u51fa, \u5373$$\\pi_1(RP^2) = \\left \\langle b \\ | \\ b^2 \\right \\rangle = Z \/ 2Z.$$Q.E.D.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6700\u8fd1\u5de5\u4f5c\u4e0a\u8fd8\u662f\u53d1\u751f\u4e86\u86ee\u591a\u4e8b\u7684, \u6700\u4e3b\u8981\u7684\u53d8\u52a8\u5c31\u662f\u81ea\u5df1\u5728\u7684\u9879\u76ee\u7ec4\u88ab&#8221;\u780d&#8221;\u4e86\u2026\u2026 \u51e1\u4e8b\u90fd\u5177 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/03\/05\/seifert_van_kampen_theorem_related_exercises\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">Seifert &#038; Van Kampen\u5b9a\u7406\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\/1565"}],"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=1565"}],"version-history":[{"count":54,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1565\/revisions"}],"predecessor-version":[{"id":3612,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1565\/revisions\/3612"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1565"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1565"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1565"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}