{"id":2007,"date":"2022-08-20T14:04:14","date_gmt":"2022-08-20T06:04:14","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=2007"},"modified":"2025-02-26T11:16:11","modified_gmt":"2025-02-26T03:16:11","slug":"fundamental_groups_covering_spaces_mark","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/08\/20\/fundamental_groups_covering_spaces_mark\/","title":{"rendered":"\u590d\u8fed\u7a7a\u95f4\u7684\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>\u597d\u50cf\u633a\u4e45\u6ca1\u6709\u5728\u5bb6\u91cc\u8fc7\u8fc7\u5468\u672b\u4e86, \u5c3d\u7ba1\u4e0a\u4e0a\u5468\u4eba\u5728\u5bb6\u91cc, \u4f46\u7531\u4e8e\u57fa\u672c\u5367\u75c5\u5728\u5e8a, \u6240\u4ee5\u57fa\u672c\u6ca1\u6709\u5468\u672b\u5b85\u5bb6\u7684\u611f\u89c9\u2026\u2026 \u6240\u4ee5\u8bf4, \u8eab\u4f53\u624d\u662f\u9769\u547d\u7684\u672c\u94b1\u5450\u56e7~ \u590d\u8fed\u7a7a\u95f4\u7684\u57fa\u672c\u7fa4\u8fd9\u4e00\u8282\u5185\u5bb9\u4e5f\u7b97\u662f\u5b66\u4e86\u633a\u4e45\u4e86, \u672c\u6587\u4e3b\u8981\u7528\u4e8e\u8bb0\u5f55\u5b66\u4e60\u8fd9\u4e00\u8282\u5185\u5bb9\u7684\u8fc7\u7a0b\u4e2d\u78b0\u5230\u7684\u4e00\u4e9b\u7591\u60d1\u70b9\u7684\u89e3\u7b54.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/math.stackexchange.com\/questions\/419137\/induced-map-between-fundamental-groups-from-covering-map-is-injective\">Induced map between fundamental groups from covering map is injective<\/a><br \/>\n2. <a href=\"http:\/\/www2.math.ou.edu\/~forester\/6813F06\/e1sol.pdf\">Exam I Solutions<\/a><\/p>\n<p>1. \u4e66\u4e0aP237\u547d\u98985.3.1\u7684\u8bc1\u660e\u4e2d, \u6784\u9020\u4e86\u4e00\u4e2a\u4ece$a^\\uparrow$\u51fa\u53d1\u7684\u4f26\u79fb$H^\\uparrow$, \u5176\u5728$t$\u65f6\u523b\u7684\u5207\u7247\u4e3a\u9053\u8def$(h^\\uparrow)_t$. \u5f53$t$\u53d8\u5316\u65f6, $(h^\\uparrow)_t$\u7684\u8d77\u70b9\u59cb\u7ec8\u53ea\u80fd\u5728\u540c\u4e00\u6839\u7ea4\u7ef4\u4e2d\u79fb\u52a8, \u56e0\u6b64\u53ea\u80fd\u59cb\u7ec8\u56fa\u5b9a\u4e0d\u52a8. \u8fd9\u662f\u56e0\u4e3a, \u4f26\u79fb$H^\\uparrow$\u662f\u4e00\u4e2a\u5173\u4e8e\u65f6\u523b$t$\u7684\u8fde\u7eed\u51fd\u6570, $a(0)$\u5728\u590d\u8fed\u6620\u5c04\u4e0b\u7684\u539f\u50cf\u5305\u542b\u53ef\u80fd\u4e0d\u6b62\u4e00\u4e2a\u5143\u7d20. \u6b64\u65f6, \u82e5$(h^\\uparrow)_t$\u7684\u8d77\u70b9\u5728\u540c\u4e00\u6839\u7ea4\u7ef4\u4e2d\u7684\u4e0d\u540c\u70b9\u4e4b\u95f4\u79fb\u52a8, \u800c\u7ea4\u7ef4\u90fd\u662f\u79bb\u6563\u7a7a\u95f4, \u8fd9\u5c06\u4e0e\u4f26\u79fb$H^\\uparrow$\u662f\u4e00\u4e2a\u5173\u4e8e\u65f6\u523b$t$\u7684\u8fde\u7eed\u51fd\u6570\u7684\u4e8b\u5b9e\u77db\u76fe.<\/p>\n<p>2. \u8bbe$p: E \\to B$\u662f\u590d\u8fed\u6620\u5c04, \u5219$\\forall e \\in E$,$$p_\\pi : \\pi_1(E, e) \\to \\pi_1(B, p(e))$$\u662f\u5355\u540c\u6001.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u8bb0$c_e$\u4e3a$e$\u7684\u70b9\u9053\u8def, $c_{p(e)}$\u4e3a$p(e)$\u7684\u70b9\u9053\u8def, \u95ee\u9898\u5373\u8981\u8bc1$ker \\ p_\\pi = $$ [c_e] \\Leftrightarrow $\u4f7f\u5f97$p_\\pi([a]) = [c_{p(e)}]$\u6210\u7acb\u7684$[a]$\u4ec5\u6709$[c_e]$.<br \/>\n$\\\\$ \u56de\u987e\u4e00\u4e0b\u590d\u8fed\u6620\u5c04\u7684\u540c\u4f26\u63d0\u5347\u6027\u8d28: \u8bbe$p: E \\to B$\u662f\u590d\u8fed\u6620\u5c04. \u8bbe$F $$ : X \\times [0, $$ 1] \\to B$\u662f\u4ece$f: X \\to B$\u5f00\u59cb\u7684\u4f26\u79fb, \u800c$f^\\uparrow : $$ X \\to E$\u662f$f$\u5173\u4e8e$p$\u7684\u63d0\u5347, \u5219\u5b58\u5728\u552f\u4e00\u4e00\u4e2a\u4ece$f^\\uparrow$\u5f00\u59cb\u7684\u4f26\u79fb$F^\\uparrow : X \\times [0, 1] $$ \\to E$, \u4f7f\u5f97$F^\\uparrow$\u662f$F$\u5173\u4e8e$p$\u7684\u63d0\u5347.<br \/>\n$\\\\$ \u4e0d\u59a8\u4ee4$X = I$, $f^\\uparrow_0$\u662f$E$\u4e2d\u7684\u4e00\u6761\u9053\u8def, \u5176\u5728\u590d\u8fed\u6620\u5c04$p$\u4e0b\u7684\u50cf$p \\circ f^\\uparrow_0 $$ = f_0$\u662f$\\pi_1( $$ B, p(e))$\u4e2d\u7684\u96f6\u4f26\u6620\u5c04, \u5219\u6211\u4eec\u53ef\u5f97\u4e00\u4e2a\u4ece\u9053\u8def$f_0$\u5f00\u59cb, \u5230\u4e00\u6761\u70b9\u9053\u8def$f_1$\u7ed3\u675f\u7684\u4f26\u79fb$F$. \u6839\u636e\u590d\u8fed\u6620\u5c04\u7684\u540c\u4f26\u63d0\u5347\u6027\u8d28, \u6211\u4eec\u53ef\u5f97\u4e00\u4e2a\u4ece\u9053\u8def$f^\\uparrow_0$\u5f00\u59cb, \u5230\u70b9\u9053\u8def$f_1$\u7684\u63d0\u5347\u7ed3\u675f\u7684\u4f26\u79fb$F^\\uparrow$. \u53c8\u7531\u4f26\u79fb$F^\\uparrow$\u7684\u552f\u4e00\u6027, $B$\u4e2d\u7684\u70b9\u9053\u8def$f_1$\u7684\u63d0\u5347\u662f\u552f\u4e00\u7684, \u6545\u70b9\u9053\u8def$f_1$\u7684\u63d0\u5347\u4ea6\u4e3a$E$\u4e2d\u7684\u70b9\u9053\u8def, \u4ece\u800c\u9053\u8def$f^\\uparrow_0$\u4ea6\u4e3a\u96f6\u4f26\u6620\u5c04, \u547d\u9898\u5f97\u8bc1.<\/p>\n<p>3. \u4e66\u4e0aP238\u4f8b1\u7684\u8bc1\u660e\u4e2d, \u6307\u51fa\u5f53$n \\ge 2$\u65f6, \u5706\u4e0a\u9576\u5d4c(\u5373\u4e00\u70b9\u5e76) $n &#8211; 1$\u4e2a\u5c0f\u5706\u540e\u5f97\u5230\u7684\u7a7a\u95f4\u662f$S^1 \\vee S^1$\u7684\u590d\u8fed\u7a7a\u95f4, \u5373$n$\u4e2a\u5706\u7684\u4e00\u70b9\u5e76\u662f2\u4e2a\u5706\u7684\u4e00\u70b9\u5e76\u7684\u590d\u8fed\u7a7a\u95f4. \u4e8b\u5b9e\u4e0a, \u8fd9\u5728\u4e0a\u4e00\u7bc7\u6587\u7ae0\u4e2d\u5df2\u7ecf\u63d0\u53ca, \u53ea\u4e0d\u8fc7\u4e0a\u4e00\u7bc7\u6587\u7ae0\u4e2d\u672a\u5217\u51fa\u76f8\u5e94\u7684\u591a\u8fb9\u5f62\u8868\u793a, \u8865\u5145\u5982\u4e0b.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/07\/3-sheeted-coverings.png\" alt=\"\" width=\"712\" height=\"170\" class=\"aligncenter size-full wp-image-2039\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/07\/3-sheeted-coverings.png 712w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/07\/3-sheeted-coverings-300x72.png 300w\" sizes=\"(max-width: 712px) 100vw, 712px\" \/><\/p>\n<p>4. \u8bbe$p: E \\to B$\u4e3a\u4e00\u590d\u8fed\u6620\u5c04,<br \/>\n$\\\\$ (1) \u9053\u8def\u7684\u63d0\u5347\u7684\u9006\u7b49\u4e8e\u9053\u8def\u7684\u9006\u7684\u63d0\u5347, \u5373\u5bf9\u4e8e\u5e95\u7a7a\u95f4$B$\u4e2d\u7684\u4efb\u610f\u9053\u8def$a$, \u5747\u6709$(a^\\uparrow)^{-1} = (a^{-1})^\\uparrow$.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u9996\u5148\u53d6\u4efb\u610f$t \\in [0, 1]$, \u7531\u9006\u9053\u8def\u7684\u5b9a\u4e49\u6709$a^{-1}(t) = a(1 &#8211; t)$, $( $$ a^\\uparrow)^{-1}(t) = $$ a^\\uparrow(1 &#8211; t)$. \u53c8$a^\\uparrow$\u4e3a$a$\u5173\u4e8e$p$\u7684\u63d0\u5347, \u6545\u5b58\u5728\u6620\u5c04$f$(\u9700\u8981\u4fdd\u8bc1\u8fde\u7eed\u6027?) \u4f7f\u5f97$$a^\\uparrow = f \\circ a, \\\\ (a^{-1})^\\uparrow = f \\circ a^{-1}.$$\u4ece\u800c\u6211\u4eec\u6709$$(a^\\uparrow)^{-1}(t) = a^\\uparrow(1 &#8211; t) = (f \\circ a)(1 &#8211; t) = \\\\ f \\circ a(1 &#8211; t) = f \\circ a^{-1}(t) = (a^{-1})^\\uparrow(t).$$\u6545\u547d\u9898\u5f97\u8bc1.<br \/>\n$\\\\$ (2) \u5bf9\u4e8e\u5e95\u7a7a\u95f4$B$\u4e2d\u7684\u4efb\u610f\u9053\u8def$a, b$, \u9053\u8def$a$\u7684\u7ec8\u70b9\u4e3a\u9053\u8def$b$\u7684\u8d77\u70b9(\u5373\u9053\u8def\u4e58\u79ef$ab$\u662f\u6709\u610f\u4e49\u7684), \u5747\u6709$a^\\uparrow b^\\uparrow = (ab)^\\uparrow$.<br \/>\n<strong>\u8bc1:<\/strong> \u7531\u4e8e$a^\\uparrow, b^\\uparrow$\u5206\u522b\u4e3a$a, b$\u5173\u4e8e$p$\u7684\u63d0\u5347, \u4e14\u9053\u8def$a$\u7684\u7ec8\u70b9\u4e3a\u9053\u8def$b$\u7684\u8d77\u70b9, \u6545\u5b58\u5728\u6620\u5c04$f$\u4f7f\u5f97$a^\\uparrow = f \\circ a$, $b^\\uparrow = f \\circ b$, \u4ece\u800c$$a^\\uparrow b^\\uparrow = (f \\circ a)(f \\circ b) = f \\circ (ab) = (ab)^\\uparrow.$$\u6545\u547d\u9898\u5f97\u8bc1.<\/p>\n<p>5. \u5728\u6cdb\u590d\u8fed\u7a7a\u95f4\u7684\u60c5\u5f62, $\\pi_1(B, b)$\u7684\u5143\u7d20\u548c$p^{-1}(b)$\u7684\u5143\u7d20\u4e00\u4e00\u5bf9\u5e94.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u7531\u4e66\u4e0aP240\u5b9a\u74065.3.1\u53ef\u77e5, $H_e$\u5728$\\pi_1(B, b)$\u4e2d\u7684\u53f3\u966a\u96c6\u548c$p^{-1}(b)$\u7684\u5143\u7d20\u4e00\u4e00\u5bf9\u5e94. \u5f53$E$\u4e3a\u4e00\u4e2a\u6cdb\u590d\u8fed\u7a7a\u95f4\u65f6, \u7531$H_e$\u5b9a\u4e49$H_e := p_\\pi( $$ \\pi_1(E, e))$\u53ef\u77e5$H_e$\u4e3a\u4e00\u4e2a\u5e73\u51e1\u7fa4($\\pi_1(E, e)$\u4e3a\u4e00\u4e2a\u5e73\u51e1\u7fa4, $p_\\pi$\u4e3a\u4e00\u4e2a\u5355\u540c\u6001), \u6545$H_e$\u5728$\\pi_1(B, b)$\u4e2d\u7684\u53f3\u966a\u96c6\u4e0e$\\pi_1(B, b)$\u7684\u5143\u7d20\u4e00\u4e00\u5bf9\u5e94, \u547d\u9898\u5f97\u8bc1.<\/p>\n<p>6. \u8bbe$p: E \\to B$\u662f\u6cdb\u590d\u8fed\u6620\u5c04. \u8bc1\u660e\u4efb\u53d6$b \\in B$, \u5b58\u5728$b$\u7684\u90bb\u57df$U$, \u4f7f\u5f97\u5305\u542b\u6620\u5c04\u8bf1\u5bfc\u7684\u57fa\u672c\u7fa4\u540c\u6001$i_\\pi : \\pi_1(U) \\to \\pi_1(B)$\u662f\u5e73\u51e1\u540c\u6001.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u53d6\u4e00\u4e2a\u5747\u5300\u590d\u8fed\u90bb\u57df$U$, \u7136\u540e\u53d6$i: U \\hookrightarrow B$\u7684\u4e00\u4e2a\u63d0\u5347$i^\\uparrow : U $$ \\to E$, \u5219$i_\\pi $$ = p_\\pi \\circ (i^\\uparrow)_\\pi$, \u800c$E$\u5355\u8fde\u901a\u65f6\u5176\u57fa\u672c\u7fa4\u4e3a\u4e00\u4e2a\u5e73\u51e1\u7fa4, \u6545$(i^\\uparrow)_\\pi$\u662f\u5e73\u51e1\u540c\u6001, \u4ece\u800c$i_\\pi$\u662f\u5e73\u51e1\u540c\u6001.<\/p>\n<p>7. \u8bbe$a: [0, 1] \\to S^1$\u662f\u4e00\u6761\u5708\u6570\u4e3a$n$\u7684\u95ed\u9053\u8def. \u8bc1\u660e\u4efb\u53d6$x \\in S^1$, $a^{-1}(x)$\u81f3\u5c11\u5305\u542b$n$\u4e2a\u70b9.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u8bbe$p: E^1 \\to S^1, t \\mapsto (cos(2\\pi t), sin(2\\pi t))$\u662f\u6cdb\u590d\u8fed\u6620\u5c04. \u53d6$a$\u7684\u63d0\u5347$a^\\uparrow$, \u5219\u7531$a = p \\circ a^\\uparrow$\u53ef\u77e5$a^{-1} = (a^\\uparrow)^{-1} \\circ p^{-1}$. \u53c8\u5bf9\u4e8e\u6cdb\u590d\u8fed\u7a7a\u95f4$E^1$, $\\pi_1(S^1, x)$\u7684\u5143\u7d20\u4e0e$p^{-1}(x)$\u7684\u5143\u7d20\u4e00\u4e00\u5bf9\u5e94.<br \/>\n$\\\\$ $\\because a$\u662f\u4e00\u6761\u5708\u6570\u4e3a$n$\u7684\u95ed\u9053\u8def, \u6545$\\pi_1(S^1, x)$\u7684\u5143\u7d20\u6570\u91cf\u81f3\u5c11\u4e3a$n$, \u4ece\u800c$p^{-1}(x)$\u7684\u5143\u7d20\u6570\u91cf\u81f3\u5c11\u4e3a$n$. \u5728$(a^\\uparrow)^{-1}$\u7684\u590d\u5408\u4f5c\u7528\u4e0b, $a^{-1}(x)$\u7684\u5143\u7d20\u6570\u91cf\u4ea6\u81f3\u5c11\u4e3a$n$, \u547d\u9898\u5f97\u8bc1.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u597d\u50cf\u633a\u4e45\u6ca1\u6709\u5728\u5bb6\u91cc\u8fc7\u8fc7\u5468\u672b\u4e86, \u5c3d\u7ba1\u4e0a\u4e0a\u5468\u4eba\u5728\u5bb6\u91cc, \u4f46\u7531\u4e8e\u57fa\u672c\u5367\u75c5\u5728\u5e8a, \u6240\u4ee5\u57fa\u672c\u6ca1\u6709\u5468\u672b\u5b85\u5bb6\u7684\u611f\u89c9\u2026\u2026 \u6240 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/08\/20\/fundamental_groups_covering_spaces_mark\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u590d\u8fed\u7a7a\u95f4\u7684\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":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/2007"}],"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=2007"}],"version-history":[{"count":67,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/2007\/revisions"}],"predecessor-version":[{"id":3625,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/2007\/revisions\/3625"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2007"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2007"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2007"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}