{"id":2214,"date":"2022-10-03T13:50:53","date_gmt":"2022-10-03T05:50:53","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=2214"},"modified":"2025-02-26T11:14:53","modified_gmt":"2025-02-26T03:14:53","slug":"lifting_criterion_mark","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/10\/03\/lifting_criterion_mark\/","title":{"rendered":"\u6620\u5c04\u63d0\u5347\u5b9a\u7406\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>\u7ec8\u4e8e\u8fce\u6765\u4e86\u56fd\u5e86\u5047\u671f, \u5927\u6982\u6709\u4e00\u5468\u7684\u65f6\u95f4\u662f&#8221;\u8d4b\u95f2\u5728\u5bb6&#8221;\u7684, \u8bf4\u5b9e\u8bdd, \u8fd8\u662f\u86ee\u723d\u7684~ \u76ee\u524d\u4ee3\u6570\u62d3\u6251\u7684\u5b66\u4e60\u5185\u5bb9\u4ec5\u5269\u4e24\u8282, \u81ea\u5df1\u76ee\u524d\u7684\u8ba1\u5212\u662f\u5728\u5047\u671f\u5185\u5b8c\u6210\u5012\u6570\u7b2c\u4e8c\u8282\u5185\u5bb9\u2014\u2014\u6620\u5c04\u63d0\u5347\u5b9a\u7406\u7684\u5b66\u4e60, \u7136\u540e\u5f00\u542f\u6700\u540e\u4e00\u8282\u5185\u5bb9\u2014\u2014\u590d\u8fed\u53d8\u6362\u7684\u5b66\u4e60. \u5c3d\u7ba1\u4ee3\u6570\u62d3\u6251\u7684\u5b66\u4e60\u5373\u5c06\u7ed3\u675f, \u4f46\u8fd8\u662f\u5e0c\u671b\u81ea\u5df1\u80fd\u591f\u4e0d\u6025\u4e8e\u5b8c\u6210\u5269\u4e0b\u7684\u5185\u5bb9, \u800c\u662f\u7ee7\u7eed\u4ee5\u4e00\u79cd\u7cbe\u76ca\u6c42\u7cbe\u7684\u6001\u5ea6\u8d70\u4e0b\u53bb~<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/math.stackexchange.com\/questions\/3381787\/the-proof-of-lifting-criterion\">the proof of lifting criterion<\/a><br \/>\n2. <a href=\"https:\/\/wuli.wiki\/online\/HomT4.html\">\u9ad8\u9636\u540c\u4f26\u7fa4<\/a><br \/>\n3. <a href=\"https:\/\/math.stackexchange.com\/questions\/1676352\/is-a-lift-of-a-topological-embedding-still-an-embedding\">Is a lift of a topological embedding still an embedding?<\/a><br \/>\n4. <a href=\"https:\/\/math.stackexchange.com\/questions\/1689624\/showing-there-is-no-covering-map-mathbbrp2-to-t2\">Showing there is no covering map RP^2 to T^2<\/a><br \/>\n5. <a href=\"https:\/\/math.ou.edu\/~forester\/5863S06\/e3sol.pdf\">Exam III Solutions Topology II Due May 2, 2006<\/a><br \/>\n6. <a href=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/10\/Classification.pdf\">Classification<\/a><br \/>\n7. <a href=\"https:\/\/math.stackexchange.com\/questions\/3176802\/homotopy-equivalence-of-covering-spaces\">Homotopy equivalence of covering spaces<\/a><br \/>\n8. <a href=\"https:\/\/math.stackexchange.com\/questions\/1587411\/homotopy-equivalent-spaces-have-homotopy-equivalent-universal-covers\">Homotopy equivalent spaces have homotopy equivalent universal covers<\/a><\/p>\n<p>1. \u4e66\u4e0aP249\u5728\u5bf9\u4e8e\u6784\u9020\u51fa\u7684$f^\\uparrow$\u4e3aWell-Defined\u7684\u8bc1\u660e\u4e2d\u63d0\u5230: \u5982\u679c\u6709\u4e24\u4e2a\u4ece$x$\u5230$y$\u7684\u9053\u8def\u7c7b$\\alpha_1$, $\\alpha_2$, \u5219$\\alpha_1 \\alpha_2^{-1} \\in \\pi_1(X, x)$, \u56e0\u6b64\u7531\u5047\u8bbe\u6761\u4ef6\u53ef\u77e5$\\beta_1 \\beta_2^{-1} \\in H_e$, \u5b83\u7684\u63d0\u5347\u4e00\u5b9a\u662f\u4e00\u4e2a\u95ed\u9053\u8def.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u6211\u4eec\u4ec5\u9700\u8bc1\u660e\u4e00\u4e2a\u4e00\u822c\u7684\u7ed3\u8bba\u5373\u53ef: \u5bf9\u4e8e\u7531\u590d\u8fed\u6620\u5c04$p: E \\to B$\u8bf1\u5bfc\u7684\u57fa\u672c\u7fa4\u540c\u6001$p_\\pi: \\pi_1(E, e) \\to \\pi_1(B, b)$, $H_e = p_\\pi(\\pi_1(E, e $$ ))$\u4e2d\u7684\u5143\u7d20(\u5373\u4e3a$B$\u4e2d\u4e00\u4e2a\u4ee5$b$\u4e3a\u57fa\u70b9\u7684\u95ed\u9053\u8def\u7c7b) \u5173\u4e8e\u590d\u8fed\u6620\u5c04$p$\u7684\u63d0\u5347\u5747\u4e3a$\\pi_1(E, e)$\u4e2d\u7684\u5143\u7d20(\u5373\u4e3a$E$\u4e2d\u4e00\u4e2a\u4ee5$e$\u4e3a\u57fa\u70b9\u7684\u95ed\u9053\u8def\u7c7b).<br \/>\n$\\\\$ \u82e5$\\left \\langle \\gamma \\right \\rangle \\in H_e$, \u5219$\\left \\langle \\gamma \\right \\rangle = p_\\pi(\\left \\langle \\gamma_0^\\uparrow \\right \\rangle)$, \u5176\u4e2d, $\\left \\langle \\gamma_0^\\uparrow \\right \\rangle \\in \\pi_1(E, e)$. \u6545\u6211\u4eec\u6709$\\gamma \\simeq p \\circ $$ \\gamma_0^\\uparrow := \\gamma_0$. \u7531\u540c\u4f26\u63d0\u5347\u6027\u8d28\u53ef\u77e5, $\\gamma^\\uparrow \\simeq \\gamma_0^\\uparrow$. \u53c8$\\gamma_0^\\uparrow$\u4e3a$E$\u4e2d\u4e00\u4e2a\u4ee5$e$\u4e3a\u57fa\u70b9\u7684\u95ed\u9053\u8def, \u6545$\\gamma^\\uparrow$\u4ea6\u4e3a$E$\u4e2d\u4e00\u4e2a\u4ee5$e$\u4e3a\u57fa\u70b9\u7684\u95ed\u9053\u8def.<br \/>\n$\\\\$ \u53cd\u8fc7\u6765, \u82e5$\\gamma_0$\u4e3a$B$\u4e2d\u4e00\u4e2a\u4ee5$b$\u4e3a\u57fa\u70b9\u7684\u95ed\u9053\u8def, s.t. \u5176\u63d0\u5347$\\gamma^\\uparrow$\u4e3a$E$\u4e2d\u4e00\u4e2a\u4ee5$e$\u4e3a\u57fa\u70b9\u7684\u95ed\u9053\u8def(\u5373$\\left \\langle \\gamma^\\uparrow \\right \\rangle \\in \\pi_1(E, e)$), \u5219$\\gamma = p \\circ \\gamma^\\uparrow \\Rightarrow \\left \\langle \\gamma \\right \\rangle $$ = p_\\pi(\\left \\langle \\gamma^\\uparrow \\right \\rangle)$, \u6545$\\left \\langle \\gamma \\right \\rangle $$ \\in H_e$. \u7efc\u4e0a\u6240\u8ff0, \u547d\u9898\u5f97\u8bc1.<\/p>\n<p>2. \u6620\u5c04\u63d0\u5347\u5b9a\u7406\u4e2d\u7684$f_\\pi$\u6307\u7684\u662f\u8fde\u7eed\u6620\u5c04$f: X \\to B$\u8bf1\u5bfc\u7684\u57fa\u672c\u7fa4\u7684\u540c\u6001, \u4e5f\u5c31\u662f\u8bf4, \u6211\u4eec\u4ec5\u987b\u8003\u8651\u57fa\u672c\u7fa4\u7684\u56e0\u7d20(\u53ef\u5ffd\u7565\u9ad8\u7ef4\u540c\u4f26\u7fa4), \u5373\u53ef\u5224\u65ad\u662f\u5426\u5b58\u5728\u6ee1\u8db3\u6761\u4ef6\u7684\u63d0\u5347. \u8fd9\u6837\u4e00\u6765, \u5728\u4e66\u4e0aP250\u7684\u4f8b2\u4e2d\u4e4b\u6240\u4ee5\u8981\u6c42$n > 1$, \u662f\u56e0\u4e3a$S^1$\u7684\u57fa\u672c\u7fa4\u975e\u5e73\u51e1, \u800c$S^n(n > 1)$\u7684\u57fa\u672c\u7fa4\u5747\u4e3a\u5e73\u51e1\u7fa4, \u4ece\u800c\u53ef\u4ee5\u76f4\u63a5\u5bf9$S^n(n > 1)$\u4f7f\u7528\u6620\u5c04\u63d0\u5347\u5b9a\u7406.<br \/>\n$\\\\$ \u6b64\u5916, \u9ad8\u7ef4\u540c\u4f26\u7fa4$\\pi_n(X, x_0)$\u4e2d\u7684\u6bcf\u4e00\u4e2a\u5143\u7d20\u53ef\u4ee5\u7406\u89e3\u6210\u4e00\u4e2a\u628a$(1, $$ 0, \\cdots, 0)$\u53d8\u5230$x_0$\u7684\u8fde\u7eed\u6620\u5c04$h: S^n \\to X$\u7684\u5b9a\u7aef\u540c\u4f26\u7c7b, \u5373$\\pi_n(X, $$ x_0) = \\{ \\left \\langle h \\right \\rangle  \\}.$<\/p>\n<p>3. \u8bbe$p: E \\to B$\u662f\u4e00\u4e2a\u590d\u8fed\u6620\u5c04, \u800c$n > 1$. \u9009\u5b9a\u57fa\u70b9$b \\in B$\u4ee5\u53ca$e $$ \\in p^{-1}(b)$, \u5219$$p_\\pi : \\pi_n(E, e) \\to \\pi_n(B, b), \\left \\langle h \\right \\rangle \\mapsto \\left \\langle p \\circ h \\right \\rangle$$\u662f\u4e00\u4e2a\u4e00\u4e00\u5bf9\u5e94.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u4ee4$x = (1, 0, \\cdots, 0)$, \u4efb\u53d6\u8fde\u7eed\u6620\u5c04$f: S^n \\to B$\u4f7f\u5f97$f(x) = $$ b$, \u56e0\u4e3a$S^n$\u7684\u57fa\u672c\u7fa4\u5e73\u51e1, \u6240\u4ee5\u5b58\u5728\u552f\u4e00\u63d0\u5347$f^\\uparrow : S^n \\to E$\u6ee1\u8db3$f^\\uparrow(x $$ ) = e$.<br \/>\n$\\\\$ \u4e0d\u4ec5\u5982\u6b64, \u5982\u679c$f \\simeq g$, \u5e76\u4e14\u4f26\u79fb\u8fc7\u7a0b\u4e2d$x$\u7684\u50cf\u59cb\u7ec8\u4e0d\u52a8, \u5219\u7531\u540c\u4f26\u63d0\u5347\u6027\u8d28\u53ef\u77e5, \u4ece$f$\u5f00\u59cb, \u5230$g$\u7ed3\u675f\u7684\u4f26\u79fb$F$\u4e00\u5b9a\u5b58\u5728\u4ece$f^\\uparrow$\u5f00\u59cb\u7684\u4f26\u79fb$F^\\uparrow$\u4f5c\u4e3a\u5176\u63d0\u5347. \u53c8\u4f26\u79fb$F$\u4fdd\u6301$x$\u7684\u50cf\u59cb\u7ec8\u4e0d\u52a8, \u5373$f(0) = g(0) \\Rightarrow $$ f^\\uparrow(0) = g^\\uparrow(0)$(\u56e0\u4e3a\u7ea4\u7ef4\u90fd\u662f\u79bb\u6563\u7a7a\u95f4), \u6545\u7531\u4e66\u4e0aP237\u7684\u547d\u98985.3.1\u53ef\u77e5, $f^\\uparrow \\simeq g^\\uparrow$, \u4ece\u800c\u4f26\u79fb$F^\\uparrow$\u7684\u7ec8\u6b62\u72b6\u6001\u4e00\u5b9a\u662f$g^\\uparrow$. \u56e0\u6b64$p_\\pi$\u628a$f^\\uparrow$\u7684\u5b9a\u7aef\u540c\u4f26\u7c7b\u5bf9\u5e94\u4e3a$f$\u7684\u5b9a\u7aef\u540c\u4f26\u7c7b, \u5e76\u4e14\u662f\u4e00\u4e2a\u4e00\u4e00\u5bf9\u5e94.<\/p>\n<p>4. \u8bbe$B$\u9053\u8def\u8fde\u901a\u5e76\u4e14\u5c40\u90e8\u9053\u8def\u8fde\u901a, $p: E \\to B$\u662f\u6cdb\u590d\u8fed\u6620\u5c04, \u5219\u4efb\u53d6\u590d\u8fed\u6620\u5c04$q: F \\to B$, \u5b58\u5728\u590d\u8fed\u6620\u5c04$p^\\uparrow: E \\to F$, \u6ee1\u8db3$p = q \\circ $$ p^\\uparrow$.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u56e0\u4e3a$E$\u5355\u8fde\u901a, \u6240\u4ee5$p$\u6ee1\u8db3\u6620\u5c04\u63d0\u5347\u5b9a\u7406\u7684\u8981\u6c42, \u5b83\u5173\u4e8e\u590d\u8fed\u6620\u5c04$q$\u5b58\u5728\u63d0\u5347$p^\\uparrow$.<br \/>\n$\\\\$ \u73b0\u5728\u4efb\u53d6$x \\in F$, \u5219$q(x) \\in B$, \u53d6$q(x)$\u5173\u4e8e$(E, p)$\u7684\u5747\u5300\u590d\u8fed\u90bb\u57df$V$\u4ee5\u53ca\u5173\u4e8e$(F, q)$\u7684\u5747\u5300\u590d\u8fed\u90bb\u57df$W$, \u56e0\u4e3a$B$\u5c40\u90e8\u9053\u8def\u8fde\u901a, \u6240\u4ee5\u53ef\u4ee5\u53d6\u4e00\u4e2a\u66f4\u5c0f\u7684\u9053\u8def\u8fde\u901a\u90bb\u57df$U \\subseteq V \\cap W$. \u5219$p^\\uparrow$\u628a$p^{-1}(U)$\u7684\u6bcf\u4e2a\u9053\u8def\u5206\u652f(\u5f00\u96c6) \u540c\u80da\u5230$q^{-1}(U)$\u7684\u4e00\u4e2a\u5305\u542b$x$\u7684\u9053\u8def\u5206\u652f(\u5f00\u96c6), \u5219\u8be5\u9053\u8def\u5206\u652f\u662f$x$\u7684\u5747\u5300\u590d\u8fed\u90bb\u57df, \u4ece\u800c$p^\\uparrow$\u662f\u590d\u8fed\u6620\u5c04.<\/p>\n<p>5. \u8bbe$p: E \\to B$\u662f\u4e00\u4e2a\u590d\u8fed\u6620\u5c04, \u5e76\u4e14$E$\u548c$B$\u90fd\u662f\u9053\u8def\u8fde\u901a\u5e76\u4e14\u5c40\u90e8\u9053\u8def\u8fde\u901a\u7684\u7a7a\u95f4, \u5219\u4efb\u53d6\u4e00\u4e2a$B$\u4e2d\u7684\u5355\u8fde\u901a\u5f00\u5b50\u96c6$U$, \u542b\u5165\u6620\u5c04$i: $$ U \\hookrightarrow B$\u4e00\u5b9a\u53ef\u4ee5\u63d0\u5347, \u800c\u4e14\u5b83\u7684\u4efb\u4f55\u4e00\u4e2a\u63d0\u5347\u90fd\u662f\u5d4c\u5165.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u7531\u4e8e\u5f00\u5b50\u96c6$U$\u662f\u5355\u8fde\u901a\u7684, \u6545$i$\u6ee1\u8db3\u6620\u5c04\u63d0\u5347\u5b9a\u7406\u7684\u8981\u6c42, \u5b83\u5173\u4e8e\u590d\u8fed\u6620\u5c04$p$\u5b58\u5728\u63d0\u5347$i^\\uparrow$.<br \/>\n$\\\\$ \u4ee4$j: i(U) \\to U$\u4e3a$i$\u7684\u9006\u6620\u5c04. \u7531\u542b\u5165\u6620\u5c04\u7684\u5b9a\u4e49\u53ef\u77e5, $j$\u662f\u8fde\u7eed\u7684, \u5219$$j \\circ p : i^\\uparrow(U) \\to U$$\u4ea6\u4e3a\u4e00\u4e2a\u8fde\u7eed\u6620\u5c04, \u4e14$j \\circ p$\u4e3a$i^\\uparrow$\u7684\u9006\u6620\u5c04, \u6545\u542b\u5165\u6620\u5c04$i$\u7684\u4efb\u4f55\u4e00\u4e2a\u63d0\u5347$i^\\uparrow$\u90fd\u662f\u5d4c\u5165.<\/p>\n<p>6. \u4e66\u4e0aP253\u4ecb\u7ecd\u4e86\u4e00\u4e2a\u51e0\u4f55\u6784\u9020\u590d\u8fed\u7a7a\u95f4(\u7279\u522b\u662f\u6709\u9650\u590d\u8fed) \u7684\u65b9\u6cd5: \u6bd4\u5982\u8bf4, \u6211\u4eec\u53ef\u4ee5\u5728$B$\u91cc\u627e\u4e00\u4e2a\u6781\u5927\u7684\u5355\u8fde\u901a\u5f00\u5b50\u96c6$U$, \u5b83\u5173\u4e8e$p$\u7684\u539f\u50cf\u662f\u4e00\u4e9b\u4e92\u4e0d\u76f8\u4ea4\u5e76\u4e14\u540c\u80da\u4e8e$U$\u7684\u5b50\u96c6$V_\\lambda$\u7684\u5e76, \u518d\u628a\u8fd9\u4e9b$V_\\lambda$&#8221;\u62fc\u63a5&#8221; \u8d77\u6765, \u5c31\u53ef\u4ee5\u6784\u9020\u51fa$E$\u4e86.<\/p>\n<blockquote><p>PS: \u5df2\u77e5\u590d\u8fed\u6620\u5c04$p$, \u8fd8\u9700\u8981\u6784\u9020\u5168\u7a7a\u95f4$E$\u5417?<\/p><\/blockquote>\n<p>7. \u8bc1\u660e\u4efb\u53d6\u8fde\u7eed\u6620\u5c04$f: RP^2 \\to T^2$, $f$\u4e00\u5b9a\u96f6\u4f26.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> <em>(\u672c\u547d\u9898\u5176\u5b9e\u53ef\u4ee5\u4e0d\u901a\u8fc7\u6620\u5c04\u63d0\u5347\u5b9a\u7406\u8fdb\u884c\u8bc1\u660e.)<\/em> \u7531\u4e8e$$\\pi_1(RP^2, x_0) \\cong Z_2 = \\{ 0, 1 \\}, x_0 \\in RP^2, \\\\ \\pi_1(T^2, y_0) \\cong Z^2 = Z \\oplus Z, y_0 = f(x_0) \\in T^2,$$\u6545\u6211\u4eec\u4ec5\u9700\u8bc1\u660e\u4ece$Z_2$\u5230$Z^2$\u7684\u540c\u6001\u6620\u5c04\u53ea\u6709\u5e73\u51e1\u540c\u6001\u5373\u53ef.<br \/>\n$\\\\$ \u8bbe$f$\u4e3a\u4ece$Z_2$\u5230$Z^2$\u7684\u975e\u5e73\u51e1\u540c\u6001, s.t. $f(0) = (0, 0)$, $f(1) = (a, b)$, \u5176\u4e2d, $a \\ne $$ 0$\u6216\u8005$b \\ne 0$, \u5219$$(0, 0) = f(0) = f(1 + 1) = f(1) + f(1) = (2a, 2b) \\\\ \\Rightarrow a = 0, b = 0.$$\u8fd9\u4e0e\u5047\u8bbe\u77db\u76fe, \u6545\u4ece$Z_2$\u5230$Z^2$\u7684\u540c\u6001\u6620\u5c04\u53ea\u6709\u5e73\u51e1\u540c\u6001, \u4ece\u800c$f_\\pi: $$ \\pi_1(RP^2, x_0) $$ \\to \\pi_1(T^2, y_0)$\u4e3a\u5e73\u51e1\u540c\u6001, $f$\u4e00\u5b9a\u96f6\u4f26.<\/p>\n<p>8. \u8bbe$(E_i, p_i)$($i = 1, 2$) \u90fd\u662f$B$\u4e0a\u7684\u590d\u8fed\u7a7a\u95f4, \u82e5\u590d\u8fed\u7a7a\u95f4$E_1$\u4e0e$E_2$\u7b49\u4ef7, \u4e14\u8fde\u7eed\u6620\u5c04$h: E_1 \\to E^2$\u4e3a\u590d\u8fed\u7a7a\u95f4$E_1$\u4e0e$E_2$\u4e4b\u95f4\u7684\u540c\u6784, \u5219\u4ece\u76f4\u89c2\u4e0a\u7406\u89e3, \u82e5$y \\in $$ p^{-1}_1(b)$, \u5176\u4e2d, $b \\in B$, \u5219$h(y) \\in p^{-1}_2(b)$. \u4e24\u4e2a\u590d\u8fed\u7a7a\u95f4\u662f\u540c\u6784\u7684\u7684\u5fc5\u8981\u6761\u4ef6\u4e3a\u5b83\u4eec\u7684\u91cd\u6570\u662f\u76f8\u540c\u7684, \u9700\u8981\u6ce8\u610f\u7684\u662f, \u8be5\u6761\u4ef6\u5f80\u5f80\u4e0d\u662f\u5145\u5206\u7684.<\/p>\n<p>9. \u8bc1\u660e\u73af\u9762\u4e0a\u5b58\u5728\u4e24\u4e2a\u4e0d\u540c\u6784\u7684\u590d\u8fed\u7a7a\u95f4$p: E \\to T^2$, $q: F \\to $$ T^2$, \u4f7f\u5f97$E$\u4e0e$F$\u540c\u80da.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u6211\u4eec\u9700\u8981\u5206\u4e24\u6b65\u8fdb\u884c\u8bc1\u660e. \u9996\u5148\u8bc1\u660e: \u82e5$p: (E, e) \\to (B, b)$\u4e3a\u4e00\u4e2a$n$\u91cd\u590d\u8fed\u6620\u5c04, \u5219$p_\\pi(\\pi_1(E, e))$\u4e3a$\\pi_1(B, b)$\u7684\u5b50\u7fa4, \u4e14\u5176\u6307\u6570\u4e3a$n$.<br \/>\n$\\\\$ \u6211\u4eec\u9700\u8981\u5047\u8bbe$E$\u662f\u8fde\u901a\u7684. \u4ee4$G = \\pi_1(B, b)$, $H_e = p_\\pi(\\pi_1(E, e))$, $p^{-1}(b) = $$ \\{ b^\\uparrow_0, \\cdots, b^\\uparrow_{n &#8211; 1} \\}$. \u5bf9\u4e8e\u4efb\u610f$\\left \\langle \\gamma_1 \\right \\rangle, \\left \\langle \\gamma_2 \\right \\rangle \\in G$, \u6613\u77e5$\\left \\langle \\gamma_1 \\right \\rangle H_e $$ = \\left \\langle \\gamma_2 \\right \\rangle H_e$\u5f53\u4e14\u4ec5\u5f53$\\gamma_1^\\uparrow$\u4e0e$\\gamma_2^\\uparrow$\u5747\u4ee5$b^\\uparrow_0$\u4e3a\u8d77\u70b9, \u4ee5$b^\\uparrow_i$\u4e3a\u7ec8\u70b9. \u63a5\u4e0b\u6765, \u6211\u4eec\u4e5f\u5c06\u8bc1\u660e\u8fd9\u4e00\u70b9. \u8fd9\u6837\u4e00\u6765, \u4efb\u610f\u4e00\u6761\u4ee5$b^\\uparrow_0$\u4e3a\u8d77\u70b9, \u4ee5$b^\\uparrow_i$\u4e3a\u7ec8\u70b9\u7684\u9053\u8def$\\gamma^\\uparrow$, \u5747\u5bf9\u5e94\u4e8e\u4e00\u4e2a\u9053\u8def\u7c7b$\\left \\langle \\gamma \\right \\rangle = $$ \\left \\langle p \\circ \\gamma^\\uparrow \\right \\rangle $$ \\in G$. \u6545$H_e$\u7684\u5de6\u966a\u96c6\u4e0e$p^{-1}( $$ b)$\u4e00\u4e00\u5bf9\u5e94, \u4e14$[G : H_e] = n$.<br \/>\n$\\\\$ \u63a5\u4e0b\u6765\u8bc1\u660e\u4e0a\u8ff0\u7ed3\u8bba: \u6ce8\u610f\u5230$\\left \\langle \\gamma_1 \\right \\rangle H_e = \\left \\langle \\gamma_2 \\right \\rangle H_e$\u5f53\u4e14\u4ec5\u5f53$\\left \\langle \\gamma_1 \\right \\rangle^{-1} $$ \\left \\langle \\gamma_2 \\right \\rangle H_e = $$ H_e$, i.e. \u5f53\u4e14\u4ec5\u5f53$\\left \\langle \\gamma_1^{-1} \\gamma_2 \\right \\rangle \\in H_e$ i.e. \u5f53\u4e14\u4ec5\u5f53$\\gamma_1^{-1} \\gamma_2$\u7684\u63d0\u5347\u4e3a$E$\u4e2d\u7684\u95ed\u9053\u8def. \u6545\u82e5$\\gamma_1^\\uparrow$\u4e0e$\\gamma_2^\\uparrow$\u5747\u5177\u6709\u76f8\u540c\u7684\u7ec8\u70b9, \u5219$\\left \\langle \\gamma_1 \\right \\rangle H_e = $$ \\left \\langle \\gamma_2 \\right \\rangle H_e$. \u53cd\u8fc7\u6765, \u82e5$\\gamma_1^\\uparrow(1) \\ne<br \/>\n$$ \\gamma_2^\\uparrow(1)$, \u5219$\\gamma_1^{-1} \\gamma_2$\u7684\u63d0\u5347$(\\gamma_1^\\uparrow)^{-1} \\gamma_2^\\uparrow$\u4e0d\u4e3a$E$\u4e2d\u7684\u95ed\u9053\u8def, \u6545\u6b64\u65f6$\\left \\langle \\gamma_1 \\right \\rangle H_e \\ne \\left \\langle \\gamma_2 \\right \\rangle $$ H_e$.<br \/>\n$\\\\$ \u6700\u540e\u6784\u9020\u4e24\u4e2a\u4e0d\u540c\u6784\u7684\u590d\u8fed\u7a7a\u95f4$E, F$, \u4f7f\u5f97$E$\u4e0e$F$\u540c\u80da: \u4ee4$E = T^2 $$ = S^1 \\times S^1$,$$p_E: S^1 \\times S^1 \\to S^1 \\times S^1, \\\\ (z, w) \\mapsto (z^2, w^2).$$\u5728\u590d\u8fed\u6620\u5c04$p_E$\u4e0b, $S^1 \\times S^1$\u7684\u57fa\u672c\u7fa4\u7684\u50cf\u4e3a$Z \\times Z$\u7684\u5b50\u7fa4$2Z \\times 2Z$. \u518d\u4ee4$F = $$ T^2 = S^1 \\times S^1$,$$p_F: S^1 \\times S^1 \\to S^1 \\times S^1, \\\\ (z, w) \\mapsto (z^3, w^3).$$\u5728\u590d\u8fed\u6620\u5c04$p_F$\u4e0b, $S^1 \\times S^1$\u7684\u57fa\u672c\u7fa4\u7684\u50cf\u4e3a$Z \\times Z$\u7684\u5b50\u7fa4$3Z \\times 3Z$. \u7531\u4e8e$2Z $$ \\times 2Z$\u4e0e$3Z \\times 3Z$\u5e76\u4e0d\u5171\u8f6d, \u6545\u4f7f\u5f97$2Z \\times 2Z$\u4e0e$3Z \\times 3Z$\u540c\u6784\u7684\u57fa\u70b9\u5e76\u4e0d\u5b58\u5728, \u4ece\u800c$p_E$\u4e0e$p_F$\u51b3\u5b9a\u7684$\\pi_1(B, b)$\u7684\u5b50\u7fa4\u5171\u8f6d\u7c7b\u4e0d\u76f8\u540c. \u7531\u4e66\u4e0aP252\u7684\u547d\u98985.5.2\u53ef\u77e5, $(E, p_E)$\u4e0e$(F, p_F)$\u4e0d\u7b49\u4ef7, \u4f46\u4e8c\u8005\u663e\u7136\u662f\u540c\u80da\u7684. \u7efc\u4e0a\u6240\u8ff0, \u547d\u9898\u5f97\u8bc1.<\/p>\n<p>10. \u8bbe$p: X^\\uparrow \\to X$\u548c$q: Y^\\uparrow \\to Y$\u90fd\u662f\u6cdb\u590d\u8fed\u6620\u5c04, \u5e76\u4e14$X$\u548c$Y$\u540c\u4f26\u7b49\u4ef7. \u8bc1\u660e$X^\\uparrow$\u53ef\u7f29\u5f53\u4e14\u4ec5\u5f53$Y^\\uparrow$\u53ef\u7f29.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u7531\u4e8e$X$\u548c$Y$\u540c\u4f26\u7b49\u4ef7, \u6545\u5b58\u5728\u8fde\u7eed\u6620\u5c04$f: X \\to Y$, $g: Y \\to X$, s.t. $f \\circ g $$ \\simeq id_Y$, $g \\circ f \\simeq id_X$. \u53c8$X^\\uparrow, Y^\\uparrow$\u5747\u662f\u5355\u8fde\u901a\u7684, \u6211\u4eec\u53ef\u5f97$f \\circ p$\u7684\u63d0\u5347$f^\\uparrow : X^\\uparrow $$ \\to Y^\\uparrow$, \u4e0e$g \\circ q$\u7684\u63d0\u5347$g^\\uparrow : Y^\\uparrow \\to X^\\uparrow$. \u63a5\u4e0b\u6765\u6211\u4eec\u4ec5\u9700\u8bc1\u660e\u5b58\u5728\u8fde\u7eed\u6620\u5c04$\\phi : $$ Y^\\uparrow \\to X^\\uparrow$, s.t. $\\phi \\circ f^\\uparrow \\simeq id_X$\u5373\u53ef.<br \/>\n$\\\\$ \u56e0\u4e3a$g \\circ f \\simeq id_X$, \u6545\u53ef\u53d6\u5b9a\u4ece$g \\circ f \\circ p$\u5230$p$\u7684\u4f26\u79fb$H$, \u6839\u636e\u540c\u4f26\u63d0\u5347\u6027\u8d28, \u6211\u4eec\u53ef\u5f97\u4f26\u79fb$H$\u7684\u63d0\u5347$H^\\uparrow : X^\\uparrow \\times $$ [0, 1] \\to X^\\uparrow$, s.t. $$H^\\uparrow_0 = g^\\uparrow \\circ f^\\uparrow : X^\\uparrow \\times \\{ 0 \\} \\to X^\\uparrow.$$\u7531\u4e8e\u4e00\u4e2a\u6cdb\u590d\u8fed\u7a7a\u95f4\u4ea6\u4e3a\u4e00\u4e2aGalois\u590d\u8fed\u7a7a\u95f4(\u6216\u6b63\u89c4\u590d\u8fed\u7a7a\u95f4(Normal Covering Space)), \u6545\u5b58\u5728\u4e00\u4e2a\u590d\u8fed\u53d8\u6362$\\phi : X^\\uparrow \\to X^\\uparrow$, s.t. $H^\\uparrow_1 = \\phi : X^\\uparrow \\times $$ \\{ 1 \\} \\to X^\\uparrow$<em>(\u7531\u4e8e$g \\circ f \\circ p$\u4e0e$p$\u5728$X^\\uparrow$\u7684\u57fa\u70b9\u5904\u7684\u53d6\u503c\u4e0d\u4e00\u5b9a\u76f8\u540c, \u65e0\u6cd5\u4f7f\u7528\u4e66\u4e0aP237\u7684\u547d\u98985.3.1, \u6545$\\phi$\u4e0d\u4e00\u5b9a\u4e3a$p$\u7684\u63d0\u5347; \u4e0e\u6b64\u540c\u65f6, \u82e5$\\phi$\u4e3a\u4e00\u4e2a\u590d\u8fed\u53d8\u6362, \u5219\u7531\u590d\u8fed\u53d8\u6362\u7684\u5b9a\u4e49\u53ef\u77e5$H^\\uparrow_1$\u4e3a$p$\u7684\u63d0\u5347, \u7136\u800c\u8fd9\u662f\u5f85\u8bc1\u7684, \u4ea6\u4e3a\u6211\u6240\u7ffb\u9605\u7684\u6240\u6709\u76f8\u5173\u7b54\u6848\u4e2d\u5747\u5b58\u5728\u7684\u4e00\u4e2a\u95ee\u9898\u2026\u2026)<\/em>, \u4ece\u800c$$g^\\uparrow \\circ f^\\uparrow \\simeq \\phi \\Rightarrow \\phi^{-1} \\circ g^\\uparrow \\circ f^\\uparrow \\simeq id_{X^\\uparrow}.$$\u7c7b\u4f3c\u5730, \u5b58\u5728\u4e00\u4e2a$Y^\\uparrow$\u4e0a\u7684\u590d\u8fed\u53d8\u6362$\\varphi : X^\\uparrow \\to X^\\uparrow$, s.t. $f^\\uparrow \\circ g^\\uparrow \\circ $$ \\varphi^{-1} $$ \\simeq id_{Y^\\uparrow}$. \u7531Hatcher&#8217;s Algebraic Topology\u7b2c0\u7ae0\u7684Exercise 11\u53ef\u77e5, $X^\\uparrow$\u4e0e$Y^\\uparrow$\u540c\u4f26\u7b49\u4ef7.<\/p>\n<blockquote><p>\nShow that $f: X \\to Y$ is a homotopy equivalence if there exist maps $g, h : Y \\to X$ such that $fg \\simeq 1$ and $hf \\simeq 1$. More generally, show that $f$ is a homotopy equivalence if $fg$ and $hf$ are homotopy equivalences.\n<\/p><\/blockquote>\n<p>\u7efc\u4e0a\u6240\u8ff0, \u547d\u9898\u5f97\u8bc1.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u7ec8\u4e8e\u8fce\u6765\u4e86\u56fd\u5e86\u5047\u671f, \u5927\u6982\u6709\u4e00\u5468\u7684\u65f6\u95f4\u662f&#8221;\u8d4b\u95f2\u5728\u5bb6&#8221;\u7684, \u8bf4\u5b9e\u8bdd, \u8fd8\u662f\u86ee\u723d\u7684~ \u76ee\u524d &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/10\/03\/lifting_criterion_mark\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u6620\u5c04\u63d0\u5347\u5b9a\u7406\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\/2214"}],"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=2214"}],"version-history":[{"count":160,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/2214\/revisions"}],"predecessor-version":[{"id":3622,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/2214\/revisions\/3622"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2214"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2214"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2214"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}