{"id":838,"date":"2021-07-20T01:04:32","date_gmt":"2021-07-19T17:04:32","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=838"},"modified":"2023-05-04T17:01:16","modified_gmt":"2023-05-04T09:01:16","slug":"fudiekongjianjiqifudieyinlidanshequn","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=838","title":{"rendered":"\u590d\u53e0\u7a7a\u95f4\u53ca\u5176\u590d\u53e0\u5f15\u7406\u3001\u5355\u5c04\u7fa4"},"content":{"rendered":"

\u8be5\u8bfb\u4e66\u7b14\u8bb0\u6765\u81eaTopology and Groups<\/a>.
\n1. \u590d\u53e0\u6620\u5c04\u7684\u4e24\u4e2a\u4f8b\u5b50<\/a><\/span>\n\n

\u4f8b\u5b50 1<\/span>.<\/span> \u5047\u8bbe$S=\\mathbb{C}\\setminus\\left\\{ 0 \\right\\}$, \u8003\u5bdf\u8fde\u7ebf\u6620\u5c04$p: S\\to S$, $p(z)=z^2$. \u5b83\u662f\u4e00\u4e2a$2$\u91cd\u6ee1\u5c04, \u800c\u4e14\u5c40\u90e8\u4e0a\u6709\u9006\u6620\u5c04.<\/p>\n

\u4e8b\u5b9e\u4e0a, \u5982\u679c\u6211\u4eec\u5272\u6389\u534a\u76f4\u7ebf$B^-:=\\left\\{ z\\in S:\\mathrm{Im}(z)=0, \\mathrm{Re}(z)<0 \\right\\}$, \u5219\u53ef\u5b9a\u4e49\n \\[\n q_{\\pm}: \\mathbb{C}\\setminus B^-\\to \\mathbb{C}\\setminus\\left\\{ 0 \\right\\},\n \\]\n \u4f7f\u5f97$p(q_{\\pm}(z))=z$. \u8fd9\u91cc, $q_{-}=-q_{+}$.\n\n \u5b8c\u5168\u7c7b\u4f3c\u5730, \u6211\u4eec\u4e5f\u53ef\u4ee5\u5272\u6389$B^+:=\\left\\{ z\\in S:\\mathrm{Im}(z)=0, \\mathrm{Re}(z)>0 \\right\\}$, \u8fdb\u800c\u5f97\u5230\u4e24\u4e2a\u6620\u5c04
\n \\[
\n \\bar{q}_{\\pm}: \\mathbb{C}\\setminus B^+\\to \\mathbb{C}\\setminus\\left\\{ 0 \\right\\},
\n \\]
\n \u4f7f\u5f97$p(\\bar{q}_{\\pm}(z))=z$.
\n<\/div>
\n
\n\u4e0b\u9762\u6765\u770b\u4e00\u4e2a\u65e0\u7a77\u91cd\u7684\u4f8b\u5b50.
\n

\u4f8b\u5b50 2<\/span>.<\/span> \u4ee4$f: \\mathbb{C}\\to \\mathbb{C}\\setminus\\left\\{ 0 \\right\\}$, $f(z)=e^{iz}$. \u8be5\u6620\u5c04\u662f\u4e00\u4e2a\u65e0\u7a77\u91cd\u6ee1\u5c04(\u4f8b\u5982$f^{-1}(1)=\\left\\{ 2\\pi n: n\\in \\mathbb{Z} \\right\\}$. \u5728$\\mathbb{C}\\perp B^+$\u6216\u8005$\\mathbb{C}\\setminus B^-$\u4e0a, \u6211\u4eec\u6709\u65e0\u7a77\u591a\u4e2a\u5c40\u90e8\u9006\u6620\u5c04: $q_n(z)=2\\pi n-i\\ln z: \\mathbb{C}\\setminus B\\to \\mathbb{C}$, \u6ee1\u8db3$q_n(1)=2\\pi n$, $p(q_n(z))=z$.
\n<\/div>
\n
\u6ce8\u8bb0 1<\/span>.<\/span> \u5728\u4e0a\u8ff0\u4e24\u4e2a\u4f8b\u5b50\u4e2d, \u539f\u50cf\u96c6$\\mathbb{C}\\setminus\\left\\{ 0 \\right\\}$\u4ee5\u53ca$\\mathbb{C}$\u53ef\u4ee5\u662f\u975e\u5355\u8fde\u901a\u7684, \u4e5f\u53ef\u4ee5\u662f\u5355\u8fde\u901a\u7684. \u4f46\u662f\u9006\u6620\u5c04\u7684\u539f\u50cf\u96c6$\\mathbb{C}\\setminus B^{\\pm}$\u90fd\u662f\u5355\u8fde\u901a\u7684.
\n<\/div>
\n2. \u590d\u53e0\u6620\u5c04\u4e0e\u590d\u53e0\u7a7a\u95f4<\/a><\/span>\n\n
\u5b9a\u4e49 1<\/span>.<\/span> \u5047\u8bbe$p:Y\\to X$\u662f\u4e00\u4e2a\u8fde\u7eed\u6620\u5c04.
\n
  1. \u5b50\u96c6$U\\subset X$\u79f0\u4e3a\u4e00\u4e2a\u57fa\u7840\u90bb\u57df<\/span>(elementary neighbourhood), \u5982\u679c\u5b58\u5728\u79bb\u6563\u96c6$F$, \u4ee5\u53ca\u540c\u80da\u6620\u5c04(homeomorphism)$h:p^{-1}(U)\\to U\\times F$, \u4f7f\u5f97$\\mathrm{p}_1\\circ h=p|_{p^{-1}(U)}$, \u5176\u4e2d$p_1:U\\times F\\to U$\u662f\u5230\u7b2c\u4e00\u4e2a\u56e0\u5b50\u7684\u6295\u5c04;
    \n <\/li>
  2. \u6211\u4eec\u79f0$p$\u4e3a\u4e00\u4e2a\u8986\u76d6\u6620\u5c04<\/span>(covering map), \u5982\u679c$X$\u88ab\u4e00\u4e9b\u57fa\u7840\u90bb\u57df\u8986\u76d6, \u5373\u5bf9\u4efb\u610f\u7684$x\\in X$, \u90fd\u5b58\u5728\u4e00\u4e2a\u57fa\u7840\u90bb\u57df$U$, \u4f7f\u5f97$x\\in U$; \u6b64\u65f6, \u79f0$Y$\u4e3a$X$\u7684\u4e00\u4e2a\u590d\u53e0\u7a7a\u95f4<\/span>(covering space);
    \n <\/li>
  3. \u6211\u4eec\u79f0$V\\subset Y$\u4e3a\u4e00\u4e2a\u57fa\u7840\u5207\u7247<\/span>(elementary sheet), \u5982\u679c$V$\u662f\u9053\u8def\u8fde\u901a\u7684\u800c\u4e14$p(V)$\u662f\u4e00\u4e2a\u57fa\u7840\u90bb\u57df.
    \n <\/li><\/ol><\/div>
    \n
    \u6ce8\u8bb0 2<\/span>.<\/span> \u4e8b\u5b9e\u4e0a, \u590d\u53e0\u7a7a\u95f4$Y$\u4e5f\u53ef\u89c6\u4e3a\u7ea4\u7ef4\u4e1b$p:Y\\to X$, \u4f7f\u5f97\u7ea4\u7ef4\u4e3a\u79bb\u6563\u96c6\u5408$F$. \u540c\u80da\u6620\u5c04$h:p^{-1}(U)\\to U\\times F$\u7ed9\u51fa\u4e86\u5176\u5c40\u90e8\u5e73\u51e1\u5316.
    \n<\/div>
    \n\u4e00\u822c\u5730, $p_n(z)=z^n$\u5c06\u7ed9\u51fa\u4e00\u6761\u7ed5$n$\u5708\u7684\u87ba\u65cb\u7ebf. \u4e0b\u56fe\u7ed9\u51fa\u4e86$n=3$\u7684\u60c5\u5f62:
    \n    \"cov-01-04.png\"<\/a>
    \n3. \u5355\u5c04\u7fa4\u4e0e\u9053\u8def\u63d0\u5347\u5f15\u7406    <\/a><\/span>\n\n
    \u4f8b\u5b50 3<\/span>.<\/span> \u8003\u5bdf\u5355\u4f4d\u5706\u5468\u4e0a\u7684\u4e09\u91cd\u590d\u53e0\u6620\u5c04: $p(e^{i\\theta})=e^{3i\\theta}$, \u6bcf\u4e2a$s\\in S^1$\u90fd\u6709$p^{-1}(s)$\u4e3a\u4e09\u4e2a\u70b9. \u5047\u8bbe$p^{-1}(1)=\\left\\{ a,b,c \\right\\}$, \u8003\u5bdf\u6cbf\u7740$S^1$\u4e0a\u4ece$1\\in S^1$\u51fa\u53d1\u7684\u73af\u8def(loop), \u6211\u4eec\u770b\u5230\u539f\u50cf\u96c6\u4ece$a$\u79fb\u52a8\u5230$b$, \u7136\u540e$b$\u79fb\u52a8\u5230$c$, \u6700\u540e$c$\u79fb\u52a8\u5230$a$. \u56e0\u6b64\u5f53\u6211\u4eec\u91cd\u590d\u73af\u8def\u65f6, \u6211\u4eec\u5728\u5faa\u73af\u7f6e\u6362$(abc)\\in S_3$, \u5373\u5f97\u5230\u7fa4\u540c\u6001$\\pi_1(S^1,1)\\to \\mathrm{Perm}(p^{-1}(1))=\\mathrm{Perm}(a,b,c)=S_3$, \u5c06\u73af\u8def\u6620\u5c04\u5230$(abc)$. \u7531\u4e8e$(abc)\\in S_3$\u4e0d\u662f\u5e73\u51e1\u7684(\u975e\u6052\u7b49\u7f6e\u6362), \u56e0\u6b64\u73af\u8def\u5728$\\pi_1(S^1,1)$\u4e2d\u4e5f\u4e0d\u662f\u5e73\u51e1\u7684.
    \n<\/div>
    \n\u66f4\u4e00\u822c\u5730, \u5355\u5c04\u7fa4<\/span>\u662f$\\pi_1(X,x)$\u5728$p^{-1}(x)$\u4e0a\u4e00\u4e2a\u4f5c\u7528, \u5373\u7fa4\u540c\u6001$\\pi_1(X,x)\\to \\mathrm{Perm}(p^{-1}(x))$. \u8fd9\u7ed9\u51fa\u4e86\u4e00\u4e2a\u786e\u5b9a\u57fa\u672c\u7fa4$\\pi_1(X,x)$\u4e2d\u975e\u5e73\u51e1\u73af\u8def\u7684\u65b9\u6cd5. \u6b64\u5916, \u5b58\u5728\u4e00\u4e2a\u590d\u53e0\u7a7a\u95f4(\u79f0\u4e3a\u4e07\u6709\u590d\u53e0\u7a7a\u95f4), \u4f7f\u5f97$\\pi_1$\u4e2d\u6240\u6709\u5143\u7d20\u7684\u4f5c\u7528\u90fd\u662f\u975e\u5e73\u51e1\u7684.<\/p>\n

    \u5728\u4e0a\u8ff0\u4f8b\u5b50\u4e2d, \u6211\u4eec\u6ca1\u6709\u8bf4\u6e05\u695a\u4ec0\u4e48\u53eb\u505a\u6cbf\u7740\u73af\u8def, \u539f\u50cf\u96c6\u4ece$a$\u79fb\u52a8\u5230$b$, \u8fd9\u5c06\u7528\u9053\u8def\u63d0\u5347\u5f15\u7406(path-lifting lemma)\u6765\u4e25\u683c\u5316; \u4e5f\u6ca1\u8bf4\u6e05\u695a\u5355\u5c04\u7fa4\u7684\u5b9a\u4e49\u4e2d\u4e3a\u4ec0\u4e48\u53ea\u4f9d\u8d56\u4e8e\u73af\u8def\u7684\u540c\u4f26\u7c7b, \u8fd9\u5c06\u7528\u540c\u4f26\u63d0\u5347\u5f15\u7406(homotopy-lifting lemma)\u6765\u8bf4\u660e.
    \n

    \u5b9a\u7406 2<\/span> (\u9053\u8def\u63d0\u5347\u5f15\u7406<\/span>).<\/span> \u5047\u8bbe$p:Y\\to X$\u662f\u4e00\u4e2a\u590d\u53e0\u6620\u5c04, $\\delta:[0,1]\\to X$\u662f\u4e00\u6761\u9053\u8def, $\\delta(0)=x$, \u5e76\u4ee4$y\\in p^{-1}(x)$. \u5219\u5b58\u5728\u552f\u4e00\u7684\u9053\u8def$\\gamma:[0,1]\\to Y$, \u6ee1\u8db3$p\\circ\\gamma=\\delta$\u4ee5\u53ca$\\gamma(0)=y$. \u6211\u4eec\u79f0$\\gamma$\u662f$\\delta$\u7684\u4e00\u4e2a\u63d0\u5347<\/span>.
    \n<\/div>
    \n
    \u8bc1\u660e<\/span>.<\/span> \u5047\u8bbe$\\mathcal{U}$\u662f\u6620\u5c04$p$\u4e0b$X$\u7684\u4e00\u4e2a\u7531\u57fa\u7840\u90bb\u57df\u6784\u6210\u7684\u5f00\u8986\u76d6\u7684\u96c6\u5408. \u5219$\\left\\{ \\delta^{-1}(U): U\\in \\mathcal{U} \\right\\}$\u662f$[0,1]$\u7684\u4e00\u4e2a\u5f00\u8986\u76d6, \u56e0\u6b64\u6709\u6709\u9650\u5b50\u8986\u76d6. \u4ece\u800c\u5b58\u5728\u6709\u9650\u5206\u5272$0=t_0\\leq t_1\\leq \\cdots t_n=1$, \u4f7f\u5f97$\\delta_k:=\\delta|_{t_k,t_{k+1}}\\subset U_k$, \u5bf9\u67d0\u4e2a$U_k\\subset \\mathcal{U}$. <\/p>\n

    \u6211\u4eec\u5c06\u5229\u7528\u5bf9$k$\u7684\u5f52\u7eb3\u6cd5\u6784\u9020$\\gamma$.<\/p>\n

    \u5bf9$k=0$: \u7531\u4e8e\u8981\u6c42$\\gamma(0)=y$. \u6ce8\u610f\u5230$p$\u662f\u4e00\u4e2a\u8986\u76d6\u6620\u5c04, \u56e0\u6b64\u5b58\u5728\u5c40\u90e8\u9006\u6620\u5c04$q_0:U_0\\to Y$ \u4f7f\u5f97$q_0(x)=y$, $p\\circ q_0=\\mathrm{id}|_{U_0}$, \u5219$\\gamma=q_0\\circ \\delta_0$\u6ee1\u8db3\u8981\u6c42: $\\gamma(0)=q_0(\\delta_0(0))=q_0(x)=y$; $p\\circ \\gamma=p\\circ q_0\\circ \\delta_0=\\mathrm{id}|_{U_0}\\circ \\delta_0=\\delta$.<\/p>\n

    \u73b0\u5728\u5047\u8bbe\u6211\u4eec\u6784\u9020\u4e86$\\gamma_0,\\ldots,\\gamma_{k-1}$, \u4e0b\u9762\u6765\u6784\u9020$\\gamma_k:[t_k,t_{k+1}]\\to Y$. \u4e3a\u4e86\u4f7f\u5f97$\\gamma$\u8fde\u7eed, \u6211\u4eec\u8981\u6c42$\\gamma_k(t_k)=\\gamma_{k-1}(t_k)$. \u6ce8\u610f\u5230\u5b58\u5728$q_k:U_k\\to Y$, \u4f7f\u5f97$q_k(\\delta(t_k))=\\gamma_{k-1}(t_k)$, \u800c\u4e14$p\\circ q_k=\\mathrm{id}|_{U_k}$. \u56e0\u6b64, \u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49$\\gamma_k=q_k\\circ \\delta_k$. \u7531\u6b64\u6211\u4eec\u5c06$\\delta$\u7684\u9053\u8def\u63d0\u5347\u6620\u5c04$\\gamma$\u8fde\u7eed\u5ef6\u62d3\u5230$[t_k,t_{k+1}]$.<\/p>\n

    \u73b0\u5728, \u6211\u4eec\u5bf9$t\\in[t_k,t_{k+1}]$, \u5b9a\u4e49$\\gamma(t)=\\gamma_k(t)$. \u5219$\\gamma$\u8fde\u7eed, \u56e0\u4e3a\u5728\u6bcf\u4e2a\u5206\u6bb5\u5b9a\u4e49\u7684\u51fd\u6570\u7684\u91cd\u5408\u7aef\u70b9\u5904\u662f\u8fde\u7eed\u7684. \u800c\u4e14\u7531\u4e8e\u5728$[t_k,t_{k+1}$\u4e0a$p\\circ \\gamma=p\\circ q_k\\circ\\delta_k=\\mathrm{id}|_{U_k}\\circ\\delta_k=\\delta_k$, \u6545$\\gamma$\u8fd8\u662f$\\delta$\u7684\u4e00\u4e2a\u63d0\u5347.
    \n<\/div>
    \n\u4e3a\u4e86\u8bc1\u660e\u552f\u4e00\u6027, \u6211\u4eec\u7ed9\u51fa\u5982\u4e0b\u66f4\u4e00\u822c\u7684\u5f15\u7406:
    \n

    \u5f15\u7406 3<\/span>.<\/span> \u5047\u8bbe$p:Y\\to X$\u662f\u4e00\u4e2a\u8986\u76d6\u7a7a\u95f4. \u4ee4$T$\u662f\u4e00\u4e2a\u8fde\u901a\u7a7a\u95f4, $F:T\\to X$\u662f\u4e00\u4e2a\u8fde\u7eed\u6620\u5c04. \u5219\u4e24\u4e2a\u63d0\u5347\u6620\u5c04$\\tilde{F}_1,\\tilde{F}_2:T\\to Y$\u76f8\u7b49, \u5373\u5bf9\u4efb\u610f\u7684$t\\in T$, \u90fd\u6709$\\tilde{F}_1(t)=\\tilde{F}_2(T)$, \u5f53\u4e14\u4ec5\u5f53\u5b58\u5728\u4e00\u4e2a$t_0\\in T$, \u4f7f\u5f97$\\tilde{F}_1(t_0)=\\tilde{F}_2(t_0)$.
    \n<\/div>
    \n\u56de\u5fc6, $\\tilde{F}$\u79f0\u4e3a$F$\u7684\u4e00\u4e2a\u63d0\u5347, \u5982\u679c$p\\circ\\tilde{F}=F$.<\/p>\n
    \u8bc1\u660e<\/span> (\u5f15\u7406\u7684\u8bc1\u660e<\/span>).<\/span> \u4ee4$S=\\left\\{ t\\in T: \\tilde{F}_1(t)=\\tilde{F}_2(t) \\right\\}$. \u6211\u4eec\u5e0c\u671b\u8bc1\u660e$S$\u65e2\u5f00\u53c8\u95ed. \u7531\u4e8e$T$\u8fde\u901a, \u6545\u8fd9\u8868\u660e$S=\\emptyset$\u6216\u8005$S=T$, \u4f46$t_0\\in S$, \u56e0\u6b64$S=T$.<\/p>\n

    \u8003\u5bdf$t\\in T$, \u4ee4$x=F(t)$. \u7531\u4e8e$p:Y\\to X$\u662f\u4e00\u4e2a\u590d\u53e0\u7a7a\u95f4, \u56e0\u6b64\u5b58\u5728\u57fa\u7840\u90bb\u57df$x\\in W\\subset X$\u4ee5\u53ca\u57fa\u7840\u5207\u7247$V_1,V_2$, \u4f7f\u5f97$p(V_1)=W=p(V_2)$, \u800c\u4e14$\\tilde{F}_1(t)\\in V_1$, $\\tilde{F}_2(t)\\in V_2$.<\/p>\n

    \u7531\u4e8e$\\tilde{F}_1$, $\\tilde{F}_2$\u662f\u8fde\u7eed\u7684, \u6545\u5b58\u5728\u5f00\u96c6$U_1,U_2\\in T$\u4f7f\u5f97$t\\in U_i$, $\\tilde{F}_i(U_i)\\subset V_i$, $i=1,2$. \u4ee4$U=U_1\\cap U_2$.
    \n    \"\u63d0\u5347\u6620\u5c04\u7684\u552f\u4e00\u6027.jpeg\"<\/a>
    \n
    \n\u6211\u4eec\u5e0c\u671b\u8bc1\u660e$T\\setminus S$\u4e3a\u5f00\u96c6. \u4e8b\u5b9e\u4e0a, \u5047\u8bbe$t\\in T\\setminus S$, \u5219\u6211\u4eec\u5c06\u8bc1\u660e$U\\subset T\\setminus S$. \u8fd9\u662f\u56e0\u4e3a, $t\\in T\\setminus S$, \u6545$\\tilde{F}_1(t)\\neq \\tilde{F}_2(t)$, \u4ece\u800c$V_1\\cap V_2=\\emptyset$. \u6545\u5bf9\u6240\u6709\u7684$t’\\in U$, \u6211\u4eec\u90fd\u6709$\\tilde{F}_1(t’)\\neq \\tilde{F}_2(t’)$(\u8fd9\u662f\u56e0\u4e3a$t’\\in U\\implies \\tilde{F}_1(t’)\\in V_1,\\tilde{F}_2(t’)\\in V_2$, \u800c$V_1\\cap V_2=\\emptyset$).<\/p>\n

    \u6211\u4eec\u8fd8\u9700\u8981\u8bc1\u660e$S$\u4e3a\u5f00\u96c6. \u4e8b\u5b9e\u4e0a, \u5047\u8bbe$t\\in S$, \u5219$\\tilde{F}_1(t)=\\tilde{F}_2(t)$, \u56e0\u6b64$V_1=V_2=:V$. \u7531\u4e8e\u5b58\u5728$p$\u7684\u5c40\u90e8\u9006\u6620\u5c04$q:W\\to V$, \u5373$p\\circ q=\\mathrm{id}_W$, $q\\circ p=\\mathrm{id}_V$, \u6211\u4eec\u77e5\u9053\u5982\u679c$t’\\in U$, \u5219$\\tilde{F}_1(t’)\\in V$, $\\tilde{F}_2(t’)\\in V$, \u4ece\u800c$q(F(t’))=q(p(\\tilde{F}_1(t’)))=\\tilde{F}_1(t’)$, \u7c7b\u4f3c\u5730$q(F(t’))=\\tilde{F}_2(t’)$, \u56e0\u6b64$\\tilde{F}_1(t’)=\\tilde{F}_2(t’)$, \u5373$U\\subset S$.
    \n<\/div>
    \n4. \u5355\u5c04\u7fa4    <\/a><\/span>\n\n\u5728\u524d\u9762\u6211\u4eec\u8bc1\u660e\u4e86\u590d\u53e0\u7a7a\u95f4\u4e2d\u7684\u9053\u8def\u63d0\u5347\u5f15\u7406, \u5373: \u5bf9\u4efb\u610f\u7684\u4e00\u4e2a\u590d\u53e0\u7a7a\u95f4$p:X\\to Y$, \u4ee5\u53ca\u5176\u4e2d\u4efb\u610f\u4e00\u6761\u9053\u8def$\\gamma:[0, I]\\to X$, \u82e5$y\\in p^{-1}(\\gamma(0))$, \u5219\u5b58\u5728\u9053\u8def$\\gamma$\u552f\u4e00\u7684\u63d0\u5347$\\tilde{\\gamma}\\in Y$, \u4f7f\u5f97$p\\circ \\tilde{\\gamma}=\\gamma$\u4ee5\u53ca$\\tilde{\\gamma}(0)=y$.<\/p>\n

    \u95ee\u9898\u662f: \u5982\u679c\u6211\u4eec\u5047\u8bbe$\\gamma$\u662f\u4e00\u4e2a\u73af\u8def(loop), \u5219\u4e00\u822c\u800c\u8a00, \u6211\u4eec\u4e0d\u80fd\u4fdd\u8bc1$\\tilde{\\gamma}$\u4e5f\u662f\u4e00\u4e2a\u73af\u8def. \u4f8b\u5982, $p:S^1\\to S^1$, $p(e^{i\\theta})=e^{2i\\theta}$, \u662f\u4e8c\u91cd\u590d\u53e0\u6620\u5c04, \u5219$\\tilde\\gamma(t)=e^{it\\pi}$\u5c31\u662f\u73af\u8def$e^{2\\pi it}$, $t\\in[0,1]$, \u7684\u63d0\u5347. \u8fd9\u5c31\u5f15\u51fa\u5355\u5c04\u7fa4\u7684\u6982\u5ff5.
    \n

    \u5b9a\u4e49 4<\/span>.<\/span> \u5047\u8bbe$p:Y\\to X$\u4e3a\u4e00\u4e2a\u590d\u53e0\u7a7a\u95f4, $\\gamma:[0, 1]\\to X$\u662f\u4e00\u4e2a\u4ee5$x$\u70b9\u4e3a\u57fa\u70b9\u7684\u73af\u8def, \u5373$\\gamma(0)=x=\\gamma(1)$. \u6211\u4eec\u5c06\u7ed5\u7740$\\gamma$\u7684\u5355\u5c04\u7fa4<\/span>\u5b9a\u4e49\u4e3a\u7f6e\u6362$\\sigma_\\gamma:p^{-1}(x)\\to p^{-1}(x)$, $\\sigma_\\gamma(y)=\\tilde{\\gamma}(1)$, \u5176\u4e2d$\\tilde{\\gamma}$\u4e3a\u4ece$y\\in p^{-1}(x)$\u51fa\u53d1\u7684$\\gamma$\u7684\u63d0\u5347.
    \n<\/div>
    \n\u4e00\u4e2a\u57fa\u672c\u4e8b\u5b9e\u662f, \u5355\u5c04\u7fa4\u4e0e\u540c\u4f26\u9053\u8def\u7684\u9009\u62e9\u65e0\u5173, \u5373
    \n
    \u5f15\u7406 5<\/span>.<\/span> \u5047\u8bbe$p:Y\\to X$\u4e3a\u590d\u53e0\u7a7a\u95f4, \u82e5$\\gamma_1\\sim\\gamma_2$\u662f\u4ee5$x\\in X$\u4e3a\u57fa\u70b9\u7684\u4e24\u4e2a\u540c\u4f26\u73af\u8def, \u5219\u5355\u5c04\u7fa4$\\sigma_{\\gamma_1}=\\sigma_{\\gamma_2}$.
    \n<\/div>
    \n\u4e3a\u4e86\u8bc1\u660e\u8be5\u5f15\u7406, \u6211\u4eec\u8bc1\u660e\u5982\u4e0b\u66f4\u52a0\u4e00\u822c\u7684\u540c\u4f26\u63d0\u5347\u5f15\u7406.
    \n
    \u5f15\u7406 6<\/span> (\u540c\u4f26\u63d0\u5347\u5f15\u7406<\/span>).<\/span> \u5047\u8bbe$p:Y\\to X$\u662f\u4e00\u4e2a\u590d\u53e0\u7a7a\u95f4, $\\gamma_s$\u662f$X$\u4e2d\u56fa\u5b9a\u7aef\u70b9\u7684\u540c\u4f26\u9053\u8def, \u5373$\\gamma_s(0),\\gamma_{s}(1)$\u90fd\u662f$X$\u4e2d\u4e0e$s$\u65e0\u5173\u7684\u70b9. \u5219, \u5bf9\u7ed9\u5b9a\u7684$\\gamma_0$\u7684\u63d0\u5347$\\tilde{\\gamma}_0$, \u5b58\u5728\u56fa\u5b9a\u7aef\u70b9\u7684\u540c\u4f26\u63d0\u5347$\\tilde{\\gamma}_s$, \u5373$\\tilde{\\gamma}_s$\u662f$\\tilde{\\gamma}_0$\u7684\u4e00\u4e2a\u540c\u4f26, \u4f7f\u5f97
    \n \\[
    \n p\\circ\\tilde{\\gamma}_s=\\gamma_s,
    \n \\]
    \n \u800c\u4e14$\\tilde{\\gamma}_s(0),\\tilde{\\gamma}_s(1)$\u90fd\u4e0e$s$\u65e0\u5173.
    \n<\/div>
    \n
    \u8bc1\u660e<\/span>.<\/span> \u5047\u8bbe$H:[0,1]\\times[0,1]\\to X$\u662f$X$\u91cc\u7684\u4e00\u4e2a\u540c\u4f26, \u5373$\\gamma_s(t)=H(s,t)$. \u6211\u4eec\u53ef\u4ee5\u5c06$[0,1]\\times[0,1]$\u5206\u6210\u5f88\u591a\u5c0f\u77e9\u5f62$R_{ij}$, \u4f7f\u5f97$H(R_{ij})\\subset U_{ij}$, \u8fd9\u91cc$U_{ij}$\u4e3a$X$\u7684\u67d0\u4e2a\u57fa\u7840\u90bb\u57df. \u8fd9\u91cc, $i, j$\u6307\u6807\u5206\u522b\u8868\u793a$x,y$\u65b9\u5411\u7684\u6307\u6807.<\/p>\n

    \u6211\u4eec\u9700\u8981\u5177\u4f53\u6784\u9020\u4e00\u4e2a\u540c\u4f26\u63d0\u5347$\\tilde{H}$, \u4f7f\u5f97$\\tilde{H}(s,t)=\\tilde{\\gamma}_s(t)$, \u81ea\u7136\u5730, \u6211\u4eec\u53ef\u4ee5\u5229\u7528$p$\u7684\u5c40\u90e8\u9006\u6620\u5c04$q_{ij}:U_{ij}\\to Y$, \u6784\u9020$\\tilde{H}_{R_{ij}}=q_{ij}\\circ H|_{R_{ij}}$. \u4e00\u65b9\u9762, \u8fd9\u786e\u5b9e\u662f$H$\u7684\u4e00\u4e2a\u63d0\u5347, \u5373$p\\circ \\tilde{H}=p\\circ q_{ij}\\circ H=H$\u5728\u6bcf\u4e2a$R_{ij}$\u90fd\u6210\u7acb, \u4f46\u53e6\u4e00\u65b9\u9762, \u6211\u4eec\u9700\u8981\u6070\u5f53\u7684\u9009\u53d6$q_{ij}$, \u4f7f\u5f97\u5982\u4e0a\u6784\u9020\u7684$\\tilde{H}$\u662f\u8fde\u7eed\u7684, \u8fd9\u672c\u8d28\u4e0a\u8981\u6c42$\\tilde{H}|_{R_{ij}}=\\tilde{H}_{R_{kl}}$\u5728$R_{ij}\\cap R_{kl}$\u4e0a\u6210\u7acb.<\/p>\n

    \u4e3a\u4e86\u7b80\u5355\u8d77\u89c1, \u5047\u8bbe\u6211\u4eec\u5f97\u5230\u4e00\u4e2a$2\\times 2$\u7684\u7f51\u683c. \u9996\u5148\u6765\u6784\u9020$q_{1j}$. \u6ce8\u610f\u5230, $\\tilde{H}(0,t)=\\tilde{\\gamma}_0(t)$, \u8fd9\u91cc$\\tilde{\\gamma}_0$\u662f$\\gamma_0$\u7684\u4e00\u4e2a\u63d0\u5347(\u56fa\u5b9a). \u8fd9\u8868\u660e$q_{1j}$\u5e94\u8be5\u5982\u4f55\u5b9a\u4e49, \u5373$q_{11}:U_{11}\\to Y$\u662f$p$\u5c40\u90e8\u552f\u4e00\u7684\u9006\u6620\u5c04\u4f7f\u5f97$q_{11}(\\gamma_0(0))=\\tilde{\\gamma}_0(0)$, $q_{12}:U_{12}\\to Y$\u662f$p$\u5c40\u90e8\u552f\u4e00\u7684\u9006\u6620\u5c04\u4f7f\u5f97$q_{12}(\\gamma_0(t_1))=\\tilde{\\gamma}_0(t_1)$, \u8fd9\u91cc$t_1$\u662f$R_{11}$\u6700\u4e0a\u9762\u7684$t$\u5750\u6807(\u5728$2\\times 2$\u7684\u60c5\u5f62\u6709$t_1=1$).<\/p>\n

    \u6211\u4eec\u9700\u8981\u9a8c\u8bc1$\\tilde{H}_{11}:=q_{11}\\circ H$, $\\tilde{H}_{12}:=q_{12}\\circ H$\u5728$R_{11}\\cap R_{12}$\u4e0a\u662f\u76f8\u540c\u7684(\u8fd9\u4e2a\u4ea4\u96c6\u5176\u5b9e\u5c31\u662f\u8fb9$\\left\\{ (s,t_1):s\\in[0,s_1] \\right\\}$). \u4e8b\u5b9e\u4e0a, \u6309\u7167\u6784\u9020
    \n \\[
    \n \\tilde{H}_{11}(0,t_1)=\\tilde{H}_{12}(0,t_1)=\\tilde{\\gamma}_0(t_1),
    \n \\]
    \n \u56e0\u6b64$\\tilde{H}_{1j}(s,t_1)$\u73b0\u5728\u5230\u8fb9$R_{11}\\cap R_{12}$\u4e0a\u5b9a\u4e49\u4e86\u4e24\u6761\u8def\u5f84, \u4ed6\u4eec\u90fd\u662f$H(s,t_1)$\u7684\u63d0\u5347(\u56e0\u4e3a$p\\circ \\tilde{H}_{1j}(s,t_1)|_{[0,s_1]}=p\\circ q_{1j}\\circ H(s,t_1)|_{[0,s_1]}=H(s,t_1)|_{[0,s_1]}$), \u800c\u4e14\u6709\u76f8\u540c\u7684\u521d\u59cb\u6761\u4ef6$\\tilde{H}_{11}(0,t_1)=\\tilde{H}_{12}(0,t_1)=\\tilde{\\gamma}_0(t_1)$, \u56e0\u6b64\u7531\u9053\u8def\u63d0\u5347\u7684\u552f\u4e00\u6027, \u6211\u4eec\u77e5\u9053$\\tilde{H}_{11}(s,t_1)=\\tilde{H}_{12}(s,t_1)$\u5bf9\u4efb\u610f\u7684$s\\in[0,s_1]$\u90fd\u6210\u7acb.<\/p>\n

    \u5229\u7528\u540c\u6837\u7684\u6280\u5de7, \u6211\u4eec\u4ece$\\tilde{\\gamma}_{s_1}$\u51fa\u53d1, \u53ef\u4ee5\u5ef6\u62d3\u4e0a\u8ff0\u540c\u4f26\u5230$R_{2j}$, \u8fdb\u800c\u7531\u5f52\u7eb3\u6cd5, \u6211\u4eec\u5f97\u5230\u6574\u4e2a\u6b63\u65b9\u5f62\u4e0a\u7684\u540c\u4f26.<\/p>\n

    \u6211\u4eec\u8fd8\u9700\u8981\u9a8c\u8bc1$\\tilde{H}$\u5177\u6709\u56fa\u5b9a\u7684\u7aef\u70b9, \u4e8b\u5b9e\u4e0a$\\tilde{H}(s,0)$\u662f$H(s,0)$\u7684\u63d0\u5347\u662f\u4e00\u4e2a\u5e38\u503c. \u8fd9\u662f\u56e0\u4e3a, \u5e38\u503c\u63d0\u5347\u4e5f\u662f\u6ee1\u8db3\u6761\u4ef6\u7684\u4e00\u4e2a\u63d0\u5347, \u56e0\u6b64\u5229\u7528\u63d0\u5347\u7684\u552f\u4e00\u6027\u5f97\u5230$\\tilde{H}(s,0)$\u5fc5\u5b9a\u5c31\u662f\u5e38\u503c\u63d0\u5347. \u540c\u6837\u7684\u9053\u7406\u53ef\u4ee5\u8bf4\u5417$\\tilde{H}(s,1)$\u4e5f\u662f\u5e38\u503c.
    \n<\/div>\n","protected":false},"excerpt":{"rendered":"

    \u8be5\u8bfb\u4e66\u7b14\u8bb0\u6765\u81eaTopology and Groups. 1. \u590d\u53e0\u6620\u5c04\u7684\u4e24\u4e2a\u4f8b\u5b50 \u4f8b\u5b50 1. \u5047… \u7ee7\u7eed\u9605\u8bfb\u590d\u53e0\u7a7a\u95f4\u53ca\u5176\u590d\u53e0\u5f15\u7406\u3001\u5355\u5c04\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":[2],"tags":[165,161],"class_list":["post-838","post","type-post","status-publish","format-standard","hentry","category-math","tag-fudieyinli","tag-fudiekongjian","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/838","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=838"}],"version-history":[{"count":23,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/838\/revisions"}],"predecessor-version":[{"id":1165,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/838\/revisions\/1165"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=838"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=838"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=838"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}