{"id":1766,"date":"2022-05-29T20:17:54","date_gmt":"2022-05-29T12:17:54","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1766"},"modified":"2025-02-26T11:03:17","modified_gmt":"2025-02-26T03:03:17","slug":"group_action","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/05\/29\/group_action\/","title":{"rendered":"\u7fa4\u4f5c\u7528"},"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>\u8fd9\u5468\u56e0\u4e3a\u75ab\u60c5\u539f\u56e0\u6ca1\u6709\u53bb\u5b81\u6ce2, \u76ee\u524d\u6d59\u6c5f\u5185\u4e58\u5750\u516c\u5171\u4ea4\u901a\u6216\u8005\u51fa\u5165\u516c\u5171\u573a\u6240\u90fd\u9700\u8981\u51fa\u793a48\u5c0f\u65f6\u7684\u6838\u9178\u68c0\u6d4b\u9634\u6027\u8bc1\u660e, \u8981\u662f\u4e00\u4e0d\u5c0f\u5fc3\u8fc7\u4e86\u8fd9\u4e2a\u8bc1\u660e\u7684\u6709\u6548\u65f6\u95f4, \u5c31\u7740\u5b9e\u6709\u70b9\u9ebb\u70e6=.= \u672c\u6587\u4e3b\u8981\u7528\u4e8e\u8bb0\u5f55\u7fa4\u4f5c\u7528\u7684\u76f8\u5173\u77e5\u8bc6\u70b9, \u4e66\u4e0a\u4ecb\u7ecd\u7684\u81ea\u7531\u7fa4\u4f5c\u7528\u4e0e\u7eaf\u4e0d\u8fde\u7eed\u7fa4\u4f5c\u7528\u7684\u6982\u5ff5\u7531\u4e8e\u7f3a\u4e4f\u8db3\u591f\u7684\u4f8b\u5b50, \u7406\u89e3\u8d77\u6765\u8fd8\u662f\u6bd4\u8f83\u5403\u529b, \u56e0\u6b64\u81ea\u5df1\u4e5f\u53c2\u8003\u4e86\u5e84\u6653\u6ce2\u8001\u5e08\u7684\u89c6\u9891\u4e0e\u76f8\u5173Wiki.<\/p>\n<p><!--more--><\/p>\n\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/www.bilibili.com\/video\/BV1P7411N7fW?p=30\">P30 (29)\u7fa4\u4f5c\u7528<\/a><br \/>\n2. <a href=\"https:\/\/baike.baidu.com\/item\/%E7%BE%A4%E4%BD%9C%E7%94%A8\/9812908\">\u7fa4\u4f5c\u7528<\/a><\/p>\n<h3>1. \u7fa4\u4f5c\u7528\u7684\u76f8\u5173\u5b9a\u4e49<\/h3>\n<p>\u9996\u5148\u4ecb\u7ecd\u4e00\u4e0b\u62d3\u6251\u7fa4\u7684\u6982\u5ff5: \u82e5\u7fa4$G$\u4e0a\u5b9a\u4e49\u4e86\u4e00\u4e2a\u62d3\u6251, \u4f7f\u5f97\u5b9a\u4e49\u5728\u7fa4$G$\u4e0a\u7684\u4e58\u79ef\u6620\u5c04$$G \\times G \\to G, \\\\ (g, h) \\mapsto gh,$$\u4e0e\u9006\u6620\u5c04$$G \\to G, \\\\ g \\mapsto g^{-1},$$\u5747\u8fde\u7eed, \u5219\u79f0\u7fa4$G$\u4e3a\u4e00\u4e2a\u62d3\u6251\u7fa4. \u800c\u636e\u81ea\u5df1\u4e86\u89e3, \u7fa4\u4f5c\u7528\u7684\u5b9a\u4e49\u67092\u4e2a\u7248\u672c, \u5982\u4e0b\u6240\u793a.<\/p>\n<p><strong>\u5b9a\u4e491<\/strong> \u8bbe$X$\u662f\u4e00\u4e2a\u62d3\u6251\u7a7a\u95f4, \u7528$Aut \\ X$\u8868\u793a$X$\u7684\u6240\u6709\u81ea\u540c\u80da\u6784\u6210\u7684\u7fa4, \u7fa4\u7684\u4e58\u6cd5\u4e3a\u6620\u5c04\u7684\u590d\u5408. \u5982\u679c$G$\u662f\u4e00\u4e2a\u7fa4, $q: G \\to Aut \\ X$\u662f\u4e00\u4e2a\u540c\u6001, \u5219\u79f0$q$\u4e3a\u7fa4$G$\u5728$X$\u4e0a\u7684\u4e00\u4e2a\u7fa4\u4f5c\u7528, \u8bb0\u4e3a$q: G \\searrow X$, \u79f0\u6bcf\u4e2a$q(g)$\u4e3a$g$\u5728$X$\u4e0a\u7684\u4f5c\u7528.<\/p>\n<p><strong>\u5b9a\u4e492<\/strong> \u62d3\u6251\u7fa4$G$\u5728\u62d3\u6251\u7a7a\u95f4$X$\u4e0a\u7684\u5de6\u8fde\u7eed\u4f5c\u7528\u662f\u4e00\u4e2a\u8fde\u7eed\u6620\u5c04$$\\lambda: G \\times X \\to X$$\u4e14$\\lambda$\u6ee1\u8db3<br \/>\na) $\\lambda(1_G, x) = x$, $\\forall x \\in X$.<br \/>\nb) $\\lambda(gh, x) = \\lambda(g, \\lambda(h, x))$, $\\forall x \\in X$, $g, h \\in G$.<\/p>\n<p>\u5176\u4e2d, \u4e0d\u77e5\u9053\u662f\u5426\u4e3a\u539f\u4e66\u4f5c\u8005\u7684\u758f\u6f0f, \u5728\u81ea\u5df1\u67e5\u9605\u7684\u51e0\u7bc7\u6587\u732e\u4e2d, \u7fa4\u4f5c\u7528\u8ba8\u8bba\u7684\u5bf9\u8c61\u7fa4$G$\u90fd\u4e3a\u4e00\u4e2a\u62d3\u6251\u7fa4, \u800c\u5b9a\u4e491\u4e2d\u7684\u7fa4$G$\u5e76\u672a\u65bd\u52a0\u8fd9\u4e2a\u7ea6\u675f. \u4e0b\u6587\u8ba8\u8bba\u7684\u7fa4$G$\u5747\u4e3a\u4e00\u4e2a\u62d3\u6251\u7fa4. \u5b9a\u4e491\u4e0e\u5b9a\u4e492\u5b9e\u9645\u4e0a\u662f\u606f\u606f\u76f8\u8fde\u7684(\u4e0d\u4e00\u5b9a\u662f\u7b49\u4ef7\u7684), \u56e0\u4e3a\u4ece\u5b9a\u4e492\u4e2d\u7684\u6620\u5c04$\\lambda$\u603b\u53ef\u4ee5\u8bf1\u5bfc\u51fa\u6620\u5c04$q: G \\to Aut \\ X$, \u4e14$$q(g)(x) = \\lambda(g, x),$$\u8fd9\u662f\u4e00\u4e2a\u7fa4\u540c\u6001. \u53cd\u8fc7\u6765, \u82e5\u7fa4$G$\u662f\u79bb\u6563\u7684, \u5219\u4efb\u610f\u4e00\u4e2a\u540c\u6001$q: G \\to Aut \\ X$\u5b9a\u4e49\u4e86\u5728\u62d3\u6251\u7a7a\u95f4$X$\u4e0a\u7684\u7fa4$G$\u7684\u5de6\u8fde\u7eed\u4f5c\u7528.<br \/>\n$\\\\$ \u6211\u4eec\u901a\u5e38\u628a$q(g)(x)$\u7b80\u8bb0\u4e3a$gx$\u6216$g \\cdot x$. \u7c7b\u4f3c\u5730, \u5bf9\u4e8e\u62d3\u6251\u7a7a\u95f4$X$\u7684\u4e00\u4e2a\u5b50\u7a7a\u95f4$A$, $GA$\u6216$G \\cdot A$\u8868\u793a\u5b50\u7a7a\u95f4$A$\u5728$G$-\u4f5c\u7528\u4e0b\u7684\u8f68\u9053: $GA = $$ \\bigcup_{g \\in G}gA$.<br \/>\n\u62d3\u6251\u7a7a\u95f4$X$\u5728$G$-\u4f5c\u7528\u4e0b\u7684\u5546\u7a7a\u95f4$X \/ G$(\u4e5f\u7ecf\u5e38\u88ab\u8bb0\u4e3a$G \/ X$) \u662f\u7531\u62d3\u6251\u7a7a\u95f4$X$\u4e2d\u7684\u6240\u6709\u70b9\u7684$G$-\u8f68\u9053\u6784\u6210\u7684\u96c6\u5408, \u79f0\u5176\u4e3a<strong>\u8f68\u9053\u7a7a\u95f4<\/strong>, \u5176\u62d3\u6251\u4e3a\u5546\u62d3\u6251. \u8f68\u9053\u7a7a\u95f4$X \/ $$ G$\u4e2d\u7684\u5143\u7d20\u4e3a\u62d3\u6251\u7a7a\u95f4$X$\u4e2d\u7684\u7b49\u4ef7\u7c7b, \u5176\u4e2d$x $$ \\sim y$\u7b49\u4ef7\u5f53\u4e14\u4ec5\u5f53$Gx = Gy$, \u5373$y $$ \\in Gx$.<br \/>\n$\\\\$ \u70b9$x \\in X$\u5728$G$-\u4f5c\u7528\u4e0b\u7684\u4e0d\u52a8\u70b9\u662f\u7fa4$G$\u7684\u4e00\u4e2a\u5b50\u7fa4$G_x = \\{ g \\in G: $$ gx = x \\}$. \u82e5\u5bf9\u4e8e\u4efb\u610f$x \\in X$, $G_x = \\{ e \\}$, \u5219\u79f0\u7fa4$G$\u5728\u62d3\u6251\u7a7a\u95f4$X$\u4e0a\u7684\u4f5c\u7528\u662f<strong>\u81ea\u7531<\/strong>\u7684. \u66f4\u8fdb\u4e00\u6b65\u5730, \u82e5$X$\u4e3a\u4e00\u4e2aHausdorff\u7a7a\u95f4, \u5219\u5bf9\u4e8e\u4efb\u610f$x \\in X$, $G_x$\u5728$G$\u4e2d\u662f\u95ed\u96c6.<\/p>\n<h3>2. \u5de6\u8fde\u7eed\u4f5c\u7528\u4e0e\u53f3\u8fde\u7eed\u4f5c\u7528<\/h3>\n<p>\u4e0a\u4e00\u5c0f\u8282\u7684\u5b9a\u4e492\u4e2d\u63d0\u5230\u4e86\u5de6\u8fde\u7eed\u4f5c\u7528, \u7c7b\u4f3c\u5730, \u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u53f3\u8fde\u7eed\u4f5c\u7528\u7684\u5b9a\u4e49: \u62d3\u6251\u7fa4$G$\u5728\u62d3\u6251\u7a7a\u95f4$X$\u4e0a\u7684\u53f3\u8fde\u7eed\u4f5c\u7528\u662f\u4e00\u4e2a\u8fde\u7eed\u6620\u5c04$$\\lambda: X \\times G \\to X$$\u4e14$\\lambda$\u6ee1\u8db3<br \/>\n$\\\\$ a) $\\lambda(x, 1_G) = x$, $\\forall x \\in X$.<br \/>\n$\\\\$ b) $\\lambda(x, gh) = \\lambda(\\lambda(x, g), h)$, $\\forall x \\in X$, $g, h \\in G$.<br \/>\n$\\\\$ \u6ce8\u610f\u5de6\u548c\u53f3\u4f5c\u7528\u7684\u533a\u522b\u4ec5\u5728\u4e8e\u50cf$gh$\u8fd9\u6837\u7684\u79ef\u5728$x$\u4e0a\u4f5c\u7528\u7684\u6b21\u5e8f. \u5bf9\u4e8e\u5de6\u4f5c\u7528, $h$\u5148\u4f5c\u7528\u7136\u540e\u662f$g$; \u800c\u5bf9\u4e8e\u53f3\u4f5c\u7528, \u5148\u4f5c\u7528$g$\u7136\u540e\u662f$h$. \u4ece\u4e00\u4e2a\u53f3\u4f5c\u7528\u53ef\u4ee5\u6784\u9020\u4e00\u4e2a\u5de6\u4f5c\u7528, \u53ea\u8981\u548c\u7fa4\u4e0a\u7684\u9006\u64cd\u4f5c\u590d\u5408\u5c31\u53ef\u4ee5\u4e86. \u5982\u679c$r$\u4e3a\u4e00\u53f3\u4f5c\u7528, \u5219$$l: G \\times X \\to X, \\\\ (g, x) \\mapsto r(x, g^{-1}),$$\u662f\u4e00\u5de6\u4f5c\u7528, \u56e0\u4e3a$$l(gh, x) = r(x, (gh)^{-1}) = r(x, h^{-1}g^{-1}) = \\\\ r(r(x, h^{-1}), g^{-1}) = r(l(h, x), g^{-1}) = l(g, l(h, x)),$$\u800c$$l(e, x) = r(x, e^{-1}) = r(x, e) = x.$$\u6240\u4ee5\u63a5\u4e0b\u6765, \u6211\u4eec\u53ea\u8003\u8651\u5de6\u7fa4\u4f5c\u7528, \u56e0\u4e3a\u53f3\u4f5c\u7528\u53ef\u4ee5\u76f8\u5e94\u63a8\u7406.<\/p>\n<h3>3. \u7fa4\u4f5c\u7528\u7684\u76f8\u5173\u4f8b\u5b50<\/h3>\n<p><strong>\u4f8b1<\/strong> \u8bbe$Z \\searrow R$, $n(x) := n + x$, $\\forall n \\in Z$, $x \\in R$, \u5219$R \/ Z \\cong S^1$.<br \/>\n$\\\\$ <strong>\u8bc11<\/strong> \u53d6$R$\u4e0a\u7684\u533a\u95f4$[0, 1]$, \u5219$R \/ Z$\u4e2d\u7684\u6240\u6709\u70b9(\u8f68\u9053)\u4e0e\u533a\u95f4$[0, 1]$\u81f3\u5c11\u6709\u4e00\u4e2a\u4ea4\u70b9, \u81f3\u591a\u6709\u4e24\u4e2a\u4ea4\u70b9, \u4e14\u53ea\u6709\u4e00\u6761\u8f68\u9053\u4e0e\u533a\u95f4$[0, 1]$\u4ea4\u4e8e\u4e24\u70b9, \u8fd9\u4e24\u70b9\u5206\u522b\u4e3a\u533a\u95f4$[0, $$ 1]$\u7684\u4e24\u4e2a\u7aef\u70b9. \u7528\u6620\u5c04\u7684\u8bed\u8a00\u6765\u63cf\u8ff0\u8fd9\u4e2a\u4e8b\u5b9e, \u8bb0$$\\pi: R \\to R \/ Z, \\\\ \\pi|_{[0, 1]}: [0, 1] \\to  R \/ Z,$$\u5219$\\pi|_{[0, 1]}$\u662f\u4e00\u4e2a\u6ee1\u5c04. \u5982\u6b64\u4e00\u6765, \u4ece\u76f4\u89c2\u4e0a\u6765\u770b, \u5728\u533a\u95f4$[0, 1]$\u5185\u90e8, \u6bcf\u4e2a\u70b9\u5747\u8868\u793a\u4e0d\u540c\u7684\u8f68\u9053, \u800c\u5728\u533a\u95f4$[0, 1]$\u7684\u4e24\u4e2a\u7aef\u70b9\u5904, \u5219\u8868\u793a\u540c\u4e00\u6761\u8f68\u9053.<br \/>\n$\\\\$ \u53c8\u533a\u95f4$[0, 1]$\u662f$R$\u4e0a\u7684\u4e00\u4e2a\u7d27\u96c6, \u8f68\u9053\u7a7a\u95f4$R \/ Z$\u662f\u4e00\u4e2aHausdorff\u7a7a\u95f4, \u5219$\\pi|_{[0, 1]}$\u662f\u4e00\u4e2a\u7c98\u5408\u6620\u5c04, \u540c\u65f6\u4e5f\u662f\u4e00\u4e2a\u5546\u6620\u5c04, \u4ece\u800c$$R \/ Z \\cong [0, 1] \/ \\overset{\\pi}{\\sim} \\cong S^1,$$\u547d\u9898\u5f97\u8bc1.<br \/>\n$\\\\$ <strong>\u8bc12<\/strong> \u8003\u8651\u6620\u5c04$$f: R \\to S^1, x \\mapsto (cos(2 \\pi x), sin(2 \\pi x)),$$\u8fd9\u662f\u4e00\u4e2a\u8fde\u7eed\u6f2b\u5c04, \u5e76\u4e14\u662f\u4e00\u4e2a\u5f00\u6620\u5c04, \u56e0\u6b64\u662f\u4e00\u4e2a\u5546\u6620\u5c04. \u5e76\u4e14$f( $$ y) = f(x)$\u5f53\u4e14\u4ec5\u5f53$y = x + n, n \\in Z$, \u8fd9\u8bf4\u660e$f$\u8bf1\u5bfc\u7684\u7b49\u4ef7\u5173\u7cfb\u548c\u7fa4\u4f5c\u7528\u8bf1\u5bfc\u7684\u7b49\u4ef7\u5173\u7cfb\u76f8\u540c, \u56e0\u6b64$R \/ Z = R \/ \\overset{f}{\\sim} \\cong S^1$.<\/p>\n<p><strong>\u4f8b2<\/strong> \u8bbe$Z^2 \\searrow R^2$, $(m, n)((x, y)) := (x + mw_1, y + nw_2)$, $w_1, w_2$\u662f$R^2$\u4e0a\u4e24\u4e2a\u7ebf\u6027\u65e0\u5173\u7684\u5411\u91cf, \u5219$R^2 \/ Z^2 \\cong T^2$. \u7c7b\u4f3c\u4f8b1, \u6784\u9020\u4e00\u4e2a\u5de6\u4e0b\u89d2\u9876\u70b9\u4f4d\u4e8e\u539f\u70b9\u5904, \u90bb\u8fb9\u65b9\u5411\u5411\u91cf\u5206\u522b\u4e0e$w_1, w_2$\u5e73\u884c\u7684\u5355\u4f4d\u5e73\u884c\u56db\u8fb9\u5f62\u5373\u53ef\u8bc1\u660e\u547d\u9898.<\/p>\n<p><strong>\u4f8b3<\/strong> \u8bbe$X = R^{n + 1} \/ \\{ (0, 0, \\cdots, 0) \\}$, $G = R^*$, $R^*$\u4e3a$R$\u4e2d\u6240\u6709\u975e\u96f6\u5b9e\u6570\u6784\u6210\u7684\u4e58\u6cd5\u7fa4, $$R^* \\searrow X, \\\\ \\lambda(x_1, \\cdots, x_{n + 1}) := (\\lambda x_1, \\cdots, \\lambda x_{n + 1}).$$\u5219$X \/ R^* \\cong RP^n$.<\/p>\n<p><strong>\u4f8b4<\/strong> \u8bbe$$G = Z_2 = \\{ \\pm 1 \\} \\searrow S^n, \\\\ (-1)(p) = -p, 1(p) = p, \\forall p \\in S^n,$$\u5219$S^n \/ Z_2 \\cong RP^n$.<\/p>\n<p>\u5728\u524d\u51e0\u4e2a\u4f8b\u5b50\u8bc1\u660e\u5546\u6620\u5c04\u7684\u65f6\u5019, \u5b9e\u9645\u4e0a\u90fd\u9700\u8981\u5047\u8bbe\u5df2\u77e5\u8f68\u9053\u7a7a\u95f4\u662fHausdorff\u7a7a\u95f4, \u7136\u540e\u624d\u80fd\u4ece&#8221;\u7d27\u81f4\u7a7a\u95f4\u5230Hausdorff\u7a7a\u95f4\u7684\u8fde\u7eed\u6ee1\u5c04\u662f\u5546\u6620\u5c04&#8221; \u63a8\u5bfc\u51fa\u7ed3\u8bba. \u5728\u5f15\u5165\u81ea\u7531\u7fa4\u4f5c\u7528\u4e0e\u7eaf\u4e0d\u8fde\u7eed\u4e24\u4e2a\u6982\u5ff5\u4ee5\u540e, \u6211\u4eec\u5c31\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u8bc1\u660e\u8f68\u9053\u7a7a\u95f4\u7684Hausdorff\u6027\u8d28, \u8be6\u7ec6\u5185\u5bb9\u53ef\u53c2\u8003<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/05\/29\/free_group_action_proper_maps_and_proper_discontinuity\/\">\u81ea\u7531\u7fa4\u4f5c\u7528, Proper Maps\u4e0e\u7eaf\u4e0d\u8fde\u7eed<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u8fd9\u5468\u56e0\u4e3a\u75ab\u60c5\u539f\u56e0\u6ca1\u6709\u53bb\u5b81\u6ce2, \u76ee\u524d\u6d59\u6c5f\u5185\u4e58\u5750\u516c\u5171\u4ea4\u901a\u6216\u8005\u51fa\u5165\u516c\u5171\u573a\u6240\u90fd\u9700\u8981\u51fa\u793a48\u5c0f\u65f6\u7684\u6838\u9178\u68c0\u6d4b\u9634\u6027\u8bc1\u660e, \u8981 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/05\/29\/group_action\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u7fa4\u4f5c\u7528<\/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\/1766"}],"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=1766"}],"version-history":[{"count":50,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1766\/revisions"}],"predecessor-version":[{"id":3607,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1766\/revisions\/3607"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1766"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1766"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1766"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}