{"id":1625,"date":"2022-03-20T23:08:31","date_gmt":"2022-03-20T15:08:31","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1625"},"modified":"2025-02-26T11:04:20","modified_gmt":"2025-02-26T03:04:20","slug":"fundamental_groups_applications_mark","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/03\/20\/fundamental_groups_applications_mark\/","title":{"rendered":"\u57fa\u672c\u7fa4\u53ca\u5176\u5e94\u7528\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>\u8fd9\u5468\u5f00\u59cb, \u7b97\u662f\u6b63\u5f0f\u6295\u5165\u65b0\u9879\u76ee\u7ec4\u7684\u5de5\u4f5c\u4e2d\u4e86. \u76ee\u524d\u63a5\u89e6\u7684\u5de5\u4f5c\u5185\u5bb9\u4e2a\u4eba\u8fd8\u662f\u86ee\u611f\u5174\u8da3\u7684, \u5c24\u5176\u662f\u7a0b\u5e8f\u5316\u5173\u5361\u5236\u4f5c\u5de5\u5177\u7684\u8fed\u4ee3\u5de5\u4f5c\u4e5f\u4ea4\u63a5\u5230\u4e86\u6211\u7684\u624b\u4e0a, \u5e0c\u671b\u81ea\u5df1\u63a5\u4e0b\u6765\u80fd\u91cd\u70b9\u5b66\u4e60\u76f8\u5173\u5185\u5bb9\u5e76\u6709\u6240\u6210\u679c\u53ed~ \u4e5f\u7b97\u662f\u5b66\u4e60\u4e86\u633a\u4e45\u7684\u57fa\u672c\u7fa4\u4e86, \u672c\u6587\u4e3b\u8981\u662f\u5bf9\u5176\u5e94\u7528\u4f5c\u4e00\u4e2a\u76f8\u5173\u7684\u6ce8\u8bb0\u4e0e\u603b\u7ed3.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/math.stackexchange.com\/questions\/2062438\/equivalence-homotopy-between-bbb-r2-0-0-and-a-convex-set-without-a-po\">Equivalence homotopy between R2\\-{(0,0)} and a convex set without a point<\/a><br \/>\n2. <a href=\"https:\/\/www.zhihu.com\/question\/523581682\/answer\/2407200462\">\u6c42\u95ee\u5982\u4f55\u8bc1\u660e\u4e09\u4e2a\u4ea4\u4e8e\u4e00\u6761\u516c\u5171\u8fb9\u7684\u4e09\u89d2\u5f62\u7684\u5e76\u96c6\u4e0d\u662f\u6d41\u5f62?<\/a><\/p>\n<p>1. \u5355\u8fde\u901a\u7a7a\u95f4\u5230\u975e\u5355\u8fde\u901a\u7a7a\u95f4\u7684\u6620\u5c04\u8bf1\u5bfc\u7684\u540c\u6001\u4e0d\u53ef\u80fd\u4e3a\u6ee1\u6620\u5c04.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u5355\u8fde\u901a\u7a7a\u95f4\u7684\u57fa\u672c\u7fa4\u4e3a\u4e00\u4e2a\u5e73\u51e1\u7fa4, \u4ec5\u542b\u5355\u4f4d\u5143, \u800c\u975e\u5355\u8fde\u901a\u7a7a\u95f4\u7684\u57fa\u672c\u7fa4\u4e3a\u975e\u5e73\u51e1\u7fa4, \u5176\u5143\u7d20\u6570\u91cf\u5927\u4e8e1, \u6545\u547d\u9898\u5f97\u8bc1.<\/p>\n<p>2. \u4eff\u7167&#8221;\u5185\u70b9\u4e0d\u662f\u8fb9\u754c\u70b9&#8221;\u7684\u5b9a\u7406\u8bc1\u660e, \u8bc1\u660e\u4e09\u4e2a\u4ea4\u4e8e\u4e00\u6761\u516c\u5171\u8fb9\u7684\u4e09\u89d2\u5f62\u7684\u5e76\u96c6\u4e0d\u662f\u6d41\u5f62, \u6362\u8a00\u4e4b\u5982\u679c\u5355\u7eaf\u590d\u5f62$K$\u662f\u4e00\u4e2a\u66f2\u9762\u7684\u5355\u7eaf\u5256\u5206, \u5219\u6bcf\u4e2a\u4e00\u7ef4\u5355\u5f62\u81f3\u591a\u53ea\u80fd\u662f\u4e24\u4e2a\u4e8c\u7ef4\u5355\u5f62\u7684\u9762.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> <em>(\u8fd9\u4e2a\u547d\u9898\u5728\u5de5\u7a0b\u4e0a\u7684\u5e94\u7528\u4ef7\u503c\u662f\u5341\u5206\u5de8\u5927\u7684.)<\/em> \u8bb0$p$\u4e3a\u90a3\u6761\u516c\u5171\u8fb9\u7684\u4e2d\u70b9, \u4efb\u53d6$p$\u7684\u90bb\u57df$U$(\u540c\u80da\u4e8e$E^n$\u7684\u4e00\u4e2a\u5f00\u5b50\u96c6), \u5b58\u5728\u5145\u5206\u5c0f\u7684\u7403\u5f62\u90bb\u57df$B_\\epsilon(p)$, \u4f7f\u5f97$\\partial<br \/>\n $$ B_\\epsilon(p) = \\overline{B_\\epsilon(p)} \\backslash B_\\epsilon(p)$\u662f$U \\backslash \\{ p \\}$\u7684\u6536\u7f29\u6838, \u56e0\u6b64\u5b58\u5728\u4ece$\\pi_1(U \\backslash \\{ p \\})$\u5230$\\pi_1(\\partial $$ B_\\epsilon(p))$\u7684\u6ee1\u540c\u6001.<br \/>\n$\\\\$ \u7531\u4e8e$p$\u4e3a\u4e09\u4e2a\u4e09\u89d2\u5f62\u7684\u516c\u5171\u8fb9\u7684\u4e2d\u70b9, \u6545$\\partial B_\\epsilon(p)$\u540c\u80da\u4e8e\u4e09\u4e2a\u5171\u7528\u76f4\u5f84\u7684\u534a\u5706\u5468\u7684\u5e76\u96c6(\u4e0d\u5305\u62ec\u76f4\u5f84), \u8fd9\u65f6\u5019\u53ef\u4ee5\u628a\u5176\u4e2d\u4e00\u4e2a\u534a\u5706\u5468\u6536\u7f29\u81f3\u4e00\u70b9\uff0c\u5219\u5269\u4e0b\u4e24\u4e2a\u534a\u5706\u5468\u7684\u76f4\u5f84\u4e0a\u7684\u5bf9\u5f84\u70b9\u4e5f\u88ab\u6536\u7f29\u81f3\u540c\u4e00\u70b9\u4e0a\uff0c\u4ece\u800c$\\partial B_\\epsilon(p)$\u4e0e\u4e24\u4e2a\u5706\u5468\u7684\u4e00\u70b9\u5e76\u540c\u4f26\u7b49\u4ef7, \u5373$\\pi_1(\\partial B_\\epsilon(p)) \\cong $$ Z * Z$. \u53c8$U \\backslash \\{ p \\} \\cong E^2 \\backslash \\{ p \\} \\cong S^2$, \u6545$U \\backslash \\{ p \\}$\u4e3a\u4e00\u4e2a\u5355\u8fde\u901a\u7a7a\u95f4. \u6700\u540e\u5229\u7528\u4e0a\u9898\u7ed3\u8bba\u5f97\u5230\u77db\u76fe, \u547d\u9898\u5f97\u8bc1.<\/p>\n<p>3. \u6c42\u8bc1$U \\backslash \\{ p \\} \\cong E^n \\backslash \\{ q \\}$, \u5176\u4e2d, $U$\u4e3a$E^n$\u4e2d\u7684\u4e00\u4e2a\u51f8\u5b50\u96c6, $p \\in $$ int(U)$, $q \\in $$ E^n$.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> <em>(\u5c3d\u7ba1\u8fd9\u4e2a\u547d\u9898\u548c\u57fa\u672c\u7fa4\u7684\u5173\u7cfb\u4e0d\u5927, \u4f46\u5374\u662f\u4e0a\u9898\u89e3\u7b54\u4e2d\u95f4\u63a5\u5f15\u7528\u7684\u4e00\u4e2a\u547d\u9898.)<\/em> \u4e0d\u5931\u4e00\u822c\u6027\u5730, \u6211\u4eec\u53ef\u4ee4\u70b9$p, q$\u5747\u4e3a\u539f\u70b9(\u56e0\u4e3a\u5047\u5982\u4e0d\u662f\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u540c\u80da\u6620\u5c04$x \\to x &#8211; p$\u6216\u8005$x \\to x &#8211; q$&#8221;\u79fb\u52a8&#8221;\u6574\u4e2a\u96c6\u5408). \u56e0\u4e3a\u539f\u70b9\u5728$int(U)$\u4e2d, \u6545\u5b58\u5728\u7403\u5fc3\u4e3a\u539f\u70b9\u7684\u5f00\u7403\u5305\u542b\u4e8e$int(U)$\u4e2d, \u4ece\u800c\u5b58\u5728\u4e00\u4e2a\u95ed\u7403$D$(\u524d\u8ff0\u5f00\u7403\u7684\u5b50\u96c6) \u5305\u542b\u4e8e$int(U)$\u4e2d, \u4e14\u5176\u7403\u5fc3\u4e3a\u539f\u70b9, \u534a\u5f84\u4e3a$r$. \u63a5\u4e0b\u6765\u5b9a\u4e49\u540c\u4f26\u6620\u5c04$H: U \\times I \\to $$ U$\u5982\u4e0b\u6240\u793a,$$\\left\\{\\begin{matrix}<br \/>\nx, if \\ x \\in D. \\\\<br \/>\n(1 &#8211; t) \\cdot x + t \\cdot r \\cdot \\frac{x}{\\left \\| x \\right \\| }, otherwise.<br \/>\n\\end{matrix}\\right.$$\u6ce8\u610f\u5230\u4e0a\u8ff0\u6620\u5c04\u662fWell-Defined\u7684(\u540c\u4f26\u6620\u5c04$H$\u7684\u503c\u603b\u662f\u5904\u4e8e\u51f8\u96c6$U$\u4e2d\u7684), \u540c\u65f6\u4e5f\u662f\u8fde\u7eed\u7684. \u4e0a\u8ff0\u6620\u5c04\u4e5f\u7ed9\u51fa\u4e86\u51f8\u96c6$U$\u5230\u95ed\u7403$D$\u7684\u5f62\u53d8\u6536\u7f29, \u6545\u51f8\u96c6$U$\u4e0e\u95ed\u7403$D$\u540c\u4f26\u7b49\u4ef7. \u7279\u522b\u5730, \u4e0a\u8ff0\u6620\u5c04\u4e5f\u8bf1\u5bfc\u4e86$U \\backslash \\{ (0, $$ \\cdots, 0) \\}$\u5230\u95ed\u7403$D \\backslash \\{ (0, \\cdots ,0) \\}$\u7684\u540c\u4f26\u6620\u5c04. \u540c\u7406\u53ef\u5f97$E^n \\backslash \\{ (0, $$ \\cdots, 0) \\}$\u5230\u95ed\u7403$D \\backslash \\{ (0, \\cdots ,0) \\}$\u7684\u540c\u4f26\u6620\u5c04. \u6545\u547d\u9898\u5f97\u8bc1.<\/p>\n<p>4. <\/p>\n<blockquote><p>\u5f62\u53d8\u6536\u7f29\u548c\u540c\u80da\u4e0d\u662f\u4e00\u56de\u4e8b, \u540c\u80da\u662f\u53ef\u9006\u7684, \u5f62\u53d8\u6536\u7f29\u53ea\u80fd\u7f29\u8fc7\u53bb, \u6ca1\u6709\u6269\u5f20\u56de\u6765\u7684\u8bf4\u6cd5, \u56e0\u6b64\u662f\u5355\u5411\u7684.<\/p><\/blockquote>\n<p>5. \u8bbe\u8fde\u7eed\u6620\u5c04$f: D^2 \\to D^2$\u65e0\u4e0d\u52a8\u70b9, \u8fde\u7eed\u6620\u5c04$g: S^1 \\to S^1$, $x $$ \\mapsto \\frac{x &#8211; f(x)}{\\| x &#8211; f(x) \\|}$, \u6c42\u8bc1\u4efb\u53d6$x \\in S^1$, $g(x) \\ne -x$, \u6b64\u5904$\\| x \\|$\u8868\u793a\u4e8c\u7ef4\u5411\u91cf\u7684\u957f\u5ea6.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u4f7f\u7528\u53cd\u8bc1\u6cd5, \u82e5\u5b58\u5728$x_0 \\in S^1$, \u4f7f\u5f97$g(x_0) = -x_0$, \u5219\u6709$$\\frac{x_0 &#8211; f(x_0)}{\\| x_0 &#8211; f(x_|0) \\|} = -x_0, \\\\ \\Rightarrow (1 + \\| x_0 &#8211; f(x_0) \\|)x_0 = f(x_0),$$\u4e24\u8fb9\u53d6\u5176\u8303\u6570, \u5de6\u5f0f\u8303\u6570\u5fc5\u5927\u4e8e1, \u800c\u53f3\u5f0f\u8303\u6570\u5fc5\u4e0d\u5927\u4e8e1($\\because \\| x_0 \\| = $$ 1$, $f(x_0) \\in<br \/>\n$$ D^2$, $\\| f(x_0) \\| \\le 1$), \u5f97\u5230\u77db\u76fe, \u547d\u9898\u5f97\u8bc1.<br \/>\n$\\\\$ PS: \u8be5\u547d\u9898\u4e5f\u88ab\u5e94\u7528\u4e8e\u4e66\u4e0aP213 Brouwer\u4e0d\u52a8\u70b9\u5b9a\u7406\u7684\u8bc1\u660e\u5f53\u4e2d.<\/p>\n<p>6. \u5229\u7528Brouwer\u4e0d\u52a8\u70b9\u5b9a\u7406\u8bc1\u660e: \u4efb\u53d6\u4e00\u4e2a\u7531\u975e\u8d1f\u5b9e\u6570\u6784\u6210\u7684$3 \\times 3$\u53ef\u9006\u77e9\u9635, \u5b83\u4e00\u5b9a\u6709\u4e00\u4e2a\u6b63\u7684\u7279\u5f81\u503c.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u8bbe$A$\u662f\u4e00\u4e2a\u8fd9\u6837\u7684\u77e9\u9635. \u8bb0$M$\u4e3a\u5206\u91cf\u5168\u662f\u6b63\u6570\u5e76\u4e14\u957f\u5ea6\u4e3a1\u76843\u7ef4\u5217\u5411\u91cf\u6784\u6210\u7684\u7a7a\u95f4, \u5219\u53ef\u4ee5\u5b9a\u4e49$f: M \\to M, x \\mapsto \\frac{Ax}{\\| Ax \\|}$. \u800c$M \\cong $$ D^2$(\u60f3\u8c61\u4e00\u4e2a\u534a\u7403\u9762), \u56e0\u6b64$f$\u6709\u4e0d\u52a8\u70b9$x_0$. $\\| Ax_0 \\|$\u5c31\u662f\u4e00\u4e2a\u6b63\u7279\u5f81\u503c.<\/p>\n<p>7. \u8bbe$X$\u662f\u95ed\u5706\u76d8\u7684\u6536\u7f29\u6838, \u8bc1\u660e$X$\u4e0a\u7684\u4efb\u4f55\u8fde\u7eed\u81ea\u6620\u5c04\u4e5f\u4e00\u5b9a\u5b58\u5728\u4e0d\u52a8\u70b9.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u8bbe$r: D^2 \\to X$\u662f\u6536\u7f29\u6620\u5c04. \u4efb\u53d6\u8fde\u7eed\u6620\u5c04$f: X \\to X$, \u5219$f \\circ r $$ : D^2 \\to $$ D^2$\u4e00\u5b9a\u6709\u4e0d\u52a8\u70b9$p$, \u5373$(f \\circ r)(p) = p \\in X$. \u53c8$r(p) = p$, \u56e0\u6b64$p$\u4e5f\u662f$f$\u7684\u4e0d\u52a8\u70b9.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u8fd9\u5468\u5f00\u59cb, \u7b97\u662f\u6b63\u5f0f\u6295\u5165\u65b0\u9879\u76ee\u7ec4\u7684\u5de5\u4f5c\u4e2d\u4e86. \u76ee\u524d\u63a5\u89e6\u7684\u5de5\u4f5c\u5185\u5bb9\u4e2a\u4eba\u8fd8\u662f\u86ee\u611f\u5174\u8da3\u7684, \u5c24\u5176\u662f\u7a0b\u5e8f\u5316\u5173\u5361\u5236\u4f5c\u5de5\u5177 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/03\/20\/fundamental_groups_applications_mark\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u57fa\u672c\u7fa4\u53ca\u5176\u5e94\u7528\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\/1625"}],"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=1625"}],"version-history":[{"count":29,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1625\/revisions"}],"predecessor-version":[{"id":3611,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1625\/revisions\/3611"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1625"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1625"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1625"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}