{"id":3223,"date":"2023-11-01T00:01:06","date_gmt":"2023-10-31T16:01:06","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=3223"},"modified":"2025-02-26T10:57:58","modified_gmt":"2025-02-26T02:57:58","slug":"obtaining_mathbb_cpn_as_an_identification_space_of_d2n","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2023\/11\/01\/obtaining_mathbb_cpn_as_an_identification_space_of_d2n\/","title":{"rendered":"\u8bc1\u660e\u590d\u5c04\u5f71\u5e73\u9762CP^n\u4e3a\u5355\u4f4d\u5706\u76d8D^2n\u7684\u5546\u7a7a\u95f4"},"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>\u6700\u8fd1\u901a\u8fc7\u9605\u8bfbAllen Hatche\u7684\u300aAlgebraic Topology\u300b\u6765\u91cd\u65b0\u590d\u4e60\u4ee3\u6570\u62d3\u6251, \u5728\u590d\u5c04\u5f71\u5e73\u9762$\\mathbb{C} P^n$\u5c0f\u8282\u7684\u8bba\u8ff0\u4e0a\u5361\u4e86\u633a\u4e45\u2026\u2026 \u8c28\u4ee5\u672c\u6587\u8bb0\u5f55\u4e00\u4e2a\u5173\u952e\u547d\u9898\u7684\u8bc1\u660e, \u52a0\u6df1\u5bf9\u590d\u5c04\u5f71\u5e73\u9762$\\mathbb{C} P^n$\u7684\u7406\u89e3.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/math.stackexchange.com\/questions\/4164696\/obtaining-mathbb-cpn-as-an-identification-space-of-d2n\">Obtaining CPn  as an identification space of D2n.<\/a><br \/>\n2. <a href=\"https:\/\/math.stackexchange.com\/questions\/1493993\/universal-property-in-quotient-topology\">universal property in quotient topology<\/a><\/p>\n<p><strong>\u547d\u9898<\/strong> \u8bc1\u660e$\\mathbb{C} P^n = D^{2n} \/ \\sim$, \u5176\u4e2d, $x \\sim y$\u5f53\u4e14\u4ec5\u5f53$x = y$\u6216$q(x) = $$ q(y)$($q$\u4e3a\u5546\u6620\u5c04: $S^{2n + 1} \\overset{q}{\\rightarrow} \\mathbb{C} P^n$).<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u6613\u77e5, \u5b58\u5728\u4e00\u4e2a\u5d4c\u5165\u6620\u5c04,$$\\ell: D^{2n} \\to \\overline{S}^{2n + 1} = \\{ (z_1, \\cdots, z_{n + 1}) \\in S^{2n + 1} | z_{n + 1} \\in \\mathbb{R}, z_{n + 1} \\ge 0 \\}, \\\\<br \/>\n(z_1, \\cdots, z_n) \\mapsto (z_1, \\cdots, z_n, \\sqrt{1 &#8211; \\sum_{i = 1}^{n} |z_i|^2}).$$\u5206\u6790$D^{2n}$\u4e0a\u7684\u7b49\u4ef7\u5173\u7cfb. \u8be5\u7b49\u4ef7\u5173\u7cfb\u7684\u5b9a\u4e49\u5f62\u5f0f$q(x) = q(y)$\u5e76\u4e0d\u662f\u5341\u5206\u51c6\u786e, \u6211\u4eec\u5e94\u8be5\u628a\u5546\u6620\u5c04$q$\u7406\u89e3\u4e3a$$q: S^{2n &#8211; 1} = \\partial D^{2n} \\to \\mathbb{C} P^{n &#8211; 1}$$\u800c\u975e$q: S^{2n + 1} \\to \\mathbb{C} P^{n}$. \u540e\u8005\u9700\u8981\u5206\u522b\u8003\u8651\u70b9$\\xi \\in D^{2n}$\u4e0e\u70b9$\\xi&#8217; \\in $$ S^{2n + 1}$, \u5176\u4e2d, $\\xi&#8217; = \\ell(\\xi)$. \u8fd9\u6837\u4e00\u6765, $x \\sim y$\u7684\u5b9a\u4e49\u4f1a\u53d8\u5f97\u5341\u5206\u6666\u6da9\u62bd\u8c61, \u56e0\u4e3a\u5c06\u5546\u6620\u5c04$q$\u4f5c\u7528\u4e8e$x, y \\in D^{2n}$\u4e4b\u524d\u9700\u8981\u8fdb\u884c\u4e00\u6b21\u5d4c\u5165\u6620\u5c04. \u56e0\u6b64, \u6211\u4eec\u9700\u8981\u4f7f\u7528\u524d\u8005\u4f5c\u4e3a\u5546\u6620\u5c04$q$\u7684\u5b9a\u4e49,$$q: S^{2n &#8211; 1} \\to \\mathbb{C} P^{n &#8211; 1},$$\u8be5\u5546\u6620\u5c04\u4ec5\u4f5c\u7528\u4e8e$D^{2n}$\u7684\u8fb9\u754c\u70b9, \u4ece\u800c$x \\sim y$\u7684\u5b9a\u4e49\u4e5f\u5c06\u53d8\u5f97\u6e05\u6670\u660e\u4e86.<br \/>\n$\\\\$ \u63a5\u4e0b\u6765\u8bc1\u660e\u4e0a\u8ff0\u5546\u6620\u5c04$q$\u7684\u4e24\u79cd\u5b9a\u4e49\u5f62\u5f0f\u662f\u7b49\u4ef7\u7684. \u4e0d\u59a8\u8bb0\u5546\u6620\u5c04$q_m: S^{2m + 1} $$ \\to \\mathbb{C} P^{m}$, $\\lambda \\in S^1$\u5728$$S^{2m + 1} = \\{ (z_1, \\cdots, z_{m + 1}) \\in \\mathbb{C}^{m + 1} |  {\\textstyle \\sum_{i = 1}^{m + 1}} |z_i|^2 = 1 \\}$$\u4e0a\u7684\u4f5c\u7528\u4e3a$$\\lambda(z_1, \\cdots, z_{m + 1}) = (\\lambda z_1, \\cdots, \\lambda z_{m + 1}),$$\u5219\u6211\u4eec\u9700\u8bc1: $x \\sim y$(i.e. $x = y$\u6216$q_{n &#8211; 1}(x) = q_{n &#8211; 1}(y)$) \u5f53\u4e14\u4ec5\u5f53$q_n(\\ell(x)) =<br \/>\n$$ q_n(\\ell(y))$.<br \/>\n$\\\\$ \u8003\u8651$x \\in (u_1, \\cdots, u_n) \\in S^{2n &#8211; 1}$, $y = (v_1, \\cdots, v_n) \\in S^{2n &#8211; 1}$.<br \/>\n$\\\\$ \u4ee4$x \\sim y$.\u82e5$x \\sim y$, \u5219\u663e\u7136\u6709$q_n(\\ell(x)) = q_n(\\ell(y))$. \u6545\u4e0d\u59a8\u5047\u8bbe$q_{n &#8211; 1}(x) = $$ q_{n &#8211; 1}(y)$, \u5373\u5bf9\u4e8e$x, y \\in S^{2n &#8211; 1}$, \u5b58\u5728$\\lambda \\in S^1$, s.t. $(\\lambda u_1, $$ \\cdots, \\lambda u_n) = (v_1, \\cdots, $$ v_n)$. \u53c8\u7531$x, y \\in S^{2n &#8211; 1}$\u53ef\u77e5$\\ell(x) = (u_1, $$ \\cdots, u_n, 0)$, $\\ell(y) = (v_1, \\cdots, v_n, 0)$, \u5219$$\\lambda \\ell(x) = (\\lambda u_1, \\cdots, \\lambda u_n, 0) = (v_1, \\cdots, v_n, 0) = \\ell(y),$$\u4ece\u800c\u6211\u4eec\u6709$q_n(\\ell(x)) = q_n(\\ell(y))$.<br \/>\n$\\\\$ \u53cd\u8fc7\u6765, \u4ee4$q_n(\\ell(x)) = q_n(\\ell(y))$, \u5219\u5b58\u5728$\\lambda \\in S^1$, s.t. $\\lambda \\ell(x) = \\ell(y)$, i.e. $\\lambda x $$ = y$\u4e14$$\\lambda \\cdot \\sqrt{1 &#8211; \\sum_{i = 1}^n |u_i|^2} = \\sqrt{1 &#8211; \\sum_{i = 1}^n |v_i|^2}.$$\u7531\u4e8e\u5e73\u65b9\u6839\u603b\u4e3a\u4e00\u4e2a\u975e\u8d1f\u5b9e\u6570($\\sqrt{-1} = i$?), \u6545\u4e0a\u5f0f\u6210\u7acb\u5f53\u4e14\u4ec5\u5f53$\\lambda = 1$\u6216$\\sqrt{1 &#8211; \\sum_{i = 1}^n |u_i|^2} = \\sqrt{1 &#8211; \\sum_{i = 1}^n |v_i|^2} = 0$. \u5f53$\\lambda = 1$\u65f6$\\ell(x) = \\ell(y)$, \u4ece\u800c$x = y$; \u800c\u5bf9\u4e8e\u7b2c\u4e8c\u79cd\u60c5\u5f62, \u6211\u4eec\u6709$x, y \\in S^{2n &#8211; 1}$, $\\lambda x = y$, \u4ece\u800c$q_{n &#8211; 1}(x) = $$ q_{n &#8211; 1}(y)$.<br \/>\n$\\\\$ \u518d\u6765\u8bc1\u660e$\\ell(D^{2n}) = \\overline{S}^{2n + 1}$. \u9700\u8981\u6ce8\u610f\u7684\u662f, $D^{2n}$\u5728$\\ell$\u4e0b\u7684\u50cf$\\overline{S}^{2n + 1}$\u5e76\u975e$S^{2n + 1}$\u7684\u4e0a\u534a\u7403, $S^{2n + 1}$\u7684\u4e0a\u534a\u7403\u4e3a$D^{2n + 1}$\u7684\u62f7\u8d1d. \u6613\u77e5, $\\ell(D^{2n}) \\subset $$ \\overline{S}^{2n + 1}$. \u53cd\u8fc7\u6765, \u82e5$(z_1, \\cdot, z_{n + 1}) \\in \\overline{S}^{2n + 1}$, \u5219$z_{n + 1} = |z_{n + 1}| = $$ \\sqrt{1 &#8211; \\sum_{i = 1}^n |z_i|^2}$, \u6545$(z_1, \\cdots, $$ z_{n + 1}) = \\ell((z_1, \\cdots, z_n))$.<br \/>\n$\\\\$ \u53c8$q_n(\\ell(D^{2n})) = \\mathbb{C} P^n$. \u4ee4$\\xi = (z_1, \\cdots, z_{n + 1}) \\in S^{2n + 1}$. \u82e5$z_{n + 1} $$ = 0$, \u5219$x = (z_1, \\cdots, z_n) \\in S^{2n &#8211; 1}$, $\\xi = \\ell(x)$; \u82e5$z_{n + 1} \\ne 0$, \u5219$\\lambda = $$ \\frac{\\overline{z_{n + 1}}}{|z_{n + 1}|} \\in S^1$, $\\lambda \\xi = (z_1&#8242;, \\cdots, z_n&#8217;, |z_{n + 1}|)$. \u7531\u6b64\u53ef\u5f97, $\\lambda \\xi = \\ell(x)$, \u5176\u4e2d, $x = (z_1&#8242;, \\cdots, $$ z_n&#8217;) \\in D^{2n}$, $q_n(\\xi) = q_n(\\ell(x)) \\in q_n(\\ell(D^{2n}))$. \u4ece\u800c, \u6211\u4eec\u6709$\\ell(D^{2n}) = $$ \\overline{S}^{2n + 1}$.<br \/>\n$\\\\$ \u7efc\u4e0a\u6240\u8ff0, $\\mathbb{C} P^n = D^{2n} \/ \\sim$. \u4ece\u4e0a\u8ff0\u8bc1\u660e\u4e2d\u4e5f\u53ef\u4ee5\u770b\u51fa, $\\mathbb{C} P^n$\u53ef\u7531$\\mathbb{C} P^{n &#8211; 1}$\u901a\u8fc7$q_{n &#8211; 1}: S^{2n &#8211; 1} \\to \\mathbb{C} P^{n &#8211; 1}$\u7c98\u5408\u4e00\u4e2a$2n$-Cell$D^{2n}$\u5f97\u5230.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6700\u8fd1\u901a\u8fc7\u9605\u8bfbAllen Hatche\u7684\u300aAlgebraic Topology\u300b\u6765\u91cd\u65b0\u590d\u4e60\u4ee3\u6570\u62d3\u6251, \u5728\u590d\u5c04\u5f71\u5e73 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2023\/11\/01\/obtaining_mathbb_cpn_as_an_identification_space_of_d2n\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u8bc1\u660e\u590d\u5c04\u5f71\u5e73\u9762CP^n\u4e3a\u5355\u4f4d\u5706\u76d8D^2n\u7684\u5546\u7a7a\u95f4<\/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":[15,23],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3223"}],"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=3223"}],"version-history":[{"count":30,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3223\/revisions"}],"predecessor-version":[{"id":3599,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3223\/revisions\/3599"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=3223"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=3223"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=3223"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}