{"id":982,"date":"2021-08-13T12:52:56","date_gmt":"2021-08-13T04:52:56","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=982"},"modified":"2022-10-17T20:09:38","modified_gmt":"2022-10-17T12:09:38","slug":"closed_surface_connected_sum","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2021\/08\/13\/closed_surface_connected_sum\/","title":{"rendered":"\u95ed\u66f2\u9762\u7684\u8fde\u901a\u548c\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>\u5927\u6982\u4e09\u5468\u6ca1\u6709\u4f53\u9a8c\u5230\u5728\u4e0a\u5348\u5b85\u5bb6\u91cc\u4eab\u53d7\u64c2\u8336\u7684\u611f\u89c9\u4e86, \u4e0b\u5348\u5c31\u8981\u52a8\u8eab\u53bb\u5b81\u6ce2, \u6240\u4ee5\u8d81\u8fd8\u5728\u84b8\u5348\u9910\u7684\u7a7a\u9699\u8bb0\u5f55\u4e00\u4e0b\u95ed\u66f2\u9762\u8fde\u901a\u548c\u7684\u4e00\u4e9b\u77e5\u8bc6\u70b9\u53ca\u5176\u76f8\u5173\u4e60\u9898. \u4e00\u5f00\u59cb\u6ca1\u6709\u7406\u89e3\u4e66\u4e0a\u5bf9\u4e8e\u95ed\u66f2\u9762\u8fde\u901a\u548c\u7684\u76f4\u89c2\u89e3\u91ca: \u628a\u4e00\u4e2a\u591a\u8fb9\u5f62\u8868\u793a\u7684\u9876\u70b9&#8221;\u70b8\u5f00&#8221;\u53d8\u6210\u4e00\u6761\u4e0d\u4e0e\u4efb\u4f55\u5176\u5b83\u8fb9\u7c98\u5408\u7684\u65b0\u7684\u8fb9, \u76f8\u5f53\u4e8e\u5728<strong>\u95ed\u66f2\u9762<\/strong>\u4e0a\u6253\u4e86\u4e2a\u6d1e, \u8fd9\u4e2a\u6d1e\u7684\u8fb9\u7f18\u5c31\u662f\u90a3\u4e2a\u65b0\u5f97\u5230\u7684\u8fb9(\u628a\u4e24\u7aef\u7c98\u5408). \u56e0\u6b64\u8fde\u901a\u548c\u7684\u51e0\u4f55\u76f4\u89c2\u5c31\u662f\u5728\u4e24\u4e2a\u95ed\u66f2\u9762\u4e0a\u5404\u6316\u53bb\u4e00\u4e2a\u6d1e(\u540c\u80da\u4e8e\u5706\u76d8), \u7136\u540e\u628a\u4e24\u8005\u6cbf\u7740\u6d1e\u7684\u8fb9\u754c(\u5706\u5468) \u7c98\u5408.<\/p>\n<p><!--more--><\/p>\n<p>\u6ce8\u610f, \u4e0a\u8ff0\u76f4\u89c2\u89e3\u91ca\u7684\u91cd\u70b9\u5df2\u7ecf\u6807\u7c97\u5f3a\u8c03, \u662f\u5728\u95ed\u66f2\u9762\u4e0a\u6253\u6d1e\u800c\u975e\u5176\u591a\u8fb9\u5f62\u8868\u793a\u4e0a. \u56e0\u6b64\u5bb9\u6613\u5f97\u5230$v(M \\# N) = v(M) + v(N) &#8211; 1$, \u4ece\u800c\u7531\u4e66\u4e0aP141\u7684\u547d\u98983.4.1\u53ef\u5f97\u95ed\u66f2\u9762\u8fde\u901a\u548c\u7684Euler\u793a\u6027\u6570\u4e3a$\\chi(M \\# N) = \\chi(M) + \\chi(N) &#8211; 2$.<\/p>\n<p>PS: \u518d\u987a\u4fbf\u8bb0\u5f55\u4e00\u9053\u8bc1\u660e\u9898\u7684\u8bc1\u660e\u601d\u8def, \u9700\u8981\u8bc1\u660e$\\chi(S^n) = 1 + $$ (-1)^n$.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u53d6\u5b9a\u9876\u70b9\u96c6$V = \\{v_0, \\cdots, v_{n + 1}\\}$, $V$\u7684\u6240\u6709\u975e\u7a7a\u771f\u5b50\u96c6\u6784\u6210\u7684\u96c6\u5408\u65cf$K$\u5c31\u662f$S^n$\u7684\u6709\u9650\u5355\u7eaf\u5256\u5206. \u7531\u4e66\u4e0aP140\u7684\u5b9a\u74063.4.1\u77e5\u6211\u4eec\u4ec5\u9700\u8ba1\u7b97$\\chi(K)$\u5373\u53ef. \u53c8\u7531$n$\u7ef4\u6709\u9650\u590d\u5f62\u7684Euler\u793a\u6027\u6570\u5b9a\u4e49\u53ef\u5f97$$\\chi(K) = C^1_{n + 2} &#8211; C^2_{n + 2} + \\cdots + (-1)^{n}C^{n + 1}_{n + 2}.$$\u7531\u4e8c\u9879\u5f0f\u5b9a\u7406\u53ef\u77e5\u4e0a\u5f0f\u7ed3\u679c\u4e3a$$(1 + (-1))^{n + 2} &#8211; (-1)^{-1}C^0_{n + 2}-(-1)^{n + 1}C^{n + 2}_{n + 2} \\\\ = 0 + 1 + (-1)^{n + 2} = 1 + (-1)^n.$$\u547d\u9898\u5f97\u8bc1.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5927\u6982\u4e09\u5468\u6ca1\u6709\u4f53\u9a8c\u5230\u5728\u4e0a\u5348\u5b85\u5bb6\u91cc\u4eab\u53d7\u64c2\u8336\u7684\u611f\u89c9\u4e86, \u4e0b\u5348\u5c31\u8981\u52a8\u8eab\u53bb\u5b81\u6ce2, \u6240\u4ee5\u8d81\u8fd8\u5728\u84b8\u5348\u9910\u7684\u7a7a\u9699\u8bb0\u5f55\u4e00\u4e0b\u95ed\u66f2\u9762\u8fde &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/08\/13\/closed_surface_connected_sum\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u95ed\u66f2\u9762\u7684\u8fde\u901a\u548c\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\/982"}],"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=982"}],"version-history":[{"count":19,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/982\/revisions"}],"predecessor-version":[{"id":2522,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/982\/revisions\/2522"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=982"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=982"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=982"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}