{"id":1050,"date":"2021-08-29T23:17:09","date_gmt":"2021-08-29T15:17:09","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1050"},"modified":"2022-11-30T20:10:10","modified_gmt":"2022-11-30T12:10:10","slug":"homology","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2021\/08\/29\/homology\/","title":{"rendered":"\u540c\u8c03\u7684\u76f8\u5173\u7406\u89e3\u53ca\u8bc1\u660e"},"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\u7684\u5468\u672b\u662f\u5728\u6c55\u5c3e\u5ea6\u8fc7\u7684, \u5468\u4e94\u8bf7\u4e86\u4e2a\u5e74\u5047\u56de\u5bb6, \u5927\u6982\u5f85\u5230\u4e0b\u5468\u4e8c~ \u867d\u7136\u53ea\u6709\u77ed\u77ed\u56db\u4e94\u5929\u7684\u65f6\u95f4, \u4f46\u6211\u89c9\u5f97\u80fd\u56de\u5bb6\u966a\u966a\u7238\u5988, \u770b\u770b\u7237\u7237\u5976\u5976\u8fd8\u662f\u5f88\u503c\u5f97\u7684\u4e8b~ \u5f53\u7136, \u5de5\u4f5c\u8fd8\u662f\u8981\u505a\u7684, \u4f11\u606f? \u51e0\u4e4e\u4e0d\u5b58\u5728\u7684QAQ \u8fd9\u5468\u4e3b\u8981\u662f\u5728\u5f00\u59cb\u5b66\u4e60\u540c\u8c03, \u7b97\u662f\u6b63\u5f0f\u8fdb\u5165\u4ee3\u6570\u62d3\u6251\u7684\u8303\u7574\u4e86, \u672c\u6587\u4e3b\u8981\u662f\u8bb0\u5f55\u540c\u8c03\u7684\u76f8\u5173\u7406\u89e3\u53ca\u8bc1\u660e.<\/p>\n<p><!--more--><\/p>\n<p>\u540c\u8c03\u8fd9\u4e00\u5757\u5355\u770b\u4e66\u5176\u5b9e\u8fd8\u662f\u86ee\u96be\u7406\u89e3\u7684, \u6240\u4ee5\u8fd8\u53c2\u8003\u4e86\u5e84\u6653\u6ce2\u8001\u5e08\u76842\u4e2aB\u7ad9\u89c6\u9891:<br \/>\n1. <a href=\"https:\/\/www.bilibili.com\/video\/BV1P7411N7fW?p=56\">(55)\u5355\u7eaf\u540c\u8c03\u7684\u5b9a\u4e49<\/a><br \/>\n2. <a href=\"https:\/\/www.bilibili.com\/video\/BV1P7411N7fW?p=57\">(56)\u5355\u7eaf\u540c\u8c03\u8ba1\u7b97\u4e3e\u4f8b<\/a><\/p>\n<p><strong>1. \u95ed\u94fe\u7fa4, \u8fb9\u7f18\u94fe\u7fa4\u4e0e\u540c\u8c03<\/strong><\/p>\n<p>\u4e66\u4e0a\u7684\u5b9a\u4e49\u5176\u5b9e\u6709\u70b9\u62d7\u53e3, \u4e0d\u59a8\u6765\u770b\u770b\u4e0b\u9762\u8fd9\u79cd\u5b9a\u4e49(\u867d\u7136\u5199\u6cd5\u4e0a\u4e0d\u7b97\u5f88\u4e25\u8c28), \u4e2a\u4eba\u89c9\u5f97\u66f4\u5bb9\u6613\u7406\u89e3.<\/p>\n<p><strong>\u5b9a\u4e491<\/strong> \u8bbe$K$\u4e3a\u5355\u7eaf\u590d\u5f62, $n \\in \\mathbb{N}$, \u5b9a\u4e49$$Z_n(K) = Ker(C_n(K) \\overset{\\partial_n}{\\longrightarrow}C_{n &#8211; 1}(K)), \\\\ B_n(K) = Im(C_{n + 1}(K)\\overset{\\partial_{n + 1}}{\\longrightarrow}C_n(K)).$$\u7279\u522b\u5730, \u5f53$n = 0$\u65f6\u8bb0$Z_0(K) = C_0(K)$. \u4e14\u7531\u4e0a\u8ff0\u5b9a\u4e49\u6613\u77e5, $B_n( $$ K) \\subseteq Z_n( $$ K)$.<\/p>\n<p><strong>\u5b9a\u4e492<\/strong> $\\forall c_1, c_2 \\in Z_n(K)$, \u79f0$c_1, c_2$\u662f\u540c\u8c03(homologous) \u7684, \u82e5$\\exists d $$ \\in C_{n + 1}(K)$, s.t. $\\partial_{n + 1}(d) = c_2 &#8211; c_1$.<\/p>\n<p>\u56e0\u6b64, \u4ece\u5b9a\u4e49\u4e0a\u6765\u8bb2,<br \/>\n$\\\\$ a) \u82e5\u8981\u5224\u65ad\u4e00\u6761$n$\u7ef4\u94fe\u662f\u5426\u5728\u95ed\u94fe\u7fa4$Z_n(K)$\u4e2d, \u4ec5\u9700\u8981\u5224\u65ad\u5176\u8fb9\u754c\u662f\u5426\u4e3a0\u5373\u53ef;<br \/>\n$\\\\$ b) \u82e5\u8981\u5224\u65ad\u4e24\u4e2a\u5904\u4e8e\u95ed\u94fe\u7fa4$Z_n(K)$\u4e2d\u7684\u5143\u7d20\u662f\u5426\u540c\u8c03, \u4ec5\u9700\u8981\u627e\u5230\u4e00\u4e2a\u5904\u4e8e\u66f4\u9ad8\u7ef4\u5ea6\u7684$C_{n+1}(K)$\u4e2d\u7684\u94fe, \u4f7f\u5f97\u5176\u8fb9\u754c\u6070\u4e3a\u8fd9\u4e24\u4e2a\u5143\u7d20\u7684\u5dee(\u8fde\u7ebf) \u5373\u53ef.<\/p>\n<p><strong>2. $H_0(K)$\u76f8\u5173\u547d\u9898\u8bc1\u660e<\/strong><\/p>\n<p><strong>\u547d\u9898<\/strong> \u8bbe$K$\u4e3a\u4e00\u4e2a\u5355\u7eaf\u590d\u5f62, $H_0(K)$\u4e3a\u4e00\u4e2a\u81ea\u7531Abel\u7fa4, \u4e14\u5176\u79e9\u6070\u4e3a$|K|$\u7684\u8fde\u901a\u5206\u652f\u7684\u4e2a\u6570.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> <em>\u4e0b\u8ff0\u8bc1\u660e\u53c2\u8003\u81ea\u5e84\u6653\u6ce2\u8001\u5e08\u7684\u8bc1\u660e, \u5176\u4e3a\u4e00\u4e2a\u4e0d\u4e25\u683c\u7684\u8bc1\u660e.<\/em><br \/>\n$\\\\$ \u82e5$v$\u4e0e$w$\u4e3a$K$\u7684\u4e24\u4e2a0\u7ef4\u5355\u5f62, \u4e14\u843d\u5728$|K|$\u7684\u540c\u4e00\u4e2a\u8fde\u901a\u5206\u652f\u5185, \u5219\u5b58\u5728$K$\u4e2d\u7684\u4e00\u5217\u9876\u70b9$v = a_0, a_1, \\cdots, a_m = w$, s.t. $\\forall i, a_i, a_{i + 1}$\u4e3a$K$\u4e2d\u76841\u7ef4\u5355\u5f62.<br \/>\n$\\\\$ \u56e0\u6b64, \u4ee4$$c = (a_0, a_1) + (a_1, a_2) + \\cdots + (a_{n &#8211; 1}, a_n) \\in C_1(K),$$\u5219$\\partial_{1}(c) = a_1 &#8211; a_0 + a_2 &#8211; a_1 + \\cdots + a_n &#8211; a_{n &#8211; 1} = v &#8211; w \\Rightarrow$\u843d\u5728$|K|$\u4e2d\u540c\u4e00\u8fde\u901a\u5206\u652f\u7684\u4efb\u610f\u4e24\u4e2a0\u7ef4\u5355\u5f62\u90fd\u662f\u540c\u8c03\u7684.<br \/>\n$\\\\$ \u8bbe$\\{C_\\alpha|\\alpha \\in I\\}$\u4e3a$|K|$\u7684\u8fde\u901a\u5206\u652f\u5168\u4f53, $\\forall \\alpha \\in I$, \u53d6\u5b9a0\u7ef4\u5355\u5f62$v_\\alpha \\in $$ C_\\alpha$, \u5219\u6709\u6ee1\u540c\u6001$$\\underset{\\alpha \\in I}{\\oplus } \\mathbb{Z}v_\\alpha \\to H_0(K) \\\\ v_\\alpha \\longmapsto [v_\\alpha].$$\u5176\u4e2d, $\\underset{\\alpha \\in I}{\\oplus } \\mathbb{Z}v_\\alpha \\subseteq Z_0(K)$. \u6545\u53c8\u7531\u7fa4\u540c\u6001\u7b2cXX\u57fa\u672c\u5b9a\u7406(\u6211\u4e5f\u4e0d\u6e05\u695a\u7528\u7684\u54ea\u4e2a\u5b9a\u7406, \u62bd\u4ee3\u6e23\u6e23QAQ) \u53ef\u5f97$$H_0(K) = Z_0(K) \/ B_0(K) = C_0(K) \/ B_0(K) \\\\ = \\underset{\\alpha \\in I}{\\oplus} \\mathbb{Z} v_\\alpha \/ (\\underset{\\alpha \\in I}{\\oplus} \\mathbb{Z} v_\\alpha \\cup B_0(K)).$$\u63a5\u4e0b\u6765\u8bc1$\\underset{\\alpha \\in I}{\\oplus} \\mathbb{Z} v_\\alpha \\cup B_0(K) = 0$. $\\forall c \\in \\underset{\\alpha \\in I}{\\oplus} \\mathbb{Z} v_\\alpha \\cup B_0(K)$, \u53ef\u8bbe$$c = \\sum_\\alpha n_\\alpha v_\\alpha,$$\u5176\u4e2d$v_\\alpha \\in C_\\alpha$, $n_\\alpha$\u6784\u6210\u7684\u96c6\u5408\u4e2d\u53ea\u6709\u6709\u9650\u4e2a\u975e0\u5143\u7d20. \u53c8\u7531\u8fb9\u7f18\u94fe\u5b9a\u4e49\u53ef\u77e5$\\exists d \\in $$ C_1(K)$, s.t. $c = \\partial_1(d)$. \u4e0d\u59a8\u8bbe$d = \\sum_\\alpha m_\\alpha d_\\alpha$, \u5176\u4e2d$d_\\alpha \\in C_\\alpha$.<br \/>\n$\\\\$ \u6545$$c = \\partial_1(d) \\Leftrightarrow \\sum_\\alpha n_\\alpha v_\\alpha = \\sum_\\alpha m_\\alpha \\partial_1(d_\\alpha) \\\\ \\Leftrightarrow \\forall \\alpha \\in I, n_\\alpha v_\\alpha = m_\\alpha \\partial_1(d_\\alpha),$$\u5176\u4e2d$m_\\alpha \\partial_1(d_\\alpha)$\u4e3a$m_\\alpha(w_{\\alpha1} &#8211; w_{\\alpha2})$(\u4e24\u4e2a0\u7ef4\u5355\u5f62\u7684\u5dee\u4e58\u4e0a\u6574\u6570\u7cfb\u6570) \u7684\u5f62\u5f0f, \u6545$$n_\\alpha = 0, \\forall \\alpha \\in I \\Rightarrow c = 0,$$\u4ece\u800c$H_0(K) \\cong \\underset{\\alpha \\in I}{\\oplus } \\mathbb{Z} v_\\alpha.$<\/p>\n<p>\u4e0a\u8ff0\u547d\u9898\u76f4\u89c2\u4e0a\u6765\u8bb2, \u53ef\u4ee5\u7406\u89e3\u4e3a: \u8bbe$K$\u662f\u62d3\u6251\u7a7a\u95f4$M$\u7684\u4e00\u4e2a\u6709\u9650\u5355\u7eaf\u5256\u5206, $A, $$ B$\u662f$K$\u7684\u4e24\u4e2a\u9876\u70b9, \u5219$[A]$\u548c$[B]$\u540c\u8c03\u5f53\u4e14\u4ec5\u5f53$A, B$\u4e4b\u95f4\u6709\u4e00\u7ef4\u5355\u5f62\u6784\u6210\u7684\u6298\u7ebf\u76f8\u8fde, \u5373\u5b83\u4eec\u5728$M$\u4e2d\u7684\u540c\u4e00\u4e2a\u9053\u8def\u5206\u652f\u4e2d. \u7531\u6b64\u4e0d\u96be\u9a8c\u8bc1, \u5728\u6bcf\u4e2a\u9053\u8def\u5206\u652f\u4e2d\u53d6\u4e00\u4e2a\u9876\u70b9, \u5219\u8fd9\u4e9b\u9876\u70b9\u5bf9\u5e94\u7684\u96f6\u7ef4\u540c\u8c03\u7c7b\u5c31\u6784\u6210$H_0(K)$\u7684\u4e00\u7ec4\u57fa. \u4e5f\u5c31\u662f\u8bf4, Betti\u6570$b_0(K)$\u5c31\u662f$M$\u7684\u9053\u8def\u5206\u652f\u6570.<br \/>\n$\\\\$ \u800c\u63a8\u5e7f\u5230\u4efb\u610f$n$\u7ef4\u60c5\u5f62\u65f6, \u540c\u8c03\u5c31\u662f\u628a\u4e24\u4e2a&#8221;\u7aef\u53e3&#8221;\u7406\u89e3\u4e3a$n$\u7ef4\u5b9a\u5411\u5355\u5f62\u7684\u7ebf\u6027\u7ec4\u5408, \u628a&#8221;\u7ba1\u5b50&#8221;\u7406\u89e3\u6210$n + 1$\u7ef4\u5b9a\u5411\u5355\u5f62\u7684\u7ebf\u6027\u7ec4\u5408, \u800c\u5982\u679c\u4e24\u4e2a&#8221;\u7aef\u53e3&#8221;\u80fd\u88ab\u8fde\u63a5\u8d77\u6765, \u5c31\u79f0\u5b83\u4eec\u540c\u8c03.<\/p>\n<p><strong>3. \u540c\u8c03\u7fa4\u4e0e\u540c\u4f26\u7fa4<\/strong><\/p>\n<p>\u4e00\u7ef4\u540c\u8c03\u7fa4\u662f\u57fa\u672c\u7fa4\u7684Abel\u5316. \u73af\u8def$\\gamma$\u5728\u540c\u4f26\u7fa4\u4e2d\u4e0d\u662f\u5355\u4f4d\u5143, \u4f46\u5728\u540c\u8c03\u7fa4\u4e2d\u5374\u662f\u5355\u4f4d\u5143. \u51e0\u4f55\u4e0a\u770b, $\\gamma$\u5c06\u66f2\u9762\u5206\u6210\u4e24\u4e2a\u8fde\u901a\u5206\u652f, $\\gamma$\u662f\u5176\u4e2d\u4e00\u4e2a\u5206\u652f\u7684\u8fb9\u7f18. \u5728\u540c\u8c03\u7fa4\u4e2d, \u8fb9\u7f18\u73af\u8def\u88ab\u89c6\u4e3a\u5355\u4f4d\u5143.<br \/>\n$\\\\$ \u540c\u8c03\u7fa4\u6982\u5ff5\u7684\u8981\u4e49\u5728\u4e8e: \u8fb9\u7684\u8fb9\u4e3a\u7a7a, \u5708\u548c\u8fb9\u7684\u5dee\u522b\u5c31\u662f\u540c\u8c03.<\/p>\n<p>PS: \u6709\u4e00\u4e2a\u975e\u5e38\u5f3a\u5927\u7684\u5b9a\u7406: Betti\u6570\u662f\u62d3\u6251\u4e0d\u53d8\u91cf, \u5373\u5982\u679c\u62d3\u6251\u7a7a\u95f4$M$\u6709\u4e24\u4e2a\u6709\u9650\u5355\u7eaf\u5256\u5206$K_1,K_2$, \u5219$b_n(K_1)=b_n(K_2)$. \u8fd9\u4e2a\u5b9a\u7406\u8bf4\u660e\u4e86Betti\u6570\u662fWell-Defined\u7684, \u5e76\u4e14\u53ef\u4ee5\u5e2e\u52a9\u6211\u4eec\u628a\u62d3\u6251\u7a7a\u95f4\u4e0a\u7684\u95ee\u9898\u76f4\u63a5\u8f6c\u5316\u4e3a\u6709\u9650\u5355\u7eaf\u5256\u5206\u4e0a\u7684\u95ee\u9898.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u8fd9\u5468\u7684\u5468\u672b\u662f\u5728\u6c55\u5c3e\u5ea6\u8fc7\u7684, \u5468\u4e94\u8bf7\u4e86\u4e2a\u5e74\u5047\u56de\u5bb6, \u5927\u6982\u5f85\u5230\u4e0b\u5468\u4e8c~ \u867d\u7136\u53ea\u6709\u77ed\u77ed\u56db\u4e94\u5929\u7684\u65f6\u95f4, \u4f46\u6211\u89c9\u5f97\u80fd\u56de\u5bb6 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/08\/29\/homology\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u540c\u8c03\u7684\u76f8\u5173\u7406\u89e3\u53ca\u8bc1\u660e<\/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\/1050"}],"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=1050"}],"version-history":[{"count":76,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1050\/revisions"}],"predecessor-version":[{"id":2142,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1050\/revisions\/2142"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1050"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1050"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1050"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}