{"id":2594,"date":"2022-11-20T19:18:30","date_gmt":"2022-11-20T11:18:30","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=2594"},"modified":"2025-02-26T11:14:35","modified_gmt":"2025-02-26T03:14:35","slug":"introductory_notes_computational_topology","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/11\/20\/introductory_notes_computational_topology\/","title":{"rendered":"\u8ba1\u7b97\u62d3\u6251\u5165\u95e8\u7b14\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>\u6700\u8fd1\u4e09\u5468\u7684\u5468\u672b\u90fd\u662f\u5728\u5b85\u5bb6, \u76ee\u524d\u5176\u5b9e\u5904\u4e8e\u6709\u70b9\u5c0f\u8ff7\u832b\u7684\u72b6\u6001, \u5728\u5b66\u5b8c\u4ee3\u6570\u62d3\u6251\u4ee5\u540e\u53c8\u6682\u65f6\u4e0d\u5927\u60f3\u5f00\u65b0\u7684\u5927\u5751. \u6070\u5de7\u6700\u8fd1\u53d1\u73b0\u4e00\u4e2a\u5b9d\u85cf\u89c6\u9891\u5408\u96c6: \u5f90\u9e4f\u7a0b\u8001\u5e08\u7684\u8ba1\u7b97\u62d3\u6251\u7cfb\u5217, \u76ee\u524d\u8be5\u7cfb\u5217\u7684\u89c6\u9891\u6570\u91cf\u4e0d\u591a(\u8c8c\u4f3c\u5df2\u7ecf\u505c\u6b62\u66f4\u65b0\u4e86\u2026\u2026 \u6709\u70b9\u53ef\u60dcQAQ), \u800c\u4e14\u8001\u5e08\u5bf9\u77e5\u8bc6\u70b9\u7684\u8bb2\u89e3\u975e\u5e38\u8be6\u7ec6\u6e05\u6670, \u9002\u5408\u5165\u95e8, \u4e8e\u662f\u4fbf\u770b\u5b8c\u4e86\u8fd9\u4e2a\u89c6\u9891\u5408\u96c6, \u5e76\u4ee5\u672c\u6587\u8bb0\u5f55\u4e0b\u4e0e\u540c\u8c03\u7fa4\u7684\u8ba1\u7b97\u76f8\u5173\u7684\u77e5\u8bc6\u70b9(\u6301\u7eed\u540c\u8c03\u76ee\u524d\u4e3b\u8981\u7528\u4e8e\u70b9\u4e91\u6570\u636e\u5206\u6790, \u6545\u6682\u4e0d\u6d89\u53ca).<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/www.bilibili.com\/video\/BV1vt4y1n7Vk\/\">\u8ba1\u7b97\u62d3\u6251<\/a><br \/>\n2. Benjamin A. Burton, Ryan Budney, William Pettersson, et al., Regina: Software for low-dimensional topology, http:\/\/regina-normal.github.io\/, 1999\u20132022.<br \/>\n3. <a href=\"https:\/\/www.bilibili.com\/video\/BV1P7411N7fW\/?p=57\">P57 (56)\u5355\u7eaf\u540c\u8c03\u8ba1\u7b97\u4e3e\u4f8b<\/a><br \/>\n4. <a href=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/PC3-4_solution.pdf\">PC3-4_solution<\/a><br \/>\n5. <a href=\"https:\/\/zhuanlan.zhihu.com\/p\/136244778\">Smith \u6807\u51c6\u578b\u4e0e\u540c\u8c03\u7fa4\u7684\u4e00\u822c\u8ba1\u7b97<\/a><br \/>\n6. <a href=\"https:\/\/eric-bunch.github.io\/blog\/calculating_homology_of_simplicial_complex\">Calculating Homology of a Simplicial Complex Using Smith Normal Form<\/a><br \/>\n7. <a href=\"https:\/\/math.stackexchange.com\/questions\/388682\/homology-of-m%C3%B6bius-strip\">Homology of M\u00f6bius Strip<\/a><\/p>\n<p>\u53ef\u4e09\u89d2\u5256\u5206\u7a7a\u95f4\u7684\u540c\u8c03\u7fa4\u53ef\u4ee5\u901a\u8fc7\u4ee3\u8868\u8fb9\u7f18\u540c\u6001\u7684\u77e9\u9635\u6765\u8ba1\u7b97. \u5b83\u7684\u65e2\u7ea6\u7248\u672c\u63d0\u4f9b\u4e86\u95ed\u94fe\u7fa4\u548c\u8fb9\u7f18\u94fe\u7fa4\u7684\u79e9, \u800c\u4e24\u8005\u7684\u5dee\u5c31\u7ed9\u51fa\u4e86Betti\u6570. \u6b64\u5916, \u4e0b\u6587\u4e2d\u5bf9\u4e8e\u540c\u8c03\u7fa4\u7684\u8ba1\u7b97, \u5747\u662f\u5728\u57df$\\mathbb{Z}_2$(\u6a212\u6574\u6570\u52a0\u6cd5\u7fa4, \u4e00\u822c\u53c8\u8bb0\u4e3a$\\mathbb{Z} \/ 2\\mathbb{Z}$) \u4e2d\u8fdb\u884c\u7684. \u610f\u5373, \u6211\u4eec\u53ea\u8003\u8651\u6240\u6709\u7cfb\u6570\u90fd\u662f\u6a212\u6574\u6570\u7684$n$\u7ef4\u94fe(\u5f62\u5f0f\u7ebf\u6027\u7ec4\u5408), \u6b64\u65f6, \u6240\u6709\u7684$n$\u7ef4\u94fe\u5e76\u4e0d\u6784\u6210\u7ebf\u6027\u7a7a\u95f4, \u800c\u53ea\u662f\u6784\u6210\u4e00\u4e2a\u4ea4\u6362\u7fa4$C_n(K ; \\mathbb{Z}_2)$. \u6574\u7cfb\u6570\u7684$n$\u7ef4\u95ed\u94fe\u6784\u6210\u5b83\u7684\u4e00\u4e2a\u5b50\u7fa4$Z_n(K ; \\mathbb{Z}_2 $$ )$, \u800c\u6574\u7cfb\u6570$n$\u7ef4\u8fb9\u7f18\u94fe\u8fdb\u4e00\u6b65\u6784\u6210\u8fd9\u4e2a\u5b50\u7fa4\u91cc\u7684\u4e00\u4e2a\u5b50\u7fa4$B_n( $$ K ; \\mathbb{Z}_2)$. \u4ea4\u6362\u7fa4\u7684\u5b50\u7fa4\u90fd\u662f\u6b63\u89c4\u5b50\u7fa4, \u56e0\u6b64\u53ef\u4ee5\u4f5c\u5546\u7fa4. \u590d\u5f62$K$\u7684$n$\u7ef4\u6574\u7cfb\u6570\u540c\u8c03\u7fa4\u7684\u5b9a\u4e49$$H_n(K ; \\mathbb{Z}_2) = Z_n(K ; \\mathbb{Z}_2) \/ B_n(K ; \\mathbb{Z}_2) = Ker \\partial_n \/ Im \\partial_{n + 1}$$\u4e5f\u7531\u6b64\u800c\u6765. \u9700\u8981\u6ce8\u610f\u7684\u662f, \u672c\u6587\u8ba8\u8bba\u7684\u540c\u8c03\u7fa4\u7684\u8ba1\u7b97\u5747\u662f\u5728\u53ef\u4e09\u89d2\u5256\u5206\u7a7a\u95f4\u4e2d\u8fdb\u884c\u7684; \u6b64\u65f6, \u6211\u4eec\u901a\u5e38\u4e0d\u5bf9\u62d3\u6251\u7a7a\u95f4\u7684\u540c\u8c03\u7fa4\u4e0e\u5176\u5bf9\u5e94\u7684\u5355\u7eaf\u590d\u5f62\u7684\u540c\u8c03\u7fa4\u52a0\u4ee5\u533a\u5206, \u8fd9\u662f\u7531\u4e66\u4e0aP152\u5b9a\u74063.6.1\u6240\u4fdd\u8bc1\u7684.<\/p>\n<blockquote><p>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$b_n(K_1) = b_n(K_2)$.<\/p><\/blockquote>\n<p>\u6b64\u5916, \u6211\u4eec\u5c06$n$\u4e2a$\\mathbb{Z}$\u7684\u76f4\u548c\u7b80\u8bb0\u4e3a$\\mathbb{Z}^n$, \u4e0b\u6587\u4e0d\u518d\u8d58\u8ff0.<\/p>\n<p><strong>1. \u6b27\u62c9-\u5e9e\u52a0\u83b1\u516c\u5f0f<\/strong><\/p>\n<p>\u5355\u7eaf\u590d\u5f62\u7684\u6b27\u62c9\u7279\u5f81\u6570\u662f\u6240\u6709\u7ef4\u5ea6\u7684\u5355\u5f62\u7684\u6570\u91cf\u7684\u4ea4\u9519\u548c. \u7c7b\u4f3c\u7684, \u7b2c$p$\u4e2a\u540c\u8c03\u7fa4\u7684\u79e9\u7b49\u4e8e\u7b2c$p$\u4e2a\u95ed\u94fe\u7fa4\u7684\u79e9\u51cf\u53bb\u7b2c$p$\u4e2a\u8fb9\u7f18\u94fe\u7fa4\u7684\u79e9. \u8bb0$n_p = rank C_p(K)$, $z_p $$ = rank Z_p = dim(Ker \\partial_p)$, $b_p = rank $$ B_p(K) = dim(Im \\partial_{p + 1})$, \u5219\u6709$$n_p = z_p + b_{p &#8211; 1}.$$\u8fd9\u4e2a\u662f\u7ebf\u6027\u4ee3\u6570\u91cc\u9762\u7684\u5b9a\u7406, \u4efb\u4f55\u7ebf\u6027\u6620\u5c04$f: U \\to V$, $U$\u7684\u7ef4\u5ea6\u7b49\u4e8e$f$\u7684\u6838\u7684\u7ef4\u5ea6\u52a0\u4e0a$f$\u7684\u50cf\u7684\u7ef4\u5ea6. \u6b27\u62c9\u7279\u5f81\u6570\u662f$n_p$\u7684\u4ea4\u9519\u548c, \u6240\u4ee5$$\\sum_{p \\ge 0} (-1)^p(z_p + b_{p &#8211; 1}) = \\sum_{p \\ge 0} (z_p &#8211; b_p).$$<strong>(\u6b27\u62c9-\u5e9e\u52a0\u83b1\u5b9a\u7406)<\/strong> \u4e00\u4e2a\u62d3\u6251\u7a7a\u95f4\u7684\u6b27\u62c9\u7279\u5f81\u6570\u662fBetti\u6570\u7684\u4ea4\u9519\u548c, \u5373$$\\chi = \\sum_{p \\ge 0} (-1)^p \\beta_p,$$\u5176\u4e2d, $\\beta_p = z_p &#8211; b_p = n_p &#8211; b_{p &#8211; 1} &#8211; b_p$. \u4ee5\u4e0a\u516c\u5f0f\u53c8\u79f0\u6b27\u62c9-\u5e9e\u52a0\u83b1\u516c\u5f0f(Euler-Poincare Formula).<\/p>\n<p><strong>2. \u8fb9\u7f18\u77e9\u9635<\/strong><\/p>\n<p>\u4e3a\u4e86\u8ba1\u7b97\u540c\u8c03\u7fa4, \u6211\u4eec\u9700\u8981\u4ece\u4e24\u4e2a\u5730\u65b9\u6574\u5408\u4fe1\u606f, \u4e00\u662f\u95ed\u94fe\u7fa4, \u4e8c\u662f\u8fb9\u7f18\u94fe\u7fa4. \u5047\u8bbe$K$\u662f\u4e00\u4e2a\u5355\u7eaf\u590d\u5f62. \u5b83\u7684\u7b2c$p$\u4e2a\u8fb9\u7f18\u77e9\u9635\u662f\u4ee5$p &#8211; 1$\u7ef4\u5355\u5f62\u4e3a\u884c, $p$\u7ef4\u5355\u5f62\u4e3a\u5217. \u4efb\u610f\u9009\u5b9a\u5355\u5f62\u7684\u987a\u5e8f(\u5373\u9009\u5b9a\u5b9a\u5411), \u5bf9\u4e8e\u6bcf\u4e2a\u7ef4\u5ea6, \u8be5\u77e9\u9635\u4e3a$\\partial_p = [a^j_i]$, $i$\u7684\u53d6\u503c\u4ece1\u5230$n_p$, \u5982\u679c\u7b2c$i$\u4e2a$p &#8211; 1$\u7ef4\u5355\u5f62\u662f\u7b2c$j$\u4e2a$p$\u7ef4\u5355\u5f62\u7684\u4e00\u4e2a\u9762, \u5219\u4ee4$a^j_i = 1$, \u5426\u5219\u4ee4$a^j_i = 0$. \u7ed9\u5b9a\u4e00\u4e2a$p$\u7ef4\u94fe$c = \\sum a_i \\sigma_i$, \u6211\u4eec\u6709$$\\partial_p c = \\begin{bmatrix}<br \/>\na^1_1 &#038; a^2_1 &#038; \\cdots &#038; a^{n_p}_1 \\\\<br \/>\na^1_2 &#038; a^2_2 &#038; \\cdots &#038; a^{n_p}_2 \\\\<br \/>\n\\vdots &#038; \\vdots &#038; \\ddots &#038; \\vdots \\\\<br \/>\na^1_{n_{p &#8211; 1}} &#038; a^2_{n_{p &#8211; 1}} &#038; \\cdots &#038; a^{n_p}_{n_{p &#8211; 1}}<br \/>\n\\end{bmatrix}\\begin{bmatrix}<br \/>\na_1 \\\\<br \/>\na_2 \\\\<br \/>\n\\vdots \\\\<br \/>\na_{n_p}<br \/>\n\\end{bmatrix}.$$\u603b\u7684\u6765\u8bf4, \u5217\u7684\u96c6\u5408\u8868\u793a\u4e00\u4e2a$p$\u7ef4\u94fe, \u8fd9\u4e9b\u5217\u7684\u6c42\u548c\u7ed9\u51fa\u4e86\u5b83\u7684\u8fb9\u7f18. \u540c\u7406, \u884c\u7684\u96c6\u5408\u4ee3\u8868\u4e86\u4e00\u4e2a$p &#8211; 1$\u7ef4\u94fe, \u8fd9\u4e9b\u884c\u7684\u6c42\u548c\u7ed9\u51fa\u4e86\u5b83\u7684\u4e0a\u8fb9\u7f18(Coboundary).<\/p>\n<p><strong>3. Smith\u6807\u51c6\u578b<\/strong><\/p>\n<p>\u4f7f\u7528\u521d\u7b49\u53d8\u6362, \u6211\u4eec\u53ef\u4ee5\u5c06\u7b2c$p$\u4e2a\u8fb9\u7f18\u77e9\u9635\u5316\u7b80\u4e3aSmith\u6807\u51c6\u578b(Smith Normal Form), \u4e5f\u5c31\u662f\u8bf4\u9664\u4e86\u5bf9\u89d2\u7ebf\u4e0a\u7684\u5143\u7d20\u662f1\u4ee5\u5916\u5176\u5b83\u90fd\u662f0(\u5bf9\u89d2\u7ebf\u4e0a\u7684\u5143\u7d20\u4e5f\u4e0d\u5168\u4e3a1, \u4f46\u524d$k$\u4e2a\u8981\u4e3a1). \u7531\u4e8e$n_p = rank $$ C_p(K)$\u662f\u7b2c$p$\u4e2a\u8fb9\u7f18\u77e9\u9635\u7684\u5217\u7684\u6570\u91cf, $n_p = b_{p &#8211; 1} + z_p$, \u6240\u4ee5\u6700\u5de6\u8fb9\u7684$b_{p &#8211; 1}$\u4e2a\u5217\u7684\u5bf9\u89d2\u7ebf\u5143\u7d20\u4e3a1, \u6700\u53f3\u8fb9\u7684$z_p$\u4e2a\u5217\u5168\u4e3a0. \u524d\u8005\u4ee3\u8868\u975e0\u8fb9\u7f18\u751f\u6210\u4e86$p &#8211; 1$\u7ef4\u8fb9\u7f18\u94fe\u7fa4\u7684$p$\u7ef4\u94fe, \u540e\u8005\u5219\u4ee3\u8868\u4e86\u751f\u6210$Z_p(K)$\u7684$p$\u7ef4\u95ed\u94fe. \u53ea\u8981\u5c06\u6240\u6709\u7684\u8fb9\u7f18\u77e9\u9635\u90fd\u5316\u4e3a\u6807\u51c6\u578b, \u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230Betti\u6570$\\beta_p = $$ rank Z_p(K) &#8211; rank B_p(K)$. \u8981\u5f97\u5230\u8fb9\u7f18\u94fe\u7fa4\u548c\u95ed\u94fe\u7fa4\u7684\u57fa\u5e95, \u6211\u4eec\u53ef\u4ee5\u8ffd\u8e2a\u4ee3\u8868\u884c\u8fd0\u7b97\u548c\u5217\u8fd0\u7b97\u7684\u77e9\u9635\u7684\u4e58\u79ef. \u8bb0$U_{p &#8211; 1}$\u4e3a\u5de6\u4e58\u79ef, $V_p$\u4e3a\u53f3\u4e58\u79ef, \u6211\u4eec\u5f97\u5230\u6807\u51c6\u578b\u5982\u4e0b: $$N_p = U_{p &#8211; 1} \\partial_p V_p,$$\u5219\u95ed\u94fe\u7fa4\u7684\u65b0\u57fa\u5e95\u7531$V_p$\u7684\u6700\u540e$z_p$\u4e2a\u5217\u7ed9\u51fa. \u7c7b\u4f3c\u5730, \u8fb9\u7f18\u94fe\u7fa4\u7684\u57fa\u5e95\u5219\u7531$U_{p &#8211; 1}$\u7684\u524d$b_{p &#8211; 1}$\u4e2a\u5217\u7ed9\u51fa.<\/p>\n<p><strong>4. \u8fc7\u6ee4\u548c\u914d\u5bf9<\/strong><\/p>\n<p><strong>\u5b9a\u4e491(\u8fc7\u6ee4)<\/strong> \u4e00\u4e2a$n$\u7ef4\u5355\u7eaf\u590d\u5f62${\\textstyle \\sum}$\u7684\u8fc7\u6ee4\u662f\u4e00\u7cfb\u5217\u5d4c\u5957\u7684\u590d\u5f62:$$\\emptyset = {\\textstyle \\sum_{-1}} \\subset {\\textstyle \\sum_{0}} \\subset {\\textstyle \\sum_{1}} \\subset \\cdots \\subset {\\textstyle \\sum_{n}} = {\\textstyle \\sum}.$$\u5305\u542b\u6620\u5c04$$f: {\\textstyle \\sum_{i- 1}} \\hookrightarrow {\\textstyle \\sum_{i}}, \\\\ f(p) = p$$\u8bf1\u5bfc\u4e86\u4e0b\u540c\u8c03\u7fa4\u4e4b\u95f4\u7684\u540c\u6001$f_* : H_k(\\textstyle \\sum_{i &#8211; 1}) \\to H_k(\\textstyle \\sum_{i})$. \u590d\u5f62\u95f4\u7684\u5d4c\u5957\u5e8f\u5217\u5bf9\u5e94\u4e86\u540c\u8c03\u7fa4\u7684\u5e8f\u5217$$ 0 = H_k(\\textstyle \\sum_{-1}) \\to H_k(\\textstyle \\sum_{0}) \\to \\cdots \\to H_k(\\textstyle \\sum_{n}) = H_k(\\textstyle \\sum).$$\u6211\u4eec\u5c06${\\textstyle \\sum}$\u7684\u6240\u6709\u5355\u7eaf\u5f62\u4f9d\u7167\u7ef4\u6570\u5347\u5e8f\u6392\u5217$$\\sigma_1, \\sigma_2, \\cdots, \\sigma_{n &#8211; 1}, \\sigma_n,$$\u6bcf\u4e00\u6b21\u6dfb\u52a0\u4e00\u4e2a$k$\u7ef4\u5355\u7eaf\u5f62, \u6709\u4e24\u79cd\u60c5\u51b5\u53ef\u80fd\u53d1\u751f. \u7b2c\u4e00\u79cd\u60c5\u51b5, $\\sigma_i$\u751f\u6210\u4e00\u4e2a$k$\u7ef4\u975e\u8fb9\u754c\u95ed\u94fe$c_k$, \u540c\u8c03\u7fa4\u7684$k$\u7ef4Betti\u6570$\\beta_k$\u52a01, \u8fd9\u65f6\u6211\u4eec\u79f0$\\sigma_i$\u4e3a\u6b63\u7684\u5355\u5f62; \u7b2c\u4e8c\u79cd\u60c5\u51b5, $\\sigma_i$\u6d88\u706d\u6389\u4e00\u4e2a$k &#8211; 1$\u7ef4\u5df2\u7ecf\u5b58\u5728\u7684\u95ed\u94fe$c_{k &#8211; 1}$, \u540c\u8c03\u7fa4\u7684$k &#8211; 1$\u7ef4Betti\u6570$\\beta_{k &#8211; 1}$\u51cf1, \u8fd9\u65f6\u6211\u4eec\u79f0$\\sigma_i$\u4e3a\u8d1f\u7684\u5355\u5f62.<\/p>\n<p><strong>\u5b9a\u4e492(\u914d\u5bf9)<\/strong> \u5728\u4e0a\u8ff0\u8fc7\u6ee4\u4e2d, $\\sigma_i$\u662f\u4e00\u4e2a\u8d1f\u5355\u5f62, \u6d88\u706d\u4e86$k &#8211; 1$\u7ef4\u5df2\u7ecf\u5b58\u5728\u7684\u95ed\u94fe$c_{k &#8211; 1}$, \u6211\u4eec\u5c06\u88ab\u6d88\u706d\u6389\u7684\u95ed\u94fe$c_{k &#8211; 1}$\u4e2d\u6700\u540e\u4e00\u4e2a\u6b63\u5355\u5f62\u548c$\\sigma_i$\u914d\u5bf9.<\/p>\n<p>\u8fc7\u6ee4\u548c\u914d\u5bf9\u63d0\u4f9b\u4e86\u53e6\u5916\u4e00\u79cd\u8ba1\u7b97\u51fa\u5404\u9636Betti\u6570\u7684\u601d\u8def, \u6211\u4eec\u8fdb\u800c\u53ef\u4ee5\u8ba1\u7b97\u51fa\u5404\u9636\u540c\u8c03\u7fa4, \u5177\u4f53\u8ba1\u7b97\u8fc7\u7a0b\u53ef\u53c2\u8003\u5f90\u9e4f\u7a0b\u8001\u5e08\u7684\u89c6\u9891(\u8be6\u89c1\u53c2\u8003\u6750\u65991\u89c6\u9891\u5408\u96c6\u4e2d\u7684<a href=\"https:\/\/www.bilibili.com\/video\/BV1VU4y1z7KG\/\">\u8ba1\u7b97\u62d3\u625105<\/a>, <a href=\"https:\/\/www.bilibili.com\/video\/BV11e4y1m7gr\/\">\u8ba1\u7b97\u62d3\u625106<\/a>).<\/p>\n<p><strong>5. \u76f8\u5173\u4f8b\u5b50<\/strong><\/p>\n<p><strong>\u4f8b1<\/strong> \u8ba1\u7b97\u4e0b\u56fe\u8868\u793a\u7684\u6709\u9650\u5355\u7eaf\u590d\u5f62$K$\u7684\u5404\u9636\u540c\u8c03\u7fa4.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example1.png\" alt=\"\" width=\"1539\" height=\"592\" class=\"aligncenter size-full wp-image-2613\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example1.png 1539w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example1-300x115.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example1-768x295.png 768w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example1-1536x591.png 1536w\" sizes=\"(max-width: 1539px) 100vw, 1539px\" \/><\/p>\n<p><strong>\u89e3:<\/strong> \u7531\u4e8e\u6709\u9650\u5355\u7eaf\u590d\u5f62$K$\u542b4\u4e2a\u70b9, 4\u6761\u8fb9, \u6545$$C_0(K) \\cong \\mathbb{Z}^4, n_0 = 4, \\\\ C_1(K) \\cong \\mathbb{Z}^4, n_1 = 4.$$\u4ece\u800c, \u6211\u4eec\u6709$\\partial_1(e_1) = v_2 &#8211; v_1$(\u5728\u57df$\\mathbb{Z}$\u4e2d\u8fdb\u884c\u8ba1\u7b97, \u82e5\u5728\u57df$\\mathbb{Z}_2$\u4e2d\u8fdb\u884c\u8ba1\u7b97, \u5219\u5176\u7ed3\u679c\u4e3a$v_2 + v_1$), \u540c\u7406\u53ef\u5f97$$\\partial_1(e_2) = v_1 &#8211; v_3, \\\\ \\partial_1(e_3) = v_3 &#8211; v_2, \\\\ \\partial_1(e_4) = v_4 &#8211; v_3.$$\u53c8$\\partial_1(e_1 + e_2 + e_3) = 0$, $Ker \\partial_1 \\cong \\mathbb{Z}$, $$Im \\partial_1 = \\left \\langle v_2 &#8211; v_1, v_1 &#8211; v_3, v_3 &#8211; v_2, v_4 &#8211; v_3 \\right \\rangle \\\\ = \\left \\langle v_2 &#8211; v_1, v_1 &#8211; v_3, v_4 &#8211; v_3 \\right \\rangle \\cong \\mathbb{Z}^3,$$\u4ece\u800c$z_1 = 1$, $b_0 = 3$. \u4ece\u76f4\u89c2\u4e0a\u6765\u8bb2, \u6211\u4eec\u9700\u8981&#8221;\u53d6\u8d70&#8221; $z_1$\u6761\u8fb9(\u5982$e_1$) \u624d\u80fd\u7834\u5708(\u5373\u5f97\u5230\u4e00\u68f5\u6700\u5c0f\u751f\u6210\u6811), \u9664\u53bb\u516c\u5171\u70b9(\u5982$v_3$), \u5269\u4e0b$b_0$\u6761\u8fb9(\u5982$e_2, e_3$\u4e0e$e_4$) \u6070\u597d\u5bf9\u5e94$b_0$\u4e2a0\u7ef4\u5355\u5f62(\u5982$v_1, v_2$\u4e0e$v_4$).<\/p>\n<p><strong>\u4f8b2<\/strong> \u8ba1\u7b97\u4e0b\u56fe\u8868\u793a\u7684\u6709\u9650\u5355\u7eaf\u590d\u5f62$K$\u7684\u5404\u9636\u540c\u8c03\u7fa4.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example2.png\" alt=\"\" width=\"1626\" height=\"590\" class=\"aligncenter size-full wp-image-2624\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example2.png 1626w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example2-300x109.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example2-768x279.png 768w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example2-1536x557.png 1536w\" sizes=\"(max-width: 1626px) 100vw, 1626px\" \/><\/p>\n<p><strong>\u89e3:<\/strong> \u5148\u8ba1\u7b97$Z_1(K)$: \u7531\u4e8e$$\\partial_1(e_1 + e_2 + e_3) = v_0 &#8211; v_2 + v_1 &#8211; v_0 + v_2 &#8211; v_1 = 0,$$\u6545$e_1 + e_2 + e_3 \\in Z_1(K)$. \u53c8$\\partial_1(e_4 + e_5 &#8211; e_3) = 0$, \u6545$e_4 + e_5 &#8211; $$ e_3 \\in $$ Z_1(K)$.<br \/>\n$\\\\$ \u4e0d\u59a8\u53d6$\\forall c \\in C_1(K)$, \u5219$c = \\sum^5_{i = 1} n_i e_i$, \u4ece\u800c$c \\in Z_1(K)$, $\\Leftrightarrow \\partial_1 c $$ = 0$, $\\Leftrightarrow $$ \\left\\{\\begin{matrix}<br \/>\nn_1 = n_2 = n, \\\\<br \/>\nn_4 = n_5 = m,<br \/>\n\\end{matrix}\\right.$$c$\u5fc5\u957f\u6210$n(e_1 + e_2) + n_3 e_3 + m(e_4 $$ + e_5)$\u7684\u5f62\u5f0f, $\\Leftrightarrow $$ \\partial_1 c$\u4e2d$v_1$\u524d\u7cfb\u6570\u4e3a0, $v_2$\u524d\u7cfb\u6570\u4e3a0, i.e. $$\\left\\{\\begin{matrix}<br \/>\nn &#8211; n_3 &#8211; m = 0, \\\\<br \/>\n-n + n_3 + m = 0.<br \/>\n\\end{matrix}\\right.$$\u56e0\u6b64, $\\partial_1 c = 0$, $\\Leftrightarrow n &#8211; n_3 &#8211; m = 0$, i.e. $n = m + n_3$,$$c = (m + n_3)(e_1 + e_2) + n_3 e_3 + m(e_4 + e_5) \\\\ = (m + n_3)(e_1 + e_2 + e_3) + m(e_4 + e_5 &#8211; e_3).$$\u56e0\u800c, $Z_1(K) = \\mathbb{Z}(e_1 + e_2 + e_3) + \\mathbb{Z}(e_4 + e_5 &#8211; e_3)$.<br \/>\n$\\\\$ \u63a5\u4e0b\u6765\u8ba1\u7b97$B_1(K) = Im(C_2(K) \\overset{\\partial_2}{\\longrightarrow} C_1(K))$: \u7531\u4e8e$C_2(K) = $$ \\mathbb{Z}\\sigma$, \u6545$B_1(K) = \\mathbb{Z}(\\partial_2 \\sigma) = \\mathbb{Z}(e_4 + e_5 &#8211; e_3)$, \u4ece\u800c$$H_1(K) = Z_1(K) \/ B_1(K) \\cong \\mathbb{Z}.$$\u540c\u7406\u53ef\u5f97, $H_2(K) = Z_2(K) \/ B_2(K) = Z_2(K) = 0$, $H_0(K) \\cong $$ \\mathbb{Z}$.<\/p>\n<p><strong>\u4f8b3<\/strong> \u8003\u8651$K$\u4e3a\u73af\u9762. \u5df2\u77e5$H_0(K ; \\mathbb{Z}_2) \\cong \\mathbb{Z}_2$, $H_1(K ; \\mathbb{Z}_2) \\cong \\mathbb{Z}_2^2$, $H_2 $$ (K ; \\mathbb{Z}_2) = $$ \\mathbb{Z}_2$. \u6240\u4ee5$\\beta_0(K) = 1$, $\\beta_1(K) = 2$, $\\beta_2(K) = 1$. \u5982\u4e0b\u56fe\u6240\u793a. \u5bb9\u6613\u7b97\u51fa$$C_0(K) = \\left \\langle v \\right \\rangle \\cong \\mathbb{Z}_2, \\\\ C_1(K) = \\left \\langle a, b, c \\right \\rangle \\cong \\mathbb{Z}_2^3, \\\\ C_2(K) = \\left \\langle A, B \\right \\rangle \\cong \\mathbb{Z}_2^2.$$<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example3.png\" alt=\"\" width=\"1276\" height=\"708\" class=\"aligncenter size-full wp-image-2620\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example3.png 1276w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example3-300x166.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example3-768x426.png 768w\" sizes=\"(max-width: 1276px) 100vw, 1276px\" \/><\/p>\n<p>\u6240\u4ee5$n_0 = 1$, $n_1 = 3$, $n_2 = 2$. \u53ef\u4ee5\u5f97\u5230$D_1 = (0 \\ 0 \\ 0)$, \u5143\u7d20\u4e3a0\u662f\u56e0\u4e3a\u8fd9\u4e2a\u77e9\u9635\u4e5f\u662f$mod \\ 2$\u7684, \u53ea\u6709\u4e00\u4e2a\u9876\u70b9$v$, \u8fb9$a$\u7684\u4e24\u4e2a\u9876\u70b9$v$, \u6240\u4ee5\u5143\u7d20\u4e3a$2 \\equiv 0(mod \\ 2)$. \u4e5f\u5c31\u662f\u8bf4, $rank D_1 = 0$. \u8865\u5145\u5b9a\u4e49$rank $$ D_0 = rank D_3 = 0$.$$D_2 = \\begin{pmatrix}<br \/>\n1 &#038; 1 \\\\<br \/>\n1 &#038; 1 \\\\<br \/>\n1 &#038; 1<br \/>\n\\end{pmatrix} \\xrightarrow{r_1 + r_2, r_1 + r_3} \\begin{pmatrix}<br \/>\n1 &#038; 1 \\\\<br \/>\n0 &#038; 0 \\\\<br \/>\n0 &#038; 0<br \/>\n\\end{pmatrix} \\xrightarrow{c_1 + c_2} \\begin{pmatrix}<br \/>\n1 &#038; 0 \\\\<br \/>\n0 &#038; 0 \\\\<br \/>\n0 &#038; 0<br \/>\n\\end{pmatrix},$$\u6240\u4ee5$rank D_2 = 1$. \u6839\u636e\u516c\u5f0f, $$\\beta_0 = n_0 &#8211; rank D_0 &#8211; rank D_1 \\\\ = 1 &#8211; 0 &#8211; 0 = 1, \\\\ \\beta_1 = 3 &#8211; 0 &#8211; 1 = 2, \\\\ \\beta_2 = 2 &#8211; 1 &#8211; 0 = 1.$$<\/p>\n<p><strong>\u4f8b4<\/strong> \u8ba1\u7b97\u5c04\u5f71\u5e73\u9762$RP^2$\u7684\u5404\u9636\u540c\u8c03\u7fa4.<br \/>\n$\\\\$ <strong>\u89e3:<\/strong> \u5c04\u5f71\u5e73\u9762$RP^2$\u7684\u591a\u8fb9\u5f62\u8868\u793a\u7684\u4e09\u89d2\u5256\u5206\u5982\u4e0b\u56fe\u6240\u793a.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example4.png\" alt=\"\" width=\"1305\" height=\"718\" class=\"aligncenter size-full wp-image-2622\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example4.png 1305w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example4-300x165.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example4-768x423.png 768w\" sizes=\"(max-width: 1305px) 100vw, 1305px\" \/><\/p>\n<p>\u5176\u5404\u9636\u540c\u8c03\u7fa4\u7684\u5177\u4f53\u8ba1\u7b97\u8fc7\u7a0b\u53ef\u53c2\u8003\u5e84\u6653\u6ce2\u8001\u5e08\u7684\u89c6\u9891(\u8be6\u89c1\u53c2\u8003\u6750\u65993), \u5728\u89c6\u9891\u4e2d\u5e84\u6653\u6ce2\u8001\u5e08\u4f7f\u7528\u4e86Push to the Boundary\u7684\u65b9\u6cd5, \u8ba1\u7b97\u8fc7\u7a0b\u5341\u5206\u6709\u610f\u601d~<br \/>\n$\\\\$ \u9700\u8981\u6ce8\u610f\u7684\u662f, \u4e0b\u56fe\u6240\u793a\u7684\u4e09\u89d2\u5256\u5206\u5e76\u4e0d\u80fd\u6210\u4e3a\u5c04\u5f71\u5e73\u9762$RP^2$\u7684\u591a\u8fb9\u5f62\u8868\u793a\u7684\u4e09\u89d2\u5256\u5206.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example5.png\" alt=\"\" width=\"1305\" height=\"718\" class=\"aligncenter size-full wp-image-2623\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example5.png 1305w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example5-300x165.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/11\/Homology_Group_Calculation_Example5-768x423.png 768w\" sizes=\"(max-width: 1305px) 100vw, 1305px\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6700\u8fd1\u4e09\u5468\u7684\u5468\u672b\u90fd\u662f\u5728\u5b85\u5bb6, \u76ee\u524d\u5176\u5b9e\u5904\u4e8e\u6709\u70b9\u5c0f\u8ff7\u832b\u7684\u72b6\u6001, \u5728\u5b66\u5b8c\u4ee3\u6570\u62d3\u6251\u4ee5\u540e\u53c8\u6682\u65f6\u4e0d\u5927\u60f3\u5f00\u65b0\u7684\u5927\u5751. \u6070\u5de7\u6700 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/11\/20\/introductory_notes_computational_topology\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u8ba1\u7b97\u62d3\u6251\u5165\u95e8\u7b14\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\/2594"}],"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=2594"}],"version-history":[{"count":31,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/2594\/revisions"}],"predecessor-version":[{"id":3620,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/2594\/revisions\/3620"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2594"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2594"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2594"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}