{"id":5258,"date":"2026-06-23T16:54:08","date_gmt":"2026-06-23T08:54:08","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=5258"},"modified":"2026-06-30T15:32:05","modified_gmt":"2026-06-30T07:32:05","slug":"interpretation_paper_efficient_computation_handle_tunnel_loops_via_reeb_graphs","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2026\/06\/23\/interpretation_paper_efficient_computation_handle_tunnel_loops_via_reeb_graphs\/","title":{"rendered":"\u300aAn Efficient Computation of Handle and Tunnel Loops via Reeb Graphs\u300b\u8bba\u6587\u89e3\u8bfb"},"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>\u57285\u670823\u65e5\u7ec8\u4e8e\u7ed3\u675f\u4e862026\u5e74\u4e0a\u534a\u5e74\u7cfb\u7edf\u67b6\u6784\u8bbe\u8ba1\u5e08\u8f6f\u8003, \u8003\u5b8c\u4ee5\u540e\u5fc3\u90fd\u51c9\u4e86, \u4eca\u5e74\u7684\u6cc4\u9898\u4e8b\u4ef6\u6682\u4e14\u4e0d\u63d0, \u81ea\u5df1\u8fd8\u628a\u6848\u4f8b\u5206\u6790\u9898\u91cc\u4e00\u9053\u63a8\u8350\u7cfb\u7edf\u76f8\u5173\u7684\u9898\u76ee\u51e0\u4e4e\u5b8c\u5168\u7406\u89e3\u9519\u8befQAQ. \u611f\u89c9\u5df2\u7ecf\u53ef\u4ee5\u505a\u597d\u4e0b\u534a\u5e74\u53c2\u8003\u7684\u5fc3\u7406\u51c6\u5907\u4e86, \u65e0\u8bba\u5982\u4f55, \u7b49\u6210\u7ee9\u51fa\u6765\u540e\u518d\u4f5c\u6253\u7b97~ \u8fd9\u4e24\u4e09\u5e74\u5df2\u7ecf\u758f\u4e8e\u4ee3\u6570\u62d3\u6251\u7684\u5b66\u4e60\u4e86, \u611f\u89c9\u63a5\u4e0b\u6765\u53ea\u4f1a\u6709\u8d8a\u6765\u8d8a\u591a\u7684\u5404\u79cd\u8282\u70b9\u53bb\u963b\u788d\u81ea\u5df1\u5bf9\u4e8e\u4ee3\u6570\u62d3\u6251\u7684\u5b66\u4e60, \u62e9\u65e5\u4e0d\u5982\u649e\u65e5, \u51b3\u5b9a\u7acb\u9a6c\u91cd\u62fe\u4ee3\u6570\u62d3\u6251. \u4eca\u5e74\u53ef\u8c13\u662fLLM\u5143\u5e74, AI\u5df2\u7ecf\u80fd\u591f\u6781\u5927\u5730\u63d0\u5347\u81ea\u8eab\u7684\u5b66\u4e60\u901f\u5ea6, \u5df2\u7ecf\u4e0d\u7528\u518d\u50cf\u8fc7\u53bb\u90a3\u822c\u91c7\u7528\u53e4\u6cd5\u786c\u5543\u4e86. \u4e8e\u662f\u4e4e, \u5728AI\u7684\u5e2e\u52a9\u4e0b, \u91cd\u65b0\u5b66\u4e60\u4e86Dey T K, Fan F, Wang Y. An efficient computation of handle and tunnel loops via Reeb graphs[J]. ACM Transactions on Graphics (TOG), 2013, 32(4): 1-10.\u8fd9\u7bc7\u8bba\u6587, \u5bf9\u6b64\u6709\u4e86\u66f4\u6df1\u7406\u89e3.<\/p>\n<p><!--more--><\/p>\n<p>\u8bba\u6587\u7684\u6838\u5fc3\u76ee\u6807\u662f: \u5728\u4e09\u7ef4\u5d4c\u5165\u66f2\u9762\u4e0a\u9ad8\u6548\u8ba1\u7b97\u628a\u624b\u73af\u548c\u96a7\u9053\u73af\u7684\u4e00\u7ec4\u57fa, \u800c\u4e0d\u9700\u8981\u5bf9\u66f2\u9762\u5185\u5916\u7a7a\u95f4\u505a\u56db\u9762\u4f53\u5256\u5206. \u8bba\u6587\u63d0\u51fa\u4e86\u4e00\u79cd\u7528Reeb\u56fe + \u73af\u7ed5\u6570 + \u6700\u77ed\u8def\u5f84\u6811\u4f18\u5316\u6765\u8ba1\u7b97\u66f2\u9762\u628a\u624b\u73af\u548c\u96a7\u9053\u73af\u7684\u65b9\u6cd5, \u5b83\u6bd4\u65e7\u65b9\u6cd5\u5feb\u5f88\u591a, \u56e0\u4e3a\u65e7\u65b9\u6cd5\u9700\u8981\u6784\u9020\u66f2\u9762\u5185\u90e8 \/ \u5916\u90e8\u7684\u4e09\u7ef4\u4f53\u7f51\u683c, \u800c\u8bba\u6587\u53ea\u5728\u539f\u59cb\u8868\u9762\u7f51\u683c\u4e0a\u5de5\u4f5c.<\/p>\n<p><strong>1. \u95ee\u9898\u80cc\u666f<\/strong><\/p>\n<p>\u5728\u4e09\u7ef4\u6a21\u578b\u5904\u7406\u4e2d, \u7ecf\u5e38\u9700\u8981\u627e\u66f2\u9762\u4e0a\u7684\u975e\u5e73\u51e1\u73af\u8def, \u4f8b\u5982\u7528\u4e8e\u62d3\u6251\u4fee\u590d, \u66f2\u9762\u53c2\u6570\u5316, \u7279\u5f81\u8bc6\u522b, \u5f62\u72b6\u5bf9\u5e94\u7b49. \u4f46\u666e\u901a\u7684&#8221;\u975e\u5e73\u51e1\u73af&#8221; \u53ea\u53cd\u6620\u66f2\u9762\u7684\u62bd\u8c61\u62d3\u6251, \u4e0d\u4e00\u5b9a\u5bf9\u5e94\u771f\u5b9e\u51e0\u4f55\u4e0a\u7684&#8221;\u628a\u624b&#8221; \u6216&#8221;\u96a7\u9053&#8221;. \u8bba\u6587\u91c7\u7528Dey\u7b49\u4eba\u4e4b\u524d\u7684\u5b9a\u4e49:<br \/>\n$\\\\$ $\\cdot$ \u628a\u624b\u73af: \u5728\u66f2\u9762\u5185\u90e8\u533a\u57df$I$\u662f\u5e73\u51e1\u7684, \u4f46\u5728\u5916\u90e8\u533a\u57df$O$\u4e2d\u975e\u5e73\u51e1.<br \/>\n$\\\\$ $\\cdot$ \u96a7\u9053\u73af: \u5728\u5916\u90e8\u533a\u57df$O$\u662f\u5e73\u51e1\u7684, \u4f46\u5728\u5185\u90e8\u533a\u57df$I$\u4e2d\u975e\u5e73\u51e1.<br \/>\n$\\\\$ \u76f4\u89c2\u5730\u8bf4, \u628a\u624b\u73af\u56f4\u4f4f\u6a21\u578b\u7684&#8221;\u628a\u624b&#8221; \u7279\u5f81; \u96a7\u9053\u73af\u5219\u7a7f\u8fc7\u6216\u56f4\u7ed5&#8221;\u96a7\u9053&#8221; \u7279\u5f81. \u5bf9\u4e8e\u4e8f\u683c\u4e3a$g$\u7684\u95ed\u5408\u8fde\u901a\u66f2\u9762, \u628a\u624b\u73af\u7a7a\u95f4\u548c\u96a7\u9053\u73af\u7a7a\u95f4\u5404\u6709\u7ef4\u5ea6$g$, \u56e0\u6b64\u76ee\u6807\u662f\u5206\u522b\u6c42\u51fa$g$\u6761\u72ec\u7acb\u7684\u628a\u624b\u57fa\u548c$g$\u6761\u72ec\u7acb\u7684\u96a7\u9053\u57fa.<\/p>\n<p><strong>2. \u65e7\u65b9\u6cd5\u7684\u75db\u70b9<\/strong><\/p>\n<p>\u5df2\u6709\u80fd\u4fdd\u8bc1\u6b63\u786e\u6027\u7684\u7b97\u6cd5\u9700\u8981\u6784\u9020\u66f2\u9762\u5185\u90e8\u548c\u5916\u90e8\u7684\u4e09\u7ef4\u5256\u5206, \u6bd4\u5982\u56db\u9762\u4f53\u7f51\u683c. \u95ee\u9898\u662f:<br \/>\n$\\\\$ $\\cdot$ \u4e09\u7ef4\u5256\u5206\u6027\u80fd\u5f00\u9500\u5f88\u5927, \u5c24\u5176\u5bf9\u9ad8\u9762\u6570\u7f51\u683c;<br \/>\n$\\\\$ $\\cdot$ \u53ef\u80fd\u9700\u8981\u6539\u53d8\u539f\u59cb\u8868\u9762\u7f51\u683c;<br \/>\n$\\\\$ $\\cdot$ \u5bf9\u7a00\u758f, \u566a\u58f0, \u975e\u5149\u6ed1, \u590d\u6742\u5d4c\u5165\u751a\u81f3\u7ebd\u7ed3\u66f2\u9762\u7684\u8f93\u5165\u4e0d\u53cb\u597d;<br \/>\n$\\\\$ $\\cdot$ \u5b9e\u73b0\u548c\u5de5\u7a0b\u4f7f\u7528\u6210\u672c\u9ad8.<br \/>\n$\\\\$ \u8bba\u6587\u7684\u5173\u952e\u6539\u8fdb\u662f: \u5b8c\u5168\u907f\u514d\u6784\u9020\u4f53\u7f51\u683c, \u76f4\u63a5\u5728\u8f93\u5165\u8868\u9762\u7f51\u683c\u4e0a\u8ba1\u7b97.<\/p>\n<p><strong>3. \u6838\u5fc3\u5de5\u5177<\/strong><\/p>\n<p><strong>Reeb\u56fe<\/strong><\/p>\n<p>\u8bba\u6587\u5bf9\u66f2\u9762\u53d6\u4e00\u4e2a\u9ad8\u5ea6\u51fd\u6570$h$, \u4f8b\u5982\u6cbf\u67d0\u4e2a\u56fa\u5b9a\u65b9\u5411\u7684\u9ad8\u5ea6. Reeb\u56fe\u53ef\u4ee5\u7406\u89e3\u4e3a: \u628a\u6bcf\u4e2a\u7b49\u9ad8\u7ebf\u8fde\u901a\u5206\u652f\u538b\u7f29\u6210\u4e00\u4e2a\u70b9\u540e\u5f97\u5230\u7684\u56fe.<br \/>\n$\\\\$ \u5bf9\u4e8e\u4e8f\u683c\u4e3a$g$\u7684\u95ed\u5408\u66f2\u9762, Reeb\u56fe\u4e2d\u5b58\u5728$g$\u4e2a\u72ec\u7acb\u73af. \u8bba\u6587\u5229\u7528\u8fd9\u4e2a\u4e8b\u5b9e, \u5148\u5728Reeb\u56fe\u4e2d\u627e\u51fa$g$\u4e2a\u56fe\u73af, \u518d\u628a\u5b83\u4eec\u6620\u5c04\u56de\u66f2\u9762, \u5f97\u5230\u4e00\u7ec4\u521d\u59cb\u73af$\\gamma_i$.<br \/>\n$\\\\$ \u4f46\u8fd9\u4e9b$\\gamma_i$\u8fd8\u4e0d\u662f\u628a\u624b \/ \u96a7\u9053\u73af. \u4e8e\u662f\u8bba\u6587\u8fd8\u7ed9\u6bcf\u4e2a$\\gamma_i$\u6784\u9020\u4e00\u4e2a\u5bf9\u5e94\u7684\u5bf9\u5076\u6c34\u5e73\u96c6\u73af\u8def, \u8bb0\u4f5c$\\overline{\\gamma_i}$. \u8fd9\u6837\u4e00\u5171\u5f97\u5230$2g$\u6761\u66f2\u9762\u73af, \u5b83\u4eec\u6784\u6210$H_1(M)$\u7684\u4e00\u7ec4\u57fa.<\/p>\n<p><strong>\u73af\u7ed5\u6570<\/strong><\/p>\n<p>\u73af\u7ed5\u6570\u7528\u6765\u5224\u65ad\u4e24\u6761\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u7684\u4e0d\u76f8\u4ea4\u73af\u662f\u5426&#8221;\u94fe\u5728\u4e00\u8d77&#8221;. \u5728\u8bba\u6587\u4f7f\u7528\u7684$\\mathbb{Z}_2$\u7cfb\u6570\u57df\u4e0b, \u73af\u7ed5\u6570\u53ea\u67090\u62161:<br \/>\n$\\\\$ $\\cdot$ 0: \u6ca1\u6709\u5947\u6570\u6b21\u7f20\u7ed5;<br \/>\n$\\\\$ $\\cdot$ 1: \u6709\u5947\u6570\u6b21\u7f20\u7ed5.<br \/>\n$\\\\$ \u5b83\u7684\u4f5c\u7528\u662f\u628a&#8221;\u66f2\u9762\u4e0a\u7684\u73af&#8221; \u4e0e&#8221;\u5185\u90e8 \/ \u5916\u90e8\u7a7a\u95f4\u4e2d\u7684\u975e\u5e73\u51e1\u6027&#8221; \u8054\u7cfb\u8d77\u6765. \u4e5f\u5c31\u662f\u8bf4, \u73af\u7ed5\u6570\u5e2e\u52a9\u5224\u65ad\u67d0\u6761\u73af\u5230\u5e95\u662f\u628a\u624b, \u8fd8\u662f\u96a7\u9053.<\/p>\n<p><strong>4. \u7b97\u6cd5\u4e3b\u6d41\u7a0b<\/strong><\/p>\n<p>\u8bba\u6587\u7684\u7b97\u6cd51\u53ef\u4ee5\u6982\u62ec\u4e3a\u516d\u6b65:<br \/>\n$\\\\$ 1) \u5bf9\u8f93\u5165\u95ed\u5408\u4e09\u89d2\u7f51\u683c$M$\u8ba1\u7b97\u9ad8\u5ea6\u51fd\u6570$h$\u7684Reeb\u56fe.<br \/>\n$\\\\$ 2) \u5728Reeb\u56fe\u4e0a\u7528\u6700\u5927\u751f\u6210\u6811\u627e\u51fa$g$\u4e2a\u72ec\u7acb\u56fe\u73af.<br \/>\n$\\\\$ 3) \u628a\u8fd9\u4e9b\u56fe\u73af\u6620\u5c04\u56de\u66f2\u9762, \u5f97\u5230$\\gamma_i$; \u5e76\u4e3a\u6bcf\u4e2a$\\gamma_i$\u6784\u9020\u4e00\u4e2a\u7b49\u9ad8\u7ebf\u4e0a\u7684\u5bf9\u5076\u73af\u8def $\\overline{\\gamma_i}$.<br \/>\n$\\\\$ 4) \u6839\u636e\u978d\u70b9\u9644\u8fd1\u7684\u5c40\u90e8\u51e0\u4f55\u914d\u7f6e, \u628a\u8fd9\u4e9b\u73af\u8f7b\u5fae\u63a8\u5411\u5185\u90e8\u6216\u5916\u90e8, \u5f97\u5230\u5185\u90e8\u7a7a\u95f4$H_1(I)$\u548c\u5916\u90e8\u7a7a\u95f4$H_1(O)$\u7684\u57fa.<br \/>\n$\\\\$ 5) \u6784\u9020\u4e00\u4e2a$2g \\times 2g$\u7684\u73af\u7ed5\u6570\u77e9\u9635$L$, \u8bc1\u660e\u5b83\u53ef\u9006, \u7136\u540e\u7528$L^{-1}$\u5bf9\u521d\u59cb\u73af\u505a\u7ebf\u6027\u7ec4\u5408, \u5f97\u5230\u771f\u6b63\u7684\u628a\u624b\u57fa\u548c\u96a7\u9053\u57fa.<br \/>\n$\\\\$ 6) \u5bf9\u5f97\u5230\u7684\u73af\u8fdb\u884c\u51e0\u4f55\u4f18\u5316, \u4f7f\u5b83\u4eec\u66f4\u77ed, \u66f4\u8d34\u8fd1\u5b9e\u9645\u628a\u624b \/ \u96a7\u9053\u7279\u5f81.<\/p>\n<p><strong>5. \u7406\u8bba\u4e0a\u7684\u5173\u952e\u70b9<\/strong><\/p>\n<p>\u8bba\u6587\u6700\u91cd\u8981\u7684\u7406\u8bba\u4fdd\u8bc1\u662f: \u901a\u8fc7Reeb\u56fe\u5f97\u5230\u7684$\\gamma_i$\u4e0e\u5bf9\u5076\u73af\u8def $\\overline{\\gamma_i}$\u4e0d\u53ea\u662f\u542f\u53d1\u5f0f\u5019\u9009, \u800c\u662f\u5177\u6709\u53ef\u8bc1\u660e\u7ed3\u6784:<br \/>\n$\\\\$ $\\cdot$ \u5b83\u4eec\u5171\u540c\u6784\u6210\u66f2\u9762\u4e00\u9636\u540c\u8c03\u7fa4$H_1(M)$\u7684\u57fa;<br \/>\n$\\\\$ $\\cdot$ \u5b83\u4eec\u4e4b\u95f4\u7684\u76f8\u4ea4\u5173\u7cfb\u6709\u7279\u6b8a\u7684\u4e09\u89d2\u7ed3\u6784;<br \/>\n$\\\\$ $\\cdot$ \u8f7b\u5fae\u6270\u52a8\u540e\u53ef\u4ee5\u5f62\u6210$H_1(I)$\u548c$H_1(O)$\u7684\u57fa;<br \/>\n$\\\\$ $\\cdot$ \u73af\u7ed5\u6570\u77e9\u9635$L$\u56e0\u8fd9\u79cd\u7ed3\u6784\u800c\u53ef\u9006;<br \/>\n$\\\\$ $\\cdot$ \u7528$L^{-1}$\u7ec4\u5408\u51fa\u6765\u7684\u73af\u6ee1\u8db3\u628a\u624b \/ \u96a7\u9053\u7684\u5224\u5b9a\u6761\u4ef6.<br \/>\n$\\\\$ \u6362\u53e5\u8bdd\u8bf4, \u8bba\u6587\u4e0d\u662f\u5355\u7eaf&#8221;\u7528Reeb\u56fe\u731c\u73af&#8221;, \u800c\u662f\u7528Reeb\u56fe\u4ea7\u751f\u4e00\u4e2a\u6709\u8db3\u591f\u4ee3\u6570\u7ed3\u6784\u7684\u5019\u9009\u7cfb\u7edf, \u518d\u7528\u73af\u7ed5\u6570\u7cbe\u786e\u5206\u79bb\u628a\u624b\u548c\u96a7\u9053.<br \/>\n$\\\\$ \u5176\u4e2d, \u8981\u7406\u89e3&#8221;\u7528$L^{-1}$\u7ec4\u5408\u51fa\u6765\u7684\u73af\u6ee1\u8db3\u628a\u624b \/ \u96a7\u9053\u7684\u5224\u5b9a\u6761\u4ef6&#8221; \u8fd9\u53e5\u8bdd, \u53ef\u4ee5\u5c06\u5176\u653e\u5230\u4ee3\u6570\u62d3\u6251\u4e0e\u7ebf\u6027\u4ee3\u6570\u7684\u4ea4\u53c9\u6846\u67b6\u4e0b\u6765\u770b. \u9700\u8981\u6ce8\u610f\u7684\u662f, \u4e0b\u9762\u5173\u4e8e$H^1$\u548c\u5bf9\u5076\u57fa\u7684\u8bf4\u6cd5\u662f\u4e00\u79cd\u8f85\u52a9\u7406\u89e3\u7684\u7c7b\u6bd4; \u8bba\u6587\u539f\u6587\u4e25\u683c\u6784\u9020\u7684\u662f\u6270\u52a8\u540e\u7684\u68c0\u6d4b\u66f2\u7ebf$\\alpha_j$, \u5e76\u901a\u8fc7\u5b83\u4eec\u4e0e$\\gamma_i$\u7684\u73af\u7ed5\u6570\u77e9\u9635\u5b8c\u6210\u57fa\u53d8\u6362.<\/p>\n<p><strong>5.1 \u6270\u52a8\u57fa\u66f2\u7ebf\u4e0e\u53cc\u7ebf\u6027\u914d\u5bf9(\u5efa\u7acb\u6d4b\u91cf\u6846\u67b6)<\/strong><\/p>\n<p>\u5728\u4e8f\u683c\u4e3a$g$\u7684\u53ef\u5b9a\u5411\u95ed\u66f2\u9762$M$\u4e0a, \u5176\u7b2c\u4e00\u540c\u8c03\u7fa4$H_1(M; \\mathbb{Z})$\u662f\u4e00\u4e2a\u79e9\u4e3a$2g$\u7684\u81ea\u7531\u963f\u8d1d\u5c14\u7fa4; \u5728$\\mathbb{Z}_2$\u7cfb\u6570\u57df\u4e0b, $H_1(M; \\mathbb{Z}_2) \\cong (\\mathbb{Z}_2)^{2g}$, \u5219\u4e3a$2g$\u7ef4\u5411\u91cf\u7a7a\u95f4. \u8bc1\u660e\u8be6\u89c1\u9644\u5f55B.<br \/>\n$\\\\$ $\\cdot$ \u540c\u8c03\u7c7b(\u5411\u91cf): \u66f2\u9762\u4e0a\u7684\u95ed\u5408\u66f2\u7ebf(\u73af\u8def, \u5982$\\gamma_i$\u6216$\\beta_i$) \u4ee3\u8868\u4e86$H_1$\u4e2d\u7684\u540c\u8c03\u7c7b.<br \/>\n$\\\\$ $\\cdot$ \u6270\u52a8\u57fa\u66f2\u7ebf(\u68c0\u6d4b\u5de5\u5177): \u8bba\u6587\u628a$\\gamma_i$\u548c$\\overline{\\gamma_i}$\u8f7b\u5fae\u63a8\u5165\u5185\u90e8\u6216\u5916\u90e8, \u5f97\u5230\u6270\u52a8\u7ebf\u96c6\u5408$\\{ \\alpha_j \\}^{2g}_{j = 1}$. \u8fd9\u4e9b$\\alpha_j$\u4e25\u683c\u6765\u8bf4\u4e0d\u662f\u8bba\u6587\u663e\u5f0f\u5b9a\u4e49\u7684$H^1$\u5bf9\u5076\u57fa, \u800c\u662f\u7528\u4e8e\u73af\u7ed5\u6570\u68c0\u6d4b\u7684\u51e0\u4f55\u66f2\u7ebf; \u82e5\u501f\u7528\u5bf9\u5076\u7a7a\u95f4\u7684\u8bed\u8a00, \u5b83\u4eec\u53ef\u4ee5\u7c7b\u6bd4\u4e3a\u4e00\u7ec4&#8221;\u6d4b\u91cf\u65b9\u5411&#8221;.<br \/>\n$\\\\$ $\\cdot$ \u53cc\u7ebf\u6027\u578b(\u914d\u5bf9\u6cd5\u5219): \u66f2\u7ebf$\\gamma$\u4e0e\u68c0\u6d4b\u7ebf$\\alpha$\u4e4b\u95f4\u7684\u73af\u7ed5\u6570($Lk$, <strong>\u53ef\u89c6\u4e3a\u4e00\u79cd\u5185\u79ef\u5b9a\u4e49<\/strong>), \u5728$\\mathbb{Z}_2$\u7cfb\u6570\u4e0b\u7ed9\u51fa\u4e00\u4e2a\u53cc\u7ebf\u6027\u914d\u5bf9. \u8bba\u6587\u6b63\u662f\u901a\u8fc7\u8fd9\u4e2a\u914d\u5bf9\u6765\u8bb0\u5f55$\\gamma_i$\u76f8\u5bf9\u4e8e\u6270\u52a8\u66f2\u7ebf\u96c6\u5408\u7684\u4ee3\u6570\u4fe1\u606f.<br \/>\n$\\\\$ \u7ed3\u8bba: \u8ba1\u7b97$Lk(\\gamma_i, \\alpha_j)$, \u53ef\u4ee5\u7406\u89e3\u4e3a\u8bfb\u53d6$\\gamma_i$\u76f8\u5bf9\u4e8e\u7b2c$j$\u6761\u68c0\u6d4b\u66f2\u7ebf\u7684\u4e00\u4e2a$\\mathbb{Z}_2$\u5750\u6807; \u8fd9\u662f\u4e00\u79cd\u4fbf\u4e8e\u7406\u89e3\u77e9\u9635$L$\u7684\u7ebf\u6027\u4ee3\u6570\u89c6\u89d2.<\/p>\n<p><strong>5.2 \u914d\u5bf9\u77e9\u9635$L$\u7684\u4ee3\u6570\u610f\u4e49(\u65e7\u57fa\u7684\u77e9\u9635\u8868\u793a)<\/strong><\/p>\n<p>\u5047\u8bbe\u7b97\u6cd5\u521d\u59cb\u63d0\u53d6\u7684$2g$\u6761\u6df7\u5408\u66f2\u7ebf$\\{ \\gamma_1, \\cdots, \\gamma_{2g} \\}$\u6784\u6210\u4e86$H_1$\u7684\u4e00\u7ec4\u521d\u59cb\u57fa(\u65e7\u57fa), \u540c\u65f6\u6211\u4eec\u6709\u6270\u52a8\u68c0\u6d4b\u7ebf\u96c6\u5408$\\{ \\alpha_1, \\cdots, \\alpha_{2g} \\}$. \u77e9\u9635$L$\u901a\u8fc7\u4ee5\u4e0b\u65b9\u5f0f\u6784\u9020:$$l_{ij} = Lk(\\gamma_i, \\alpha_j).$$$\\cdot$ \u77e9\u9635\u7684\u4ee3\u6570\u672c\u8d28: \u77e9\u9635$L$\u662f\u73af\u7ed5\u6570\u53cc\u7ebf\u6027\u914d\u5bf9\u5728\u7ed9\u5b9a\u66f2\u7ebf\u96c6\u5408$\\{ \\gamma_i \\}$\u548c\u68c0\u6d4b\u66f2\u7ebf\u96c6\u5408$\\{ \\alpha_j \\}$\u4e0b\u7684\u77e9\u9635\u8868\u793a.<br \/>\n$\\\\$ $\\cdot$ \u884c\u5411\u91cf\u7684\u610f\u4e49: $L$\u7684\u7b2c$i$\u884c\u8bb0\u5f55\u4e86\u65e7\u57fa\u5143\u7d20$\\gamma_i$\u4e0e\u6240\u6709\u6270\u52a8\u68c0\u6d4b\u7ebf$\\{ \\alpha_j \\}$\u7684\u73af\u7ed5\u6570.<br \/>\n$\\\\$ $\\cdot$ \u62d3\u6251\u7269\u7406\u610f\u4e49: \u7531\u4e8e\u521d\u59cb\u63d0\u53d6\u7684\u65e7\u57fa$\\{ \\gamma_i \\}$\u901a\u5e38\u662f\u4efb\u610f\u751f\u6210\u7684(\u975e\u89c4\u8303\u7684), \u5b83\u4eec\u5728\u62d3\u6251\u4e0a\u76f8\u4e92&#8221;\u7f20\u7ed5&#8221;, \u5bfc\u81f4\u5176\u5750\u6807\u8868\u793a\u5728\u591a\u4e2a\u5bf9\u5076\u65b9\u5411\u4e0a\u5747\u6709\u975e\u96f6\u6295\u5f71. \u56e0\u6b64, \u77e9\u9635$L$\u901a\u5e38\u662f\u4e00\u4e2a\u7a20\u5bc6\u7684\u975e\u5bf9\u89d2\u77e9\u9635.<\/p>\n<p><strong>5.3 \u89c4\u8303\u57fa\u7684\u6784\u9020\u4e0e\u8fc7\u6e21\u77e9\u9635$L^{-1}$(\u57fa\u53d8\u6362\u7684\u6838\u5fc3)<\/strong><\/p>\n<p>\u7b97\u6cd5\u7684\u6700\u7ec8\u76ee\u6807, \u662f\u6784\u9020\u4e00\u7ec4\u89c4\u8303\u57fa$\\{ \\beta_1, \\cdots, \\beta_{2g} \\}$(\u5373\u65b0\u57fa).<\/p>\n<p><strong>5.3.1 &#8220;\u89c4\u8303&#8221; \u7684\u4e25\u683c\u4ee3\u6570\u5b9a\u4e49<\/strong><\/p>\n<p>\u65b0\u57fa$\\{ \\beta_i \\}$\u9700\u8981\u5bf9\u6270\u52a8\u68c0\u6d4b\u7ebf$\\{ \\alpha_j \\}$\u6ee1\u8db3\u6b63\u4ea4\u5f52\u4e00\u5316\u6761\u4ef6(Kronecker Delta, \u514b\u7f57\u5185\u514b\u51fd\u6570):$$Lk(\\beta_i, \\alpha_j) = \\delta_{ij} = \\left\\{\\begin{matrix}<br \/>\n1, &#038; if \\ i = j, \\\\<br \/>\n0, &#038; if \\ i \\ne j.<br \/>\n\\end{matrix}\\right.$$\u8fd9\u610f\u5473\u7740\u65b0\u57fa\u4e0e\u6270\u52a8\u68c0\u6d4b\u7ebf\u4e4b\u95f4\u7684\u73af\u7ed5\u6570\u77e9\u9635\u53d8\u6210\u5355\u4f4d\u77e9\u9635$I$. \u5728\u62d3\u6251\u4e0a, \u8fd9\u786e\u4fdd\u4e86\u6bcf\u4e2a$\\beta_i$\u4ec5\u4e0e\u5176\u5bf9\u5e94\u7684$\\alpha_i$\u6709\u975e\u96f6\u73af\u7ed5\u6570, \u800c\u4e0e\u5176\u5b83\u68c0\u6d4b\u7ebf\u4ee3\u6570\u89e3\u8026.<\/p>\n<p><strong>5.3.2 \u63a8\u5bfc\u8fc7\u6e21\u77e9\u9635<\/strong><\/p>\n<p>\u6613\u77e5, \u65b0\u57fa\u5fc5\u5b9a\u53ef\u4ee5\u8868\u793a\u4e3a\u65e7\u57fa\u7684\u7ebf\u6027\u7ec4\u5408:$$\\beta_i = \\sum^{2g}_{k = 1} c_{ik} \\gamma_k,$$\u5176\u4e2d, \u7cfb\u6570\u77e9\u9635$C = (c_{ik})$\u5373\u4e3a\u4ece\u65e7\u57fa\u5230\u65b0\u57fa\u7684\u8fc7\u6e21\u77e9\u9635. \u5c06\u4e0a\u8ff0\u7ebf\u6027\u7ec4\u5408\u4ee3\u5165\u53cc\u7ebf\u6027\u914d\u5bf9\u4e2d, \u5229\u7528\u73af\u7ed5\u6570\u7684\u53cc\u7ebf\u6027\u6027\u8d28, \u53ef\u5f97:$$Lk(\\beta_i, \\alpha_j) = Lk(\\sum^{2g}_{k = 1} c_{ik} \\gamma_k, \\alpha_j) = \\sum^{2g}_{k = 1} c_{ik} Lk(\\gamma_k, \\alpha_j).$$\u5c06\u5176\u8f6c\u5316\u4e3a\u77e9\u9635\u4e58\u6cd5\u5f62\u5f0f(\u8bbe\u65b0\u914d\u5bf9\u77e9\u9635\u4e3a$M$): $M = C \\cdot L$. \u7531\u4e8e\u6211\u4eec\u7684\u76ee\u6807\u662f\u4f7f\u65b0\u914d\u5bf9\u77e9\u9635$M$\u6210\u4e3a\u5355\u4f4d\u77e9\u9635$I$, \u5373:$I = C \\cdot L$. \u6839\u636e\u9006\u77e9\u9635\u7684\u5b9a\u4e49, \u6ee1\u8db3\u6b64\u65b9\u7a0b\u7684\u552f\u4e00\u89e3\u4e3a: $C = L^{-1}$. \u56e0\u6b64, \u6784\u9020\u89c4\u8303\u65b0\u57fa$\\beta_i$\u6240\u9700\u7684\u7ebf\u6027\u7ec4\u5408\u7cfb\u6570$c_{ik}$, \u5fc5\u7136\u4e14\u7cbe\u786e\u5730\u7b49\u4e8e\u65e7\u914d\u5bf9\u77e9\u9635$L$\u7684\u9006\u77e9\u9635\u7684\u5143\u7d20, \u5373$c_{ik} = (L^{-1})_{ik}$.<br \/>\n$\\\\$ \u7efc\u4e0a\u6240\u8ff0, \u8bba\u6587\u5b9e\u9645\u4f7f\u7528\u7684\u662f\u4e00\u7ec4\u7531$\\gamma_i$\u548c$\\overline{\\gamma_i}$\u6270\u52a8\u5f97\u5230\u7684\u68c0\u6d4b\u66f2\u7ebf$\\alpha_j$, \u4ee5\u53ca\u76ee\u6807\u66f2\u7ebf\u4e0e\u8fd9\u4e9b\u68c0\u6d4b\u66f2\u7ebf\u4e4b\u95f4\u7684\u73af\u7ed5\u6570. \u5982\u679c\u501f\u7528$H^1$\u548c\u5e9e\u52a0\u83b1\u5bf9\u5076\u7684\u8bed\u8a00, \u53ef\u4ee5\u628a\u8fd9\u4e00\u8fc7\u7a0b\u7c7b\u6bd4\u4e3a\u7528\u51e0\u4f55&#8221;\u6d4b\u91cf\u5668&#8221;\u8bfb\u53d6\u540c\u8c03\u7c7b\u7684\u5750\u6807; \u4f46\u4e25\u683c\u6765\u8bf4, \u8bba\u6587\u7b97\u6cd5\u7684\u53ef\u8ba1\u7b97\u5bf9\u8c61\u4ecd\u7136\u662f\u66f2\u7ebf\u4e4b\u95f4\u7684\u73af\u7ed5\u6570\u77e9\u9635$L$\u53ca\u5176\u9006\u77e9\u9635.<br \/>\n$\\\\$ PS: \u539f\u6587Remark\u8bf4, \u867d\u7136\u6bcf\u4e2a$\\gamma_i$, $\\overline{\\gamma_i}$\u662f\u8fde\u901aLoop, \u4f46\u7ecf\u8fc7\u77e9\u9635\u53cd\u6f14\u5f97\u5230\u7684$h_i$, $t_j$\u53ef\u80fd\u5305\u542b\u591a\u4e2a\u8fde\u901a\u5206\u91cf, \u53ea\u662f\u5b9e\u9a8c\u91cc\u5f88\u5c11\u51fa\u73b0, \u6545\u8f93\u51fa\u4e0d\u4e00\u5b9a\u662f\u5355\u6761\u7b80\u5355\u95ed\u66f2\u7ebf.<\/p>\n<p><strong>6. \u51e0\u4f55\u4f18\u5316<\/strong><\/p>\n<p>\u7406\u8bba\u6b65\u9aa4\u5f97\u5230\u7684\u628a\u624b \/ \u96a7\u9053\u73af\u53ef\u80fd\u5f88\u957f, \u7ed5\u5f97\u4e0d\u81ea\u7136. \u8bba\u6587\u968f\u540e\u7528\u7c7b\u4f3c\u6700\u77ed\u540c\u8c03\u57fa\u7684\u601d\u60f3\u505a\u73af\u8def\u6536\u7d27. \u4f5c\u8005\u89c2\u5bdf\u5230\u77ed\u73af\u901a\u5e38\u53ef\u4ee5\u8868\u793a\u4e3a: \u4e00\u6761\u8fb9$e = (u, v)$\u52a0\u4e0a\u4ece$u$, $v$\u5230\u67d0\u4e2a\u6839\u70b9$w$\u7684\u4e24\u6761\u6700\u77ed\u8def\u5f84.<br \/>\n$\\\\$ \u8fd9\u79cd\u73af\u79f0\u4e3a\u5178\u8303\u73af\u8def. \u5b8c\u6574\u679a\u4e3e\u6240\u6709\u5178\u8303\u73af\u8def\u6027\u80fd\u5f00\u9500\u5f88\u5927, \u56e0\u6b64\u8bba\u6587\u91c7\u7528\u8fed\u4ee3\u7b56\u7565.<br \/>\n$\\\\$ $\\cdot$ \u6bcf\u8f6e\u53ea\u9009$O(g)$\u4e2a\u57fa\u70b9;<br \/>\n$\\\\$ $\\cdot$ \u4ece\u8fd9\u4e9b\u57fa\u70b9\u6784\u9020\u6700\u77ed\u8def\u5f84\u6811;<br \/>\n$\\\\$ $\\cdot$ \u7531\u975e\u6811\u8fb9\u751f\u6210\u5019\u9009\u5178\u8303\u73af\u8def;<br \/>\n$\\\\$ $\\cdot$ \u6309\u957f\u5ea6\u6392\u5e8f;<br \/>\n$\\\\$ $\\cdot$ \u9009\u62e9\u524d$g$\u6761\u72ec\u7acb\u4e14\u4ecd\u5c5e\u4e8e\u628a\u624b \/ \u96a7\u9053\u5b50\u7a7a\u95f4\u7684\u73af;<br \/>\n$\\\\$ $\\cdot$ \u91cd\u590d\u76f4\u5230\u603b\u957f\u5ea6\u6536\u655b\u6216\u8fbe\u5230\u8fed\u4ee3\u4e0a\u9650.<br \/>\n$\\\\$ \u8fd9\u4e00\u6b65\u662f\u542f\u53d1\u5f0f\u7684, \u4f46\u5b9e\u9a8c\u4e0a\u6548\u679c\u5f88\u597d.<\/p>\n<p><strong>7. \u590d\u6742\u5ea6<\/strong><\/p>\n<p>\u7b2c\u4e00\u90e8\u5206, \u4e5f\u5c31\u662f\u8ba1\u7b97\u521d\u59cb\u628a\u624b \/ \u96a7\u9053\u57fa, \u7406\u8bba\u590d\u6742\u5ea6\u4e3b\u8981\u4e3a$O(g^2 n^2)$, \u5176\u4e2d$n$\u662f\u9876\u70b9\u6570, $g$\u662f\u4e8f\u683c. \u4f46\u4f5c\u8005\u89c2\u5bdf\u5230\u5b9e\u9645\u8fd0\u884c\u4e2d\u66f4\u63a5\u8fd1\u7ebf\u6027. \u4f18\u5316\u90e8\u5206\u6bcf\u8f6e\u5927\u7ea6\u662f: $O(gn \\log n + g^3 n)$; \u82e5\u8fed\u4ee3\u6b21\u6570\u4e3a\u5c0f\u5e38\u6570$k$, \u603b\u590d\u6742\u5ea6\u4e3a: $O(gnk \\log n + $$ g^3 nk)$. \u76f8\u6bd4\u65e7\u7b97\u6cd5, \u65e7\u7b97\u6cd5\u9700\u8981\u5bf9\u7a7a\u95f4\u505a\u56db\u9762\u4f53\u5256\u5206, \u5256\u5206\u7ed3\u679c\u9762\u6570\u53ef\u80fd\u8fdc\u5927\u4e8e\u539f\u59cb\u8868\u9762\u7f51\u683c, \u800c\u4e14\u540e\u7eed\u7b97\u6cd5\u53ef\u80fd\u6709\u5f88\u9ad8\u590d\u6742\u5ea6.<\/p>\n<p><strong>8. \u5b9e\u9a8c\u7ed3\u8bba<\/strong><\/p>\n<p>\u8bba\u6587\u5728\u591a\u79cd\u6a21\u578b\u4e0a\u6d4b\u8bd5, \u5305\u62ec\u5149\u6ed1\u66f2\u9762, CAD\u6a21\u578b, \u7ebd\u7ed3\u66f2\u9762, \u9ad8\u4e8f\u683c\u6a21\u578b, \u566a\u58f0\u6a21\u578b, \u975e\u5747\u5300\u91c7\u6837\u6a21\u578b, \u5c0f\u5b54\u6d1e\u6a21\u578b\u7b49. \u5b9e\u9a8c\u8868\u660e:<br \/>\n$\\\\$ $\\cdot$ Reeb\u56fe\u9636\u6bb5\u975e\u5e38\u5feb;<br \/>\n$\\\\$ $\\cdot$ \u5927\u90e8\u5206\u65f6\u95f4\u82b1\u5728\u73af\u8def\u6536\u7d27;<br \/>\n$\\\\$ $\\cdot$ \u76f8\u6bd4Dey at al.2008\u7684\u4f53\u7f51\u683c\u7b97\u6cd5, \u901f\u5ea6\u901a\u5e38\u5feb\u4e00\u4e2a\u6216\u591a\u4e2a\u6570\u91cf\u7ea7;<br \/>\n$\\\\$ $\\cdot$ \u5bf9\u7a00\u758f\u7f51\u683c\u7a33\u5b9a;<br \/>\n$\\\\$ $\\cdot$ \u5bf9\u566a\u58f0, \u975e\u5747\u5300\u91c7\u6837, \u5c16\u9510\u7279\u5f81, \u5c0f\u7b80\u5355\u5b54\u6d1e\u8f83\u9c81\u68d2;<br \/>\n$\\\\$ $\\cdot$ \u5bf9\u9ad8\u4e8f\u683c\u6a21\u578b\u4e5f\u80fd\u5904\u7406, \u4f8b\u5982\u7ed3\u80a0\u6a21\u578b\u4e8f\u683c\u8fbe\u5230160.<br \/>\n$\\\\$ \u8868\u683c\u4e2d\u4e00\u4e2a\u5178\u578b\u5bf9\u6bd4\u662f: \u4f5b\u50cf\u6a21\u578b\u8bba\u6587\u603b\u65f6\u95f4\u7ea65.86s, \u800c\u65e7\u7b97\u6cd5\u9884\u5904\u7406\u7ea6697.3s, \u540e\u7eed\u73af\u8ba1\u7b97\u8fd8\u5931\u8d25. \u5bf9\u4e8e\u66f4\u9ad8\u9762\u6570\u7684\u6a21\u578b, \u65e7\u65b9\u6cd5\u751a\u81f3\u8d85\u8fc710\u5c0f\u65f6\u672a\u5b8c\u6210.<\/p>\n<p><strong>9. \u5c40\u9650\u6027<\/strong><\/p>\n<p>\u8bba\u6587\u660e\u786e\u6307\u51fa\u4e24\u4e2a\u4e3b\u8981\u9650\u5236:<br \/>\n$\\\\$ 1) \u5e26\u8fb9\u754c\u66f2\u9762\u7684\u95ee\u9898\u6ca1\u6709\u6839\u672c\u89e3\u51b3. \u628a\u624b \/ \u96a7\u9053\u73af\u7684\u5b9a\u4e49\u4f9d\u8d56\u95ed\u5408\u66f2\u9762\u628a\u7a7a\u95f4\u5206\u6210\u5185\u90e8\u548c\u5916\u90e8. \u82e5\u66f2\u9762\u6709\u8fb9\u754c, \u5982\u4f55\u5c01\u6d1e\u4f1a\u5f71\u54cd\u5185\u90e8 \/ \u5916\u90e8\u7ed3\u6784, \u4ece\u800c\u5f71\u54cd\u67d0\u6761\u73af\u5230\u5e95\u662f\u628a\u624b, \u96a7\u9053\u8fd8\u662f\u4e8c\u8005\u5747\u4e0d\u662f. \u8bba\u6587\u53ea\u5bf9\u5c0f\u7684, \u53ef\u7528\u5706\u76d8\u5c01\u4f4f\u7684\u6d1e\u91c7\u7528\u7b80\u5355\u542f\u53d1\u5f0f\u5904\u7406.<br \/>\n$\\\\$ 2) \u65e0\u6cd5\u4fdd\u8bc1\u5f97\u5230\u6700\u77ed\u628a\u624b \/ \u96a7\u9053\u57fa. \u521d\u59cb\u57fa\u6709\u7406\u8bba\u4fdd\u8bc1, \u4f46\u73af\u8def\u6536\u7d27\u662f\u542f\u53d1\u5f0f. \u6c42\u6700\u77ed\u628a\u624b \/ \u96a7\u9053\u57fa\u53ef\u80fd\u7c7b\u4f3c\u56fa\u5b9a\u540c\u8c03\u7c7b\u4e2d\u6c42\u6700\u77ed\u73af\u7684\u95ee\u9898, \u540e\u8005\u4e00\u822c\u5177\u6709NP-Hard\u5473\u9053. \u4f5c\u8005\u628a\u662f\u5426\u5b58\u5728\u9ad8\u6548\u6700\u4f18\u7b97\u6cd5\u7559\u4f5c\u672a\u6765\u95ee\u9898.<\/p>\n<p><strong>10. \u6574\u4f53\u8bc4\u4ef7<\/strong><\/p>\n<p>\u8fd9\u7bc7\u8bba\u6587\u7684\u8d21\u732e\u5f88\u6e05\u6670: \u5b83\u628a\u628a\u624b \/ \u96a7\u9053\u73af\u7684\u8ba1\u7b97\u4ece\u6027\u80fd\u5f00\u9500\u6602\u8d35\u7684\u4e09\u7ef4\u4f53\u7f51\u683c\u95ee\u9898, \u8f6c\u5316\u4e3a\u8868\u9762\u4e0a\u7684Reeb\u56fe\u4e0e\u73af\u7ed5\u6570\u8ba1\u7b97\u95ee\u9898. \u7406\u8bba\u4e0a, \u5b83\u4fdd\u8bc1\u80fd\u5f97\u5230\u6b63\u786e\u7684\u628a\u624b \/ \u96a7\u9053\u57fa; \u5b9e\u8df5\u4e0a, \u5b83\u663e\u8457\u52a0\u901f, \u5e76\u4e14\u907f\u514d\u4fee\u6539\u539f\u59cb\u8f93\u5165\u7f51\u683c. \u5b83\u6700\u9002\u5408\u7684\u573a\u666f\u662f\u9700\u8981\u5728\u590d\u6742\u4e09\u7ef4\u6a21\u578b\u4e2d\u5feb\u901f, \u7a33\u5b9a\u5730\u8bc6\u522b\u628a\u624b\u548c\u96a7\u9053\u62d3\u6251\u7279\u5f81.<\/p>\n<p><strong>11. QA<\/strong><\/p>\n<p><strong>Q1:<\/strong> Reeb\u56fe\u4e3a\u4ec0\u4e48\u591f\u7528? \u4e3a\u4ec0\u4e48\u8fd9\u4e00\u6b65\u8fd8\u4e0d\u8db3\u4ee5\u76f4\u63a5\u5f97\u5230\u628a\u624b \/ \u96a7\u9053\u73af?<br \/>\n<strong>A1:<\/strong> \u7ed9\u5b9a\u4e8f\u683c\u4e3a$g$\u7684\u95ed\u5408\u8fde\u901a\u66f2\u9762$M$, \u5bf9\u9ad8\u5ea6\u51fd\u6570$h$\u8ba1\u7b97Reeb\u56fe$Rb_M( $$ h)$. \u8bba\u6587\u4e0d\u662f\u76f4\u63a5\u5728\u66f2\u9762\u4e0a\u73af, \u800c\u662f\u5148\u5728Reeb\u56fe\u4e0a\u627e\u73af, \u56e0\u4e3a: Reeb\u56fe\u4e2d\u5b58\u5728$g$\u4e2a\u72ec\u7acb\u73af; \u8fd9\u4e9b\u73af\u53ef\u4ee5\u6620\u5c04\u56de\u66f2\u9762, \u5f97\u5230\u4e00\u7ec4\u66f2\u9762\u73af$\\gamma_i$, \u5b83\u4eec\u6355\u6349\u4e86$H_1(M)$\u4e2d\u4e0e\u4e8f\u683c\u5bf9\u5e94\u7684\u4e00\u90e8\u5206\u62d3\u6251\u7ed3\u6784.<br \/>\n$\\\\$ \u4f46\u5b83\u4eec\u8fd8\u4e0d\u80fd\u76f4\u63a5\u5f53\u4f5c\u628a\u624b \/ \u96a7\u9053\u73af, \u56e0\u4e3a\u666e\u901a\u975e\u5e73\u51e1\u73af\u53ea\u8bf4\u660e\u5b83\u5728\u66f2\u9762$M$\u4e0a\u975e\u5e73\u51e1, \u4e0d\u8bf4\u660e\u5b83\u5728\u5185\u90e8$I$\u6216\u5916\u90e8$O$\u4e2d\u975e\u5e73\u51e1. \u628a\u624b \/ \u96a7\u9053\u7684\u5b9a\u4e49\u4f9d\u8d56\u4e09\u7ef4\u5d4c\u5165, \u800cReeb\u56fe\u73af\u672c\u8eab\u8fd8\u6ca1\u6709\u5b8c\u6210\u8fd9\u79cd\u5185\u5916\u5206\u7c7b.<\/p>\n<p><strong>Q2:<\/strong> \u6700\u5927\u751f\u6210\u6811\u7684\u7279\u6b8a\u4f5c\u7528\u662f\u4ec0\u4e48? \u4e3a\u4ec0\u4e48&#8221;\u6700\u4f4e\u70b9\u662f\u67d0\u7c7b\u978d\u70b9&#8221; \u5bf9\u540e\u7eed\u6784\u9020\u5bf9\u5076\u73af\u8def\u5f88\u91cd\u8981?<br \/>\n<strong>A2:<\/strong> \u8bba\u6587\u5728Reeb\u56fe\u4e2d\u7ed9\u6bcf\u6761\u8fb9\u8bbe\u7f6e\u6743\u91cd, \u6743\u91cd\u7b49\u4e8e\u8be5\u8fb9\u6700\u4f4e\u70b9\u7684\u9ad8\u5ea6\u503c, \u7136\u540e\u8ba1\u7b97\u6700\u5927\u6743\u91cd\u751f\u6210\u6811. \u6bcf\u6761\u4e0d\u5728\u6811\u4e2d\u7684\u8fb9\u90fd\u4f1a\u8bf1\u5bfc\u4e00\u4e2a\u5178\u8303\u5708. \u8fd9\u6837\u505a\u7684\u5173\u952e\u597d\u5904\u662f: \u6bcf\u4e2aReeb\u56fe\u73af$c_e$\u7684\u6700\u4f4e\u70b9\u662f$e$\u7684\u6700\u4f4e\u70b9, \u5e76\u4e14\u8fd9\u4e2a\u70b9\u5bf9\u5e94\u66f2\u9762\u4e0a\u7684\u4e00\u4e2a\u4e0a\u5206\u53c9\u978d\u70b9, \u4e5f\u5c31\u662f\u66f2\u9762\u9ad8\u5ea6\u51fd\u6570\u4e0a\u7684\u5206\u88c2\u978d\u70b9.<br \/>\n$\\\\$ &#8220;\u6700\u4f4e\u70b9\u662f\u67d0\u7c7b\u978d\u70b9&#8221; \u5bf9\u540e\u7eed\u6784\u9020\u5bf9\u5076\u73af\u8def\u5f88\u91cd\u8981, \u56e0\u4e3a\u5728\u5206\u88c2\u978d\u70b9\u7565\u9ad8\u5904, \u7b49\u9ad8\u7ebf\u4f1a\u5206\u88c2\u6210\u4e24\u4e2a\u7b49\u9ad8\u7ebf, \u8bba\u6587\u6b63\u662f\u5229\u7528\u8fd9\u4e2a\u5c40\u90e8\u7ed3\u6784\u4e3a$\\gamma_i$\u6784\u9020\u5bf9\u5076\u73af\u8def $\\overline{\\gamma_i}$.<\/p>\n<p><strong>Q3:<\/strong> \u5bf9\u5076\u73af\u8def\u7684\u89d2\u8272\u662f\u4ec0\u4e48? \u5bf9\u5076\u73af\u8def\u4e3a\u4ec0\u4e48\u4e0d\u662f\u88c5\u9970\u54c1, \u800c\u662f\u540e\u9762\u533a\u5206\u628a\u624b \/ \u96a7\u9053\u7684\u5fc5\u8981\u6750\u6599?<br \/>\n<strong>A3:<\/strong> \u5bf9\u6bcf\u4e2a\u6620\u5c04\u56de\u66f2\u9762\u7684\u73af$\\gamma_i$, \u8bba\u6587\u5728\u5176\u6700\u4f4e\u978d\u70b9$p$\u7684\u7565\u9ad8\u7b49\u9ad8\u9762$h(p) + $$ \\varepsilon$\u4e0a\u53d6\u4e00\u6761\u7b49\u9ad8\u7ebf, \u4f5c\u4e3a\u5bf9\u5076\u73af\u8def $\\overline{\\gamma_i}$. $\\gamma_i$\u4e0e$\\overline{\\gamma_i}$\u67091\u4e2a\u4ea4\u70b9; \u6240\u6709$\\overline{\\gamma_i}$\u5f7c\u6b64\u4e0d\u76f8\u8fde, \u5e76\u4e14\u4f4d\u4e8e\u4e0d\u540c\u7684\u6c34\u5e73\u96c6. \u56e0\u6b64$\\gamma_i \\cup \\overline{\\gamma_i}$\u53ef\u4ee5\u4f5c\u4e3a$H_1(M)$\u7684\u4e00\u7ec4\u57fa.<br \/>\n$\\\\$ \u5bf9\u5076\u73af\u8def\u4e0d\u662f\u9644\u5c5e\u7269. \u53ea\u6709$\\gamma_i$\u4e0d\u591f\u5f62\u6210\u5b8c\u6574\u7684$2g$\u7ef4\u66f2\u9762\u540c\u8c03\u57fa; \u52a0\u5165\u5bf9\u5076\u73af\u8def\u540e, \u7b97\u6cd5\u624d\u6709\u8db3\u591f\u7684\u73af\u6765\u6784\u9020\u73af\u7ed5\u6570\u77e9\u9635, \u5e76\u901a\u8fc7\u77e9\u9635\u53cd\u6f14\u5206\u79bb\u51fa\u628a\u624b\u57fa\u548c\u96a7\u9053\u57fa.<\/p>\n<p><strong>Q4:<\/strong> \u5c40\u90e8Push\u7684\u76ee\u7684\u662f\u4ec0\u4e48? \u4e3a\u4ec0\u4e48\u5fc5\u987b\u6270\u52a8, \u800c\u4e0d\u662f\u76f4\u63a5\u62ff\u539f\u6765\u7684\u73af\u7b97\u73af\u7ed5\u6570?<br \/>\n<strong>A4:<\/strong> \u8bba\u6587\u6839\u636e\u6700\u4f4e\u978d\u70b9\u9644\u8fd1\u4e09\u4e2a\u7b49\u9ad8\u7ebf$\\beta_0$, $\\beta_1$, $\\beta_2$\u7684\u5305\u542b\u5173\u7cfb, \u4ee5\u53ca\u5b83\u4eec\u8f7b\u5faePush\u540e\u843d\u5728$I$\u8fd8\u662f$O$, \u628a$\\gamma_i$\u4e0e$\\overline{\\gamma_i}$\u5206\u6210\u4e24\u7ec4. \u8fd9\u4e00\u6b65\u7684\u76ee\u6807\u4e0d\u662f\u76f4\u63a5\u5f97\u5230\u628a\u624b \/ \u96a7\u9053\u73af, \u800c\u662f\u5f97\u5230\u4e24\u7ec4\u6270\u52a8\u540e\u7684\u73af$A_1$\u548c$A_2$, \u5b83\u4eec\u5206\u522b\u6784\u6210$H_1(I)$\u548c$H_1(O)$\u7684\u5708\u57fa, \u5e76\u4e14\u8fd9\u4e9b\u6270\u52a8\u540e\u7684\u73af\u4e0e\u539f\u66f2\u9762\u4e0a\u7684$\\gamma_i$, $\\overline{\\gamma_i}$\u4fdd\u6301\u4e0d\u76f8\u8fde.<br \/>\n$\\\\$ \u6270\u52a8\u662f\u56e0\u4e3a\u73af\u7ed5\u6570\u53ea\u5bf9\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u4e24\u6761\u4e0d\u76f8\u4ea4\u7684\u73af\u5b9a\u4e49. \u800c\u539f\u6765\u7684$\\gamma_i$\u548c$\\overline{\\gamma_i}$\u5728\u66f2\u9762\u4e0a\u53ef\u80fd\u76f8\u4ea4, \u4f8b\u5982\u5bf9\u5e94\u7684\u4e00\u5bf9\u6709\u4e00\u4e2a\u4ea4\u70b9, \u6240\u4ee5\u9700\u8981\u628a\u5176\u4e2d\u4e00\u7ec4\u8f7b\u5fae\u63a8\u5165\u5185\u90e8\u6216\u5916\u90e8.<\/p>\n<p><strong>Q5:<\/strong> \u73af\u7ed5\u6570\u77e9\u9635\u4e3a\u4ec0\u4e48\u53ef\u9006? \u73af\u7ed5\u6570\u77e9\u9635$L$\u53ef\u9006\u5728\u7b97\u6cd5\u4e2d\u5230\u5e95\u5141\u8bb8\u6211\u4eec\u505a\u4ec0\u4e48?<br \/>\n<strong>A5:<\/strong> \u8bba\u6587\u6784\u9020$2g \\times 2g$\u7684\u73af\u7ed5\u6570\u77e9\u9635$L$, \u5176\u4e2d\u5143\u7d20\u662f:$$L_{ij} = Lk(\\gamma_i, \\alpha_j), i, j \\in [1, 2g],$$\u5176\u4e2d, $\\gamma_{g + i} = \\overline{\\gamma_i}$, $\\alpha_j$\u662f$\\gamma_j$\u6216$\\overline{\\gamma_j}$\u7684\u6270\u52a8\u7248\u672c, $\\alpha_{g + i} = \\overline{\\alpha_i}$. \u8bc1\u660e\u53ef\u9006\u65f6, \u8bba\u6587\u901a\u8fc7\u884c\u5217\u7f6e\u6362\u5f97\u5230\u77e9\u9635$L&#8217;$, \u5176\u4e2d\u67d0\u4e2a\u5b50\u77e9\u9635$C = 0$, \u53e6\u5916\u4e24\u4e2a\u5173\u952e\u5b50\u77e9\u9635$A$\u548c$D$\u662f\u4e0a\u4e09\u89d2\u77e9\u9635, \u4e14\u5bf9\u89d2\u7ebf\u5143\u7d20\u5747\u975e0(\u5728$\\mathbb{Z}_2$\u4e0b\u5c31\u662f1). \u56e0\u6b64\u6574\u4e2a\u77e9\u9635\u975e\u5947\u5f02.<br \/>\n$\\\\$ $L$\u53ef\u9006\u5141\u8bb8\u7b97\u6cd5\u7528$L^{-1}$\u5bf9$(\\gamma_i, \\overline{\\gamma_i})$\u505a\u7ebf\u6027\u7ec4\u5408, \u6784\u9020\u65b0\u7684\u73af, \u4f7f\u5b83\u4eec\u4e0e$A_1$ \/ $A_2$\u7684\u73af\u7ed5\u6570\u5448\u73b0\u6307\u5b9a\u6a21\u5f0f, \u8fd9\u4e2a\u6a21\u5f0f\u6b63\u662f\u5224\u65ad\u628a\u624b \/ \u96a7\u9053\u7684\u4f9d\u636e.<\/p>\n<p><strong>Q6:<\/strong> \u4ece\u57fa\u5230\u771f\u6b63\u7684\u628a\u624b \/ \u96a7\u9053\u7684\u8f6c\u6362\u6d41\u7a0b\u662f\u4ec0\u4e48? \u4e3a\u4ec0\u4e48\u8fd9\u4e00\u6b65\u6bd4&#8221;\u53ea\u77e5\u9053\u5b83\u5728$H_1(M)$\u4e2d\u975e\u5e73\u51e1&#8221; \u66f4\u5f3a?<br \/>\n<strong>A6:<\/strong> \u8bba\u6587\u7528$L^{-1}$\u5bf9$\\{ \\gamma_i, \\overline{\\gamma_i} \\}$\u4f5c\u7ebf\u6027\u7ec4\u5408, \u5f97\u5230\u65b0\u7684\u73af$\\beta_i$. \u8fd9\u4e9b\u73af\u4e4b\u6240\u4ee5\u80fd\u88ab\u5224\u5b9a\u4e3a\u628a\u624b\u6216\u96a7\u9053, \u662f\u56e0\u4e3a\u5b83\u4eec\u548c$A_1$ \/ $A_2$\u7684\u73af\u7ed5\u6570\u6ee1\u8db3\u4e00\u79cd&#8221;\u9009\u62e9\u6027\u6d88\u5931&#8221; \u7684\u6a21\u5f0f. \u5982\u679c\u67d0\u4e2a\u73af\u8def\u4e0e$A_1$\u4e2d\u6240\u6709\u73af\u7684\u73af\u7ed5\u6570\u90fd\u4e3a0, \u4f46\u4e0e$A_2$\u4e2d\u81f3\u5c11\u4e00\u4e2a\u73af\u7684\u73af\u7ed5\u6570\u4e3a1, \u5219\u5b83\u662f\u96a7\u9053\u73af. \u5bf9\u79f0\u5730, \u5982\u679c\u5b83\u4e0e$A_2$\u5168\u90e8\u73af\u7ed5\u6570\u4e3a0, \u4f46\u4e0e$A_1$\u81f3\u5c11\u4e00\u4e2a\u73af\u7ed5\u6570\u4e3a1, \u5219\u5b83\u662f\u628a\u624b\u73af.<br \/>\n$\\\\$ \u8fd9\u6bd4&#8221;\u5728$H_1(M)$\u4e2d\u975e\u5e73\u51e1&#8221; \u66f4\u5f3a, \u56e0\u4e3a$H_1(M)$\u7684\u975e\u5e73\u51e1\u6027\u53ea\u8bf4\u660e\u5b83\u4e0d\u662f\u66f2\u9762\u4e0a\u7684\u8fb9\u754c; \u628a\u624b \/ \u96a7\u9053\u8fd8\u8981\u6c42\u5b83\u5728\u5185\u90e8\u6216\u5916\u90e8\u5176\u4e2d\u4e00\u4fa7\u53d8\u6210\u5e73\u51e1, \u5728\u53e6\u4e00\u4fa7\u4fdd\u6301\u975e\u5e73\u51e1. \u73af\u7ed5\u6570\u7ed9\u51fa\u4e86\u8fd9\u79cd\u76f8\u5bf9\u5185\u5916\u7a7a\u95f4\u7684\u5224\u522b\u4fe1\u606f.<\/p>\n<p><strong>Q7:<\/strong> \u73af\u8def\u6536\u7d27\u4e3a\u4ec0\u4e48\u53ea\u662f\u542f\u53d1\u5f0f? \u4e3a\u4ec0\u4e48\u8bba\u6587\u4e0d\u80fd\u58f0\u79f0\u6700\u7ec8\u5f97\u5230\u7684\u662f\u6700\u77ed\u7684\u628a\u624b \/ \u96a7\u9053\u57fa?<br \/>\n<strong>A7:<\/strong> \u521d\u59cb\u628a\u624b \/ \u96a7\u9053\u57fa\u6709\u7406\u8bba\u4fdd\u8bc1, \u4f46\u51e0\u4f55\u4e0a\u53ef\u80fd\u5f88\u957f. \u8bba\u6587\u7528\u6700\u77ed\u8def\u5f84\u6811\u4ea7\u751f\u5178\u8303\u73af\u8def, \u5e76\u8fed\u4ee3\u9009\u51fa\u8f83\u77ed\u7684\u72ec\u7acb\u628a\u624b \/ \u96a7\u9053\u73af. \u6bcf\u4e2a\u5178\u8303\u73af\u8def\u7531\u4e00\u6761\u975e\u6811\u8fb9$(u, v)$\u52a0\u4e0a\u4e24\u6761\u4ece$u$, $v$\u5230\u540c\u4e00\u6839\u70b9$w$\u7684\u6700\u77ed\u8def\u5f84\u7ec4\u6210. \u7b97\u6cd5\u6bcf\u8f6e\u53ea\u4ece$O(g)$\u4e2a\u57fa\u70b9\u6784\u5efa\u6700\u77ed\u8def\u5f84\u6811, \u800c\u4e0d\u662f\u4ece$n$\u4e2a\u9876\u70b9\u5168\u90e8\u5efa\u6811, \u4ee5\u63d0\u9ad8\u6548\u7387.<br \/>\n$\\\\$ \u8bba\u6587\u4e0d\u80fd\u58f0\u79f0\u6700\u7ec8\u5f97\u5230\u6700\u77ed\u628a\u624b \/ \u96a7\u9053\u57fa, \u56e0\u4e3a\u73af\u8def\u6536\u7d27\u662f\u542f\u53d1\u5f0f. \u867d\u7136\u6700\u77ed\u540c\u8c03\u57fa\u6709\u591a\u9879\u5f0f\u7b97\u6cd5, \u4f46\u6700\u77ed\u628a\u624b \/ \u96a7\u9053\u57fa\u8981\u5728\u628a\u624b \/ \u96a7\u9053\u5b50\u7a7a\u95f4\u8fd9\u79cd\u989d\u5916\u7ea6\u675f\u4e0b\u4f18\u5316, \u73b0\u6709\u6700\u77ed\u540c\u8c03\u57fa\u65b9\u6cd5\u4e0d\u80fd\u76f4\u63a5\u5957\u7528; \u4f5c\u8005\u8fd8\u6307\u51fa\u8be5\u95ee\u9898\u53ef\u80fd\u5177\u6709\u7c7b\u4f3cNP-Hard\u95ee\u9898\u7684\u56f0\u96be\u6027.<\/p>\n<p><strong>12. \u9644\u5f55<\/strong><\/p>\n<p><strong>12.1 \u9644\u5f55A<\/strong><\/p>\n<p>\u672c\u9644\u5f55\u4f7f\u7528\u5bf9\u5076\u7a7a\u95f4$H^1$\u548c\u5e9e\u52a0\u83b1\u5bf9\u5076\u6765\u89e3\u91ca\u73af\u7ed5\u6570\u77e9\u9635\u4e3a\u4ec0\u4e48\u50cf\u4e00\u5957&#8221;\u6d4b\u91cf\u5750\u6807&#8221;\u90a3\u6837\u5de5\u4f5c. \u8fd9\u662f\u4e00\u79cd\u5e2e\u52a9\u7406\u89e3\u7684\u6570\u5b66\u7c7b\u6bd4, \u4e0d\u662f\u8bba\u6587\u5bf9$\\alpha_j$\u7684\u539f\u6587\u5b9a\u4e49; \u8bba\u6587\u4e2d\u7684$\\alpha_j$\u4e25\u683c\u6765\u8bf4\u662f\u7531$\\gamma_i$\u548c$\\overline{\\gamma_i}$\u6270\u52a8\u5f97\u5230\u7684\u4e09\u7ef4\u66f2\u7ebf. \u5728\u8fd9\u4e2a\u524d\u63d0\u4e0b, \u6211\u4eec\u53ef\u4ee5\u4ece\u4ee5\u4e0b\u56db\u4e2a\u7ef4\u5ea6\u7406\u89e3\u8fd9\u79cd\u7c7b\u6bd4.<\/p>\n<p><strong>12.1.1 \u52a8\u673a: \u4e3a\u4ec0\u4e48\u540c\u8c03\u7c7b$H_1$\u9700\u8981\u88ab&#8221;\u523b\u753b&#8221;?<\/strong><\/p>\n<p>\u5728\u62d3\u6251\u5b66\u4e2d, \u7b2c\u4e00\u540c\u8c03\u7fa4$H_1$\u7684\u5143\u7d20\u662f\u540c\u8c03\u7c7b.<br \/>\n$\\\\$ $\\cdot$ \u51e0\u4f55\u672c\u8d28: \u4e00\u4e2a\u540c\u8c03\u7c7b\u5e76\u4e0d\u662f\u6307\u67d0\u4e00\u6761\u5177\u4f53\u7684\u95ed\u5408\u66f2\u7ebf, \u800c\u662f\u6307\u65e0\u6570\u6761\u76f8\u5dee\u4e00\u4e2a\u8fb9\u754c\u7684\u95ed\u5408\u66f2\u7ebf\u7684&#8221;\u7b49\u4ef7\u7c7b&#8221;.<br \/>\n$\\\\$ $\\cdot$ \u8ba1\u7b97\u56f0\u5883: \u7531\u4e8e\u540c\u8c03\u7c7b\u662f\u4e00\u4e2a\u65e0\u9650\u96c6\u5408(\u5305\u542b\u65e0\u6570\u6761\u5f62\u72b6\u5404\u5f02\u7684\u66f2\u7ebf), \u8ba1\u7b97\u673a\u65e0\u6cd5\u76f4\u63a5\u5b58\u50a8\u548c\u5904\u7406&#8221;\u7b49\u4ef7\u7c7b&#8221; \u672c\u8eab. \u6211\u4eec\u9700\u8981\u4e00\u79cd\u673a\u5236, \u7ed9\u6bcf\u4e00\u4e2a\u540c\u8c03\u7c7b\u5206\u914d\u4e00\u4e2a\u552f\u4e00\u7684, \u6709\u9650\u7684\u4ee3\u6570\u6307\u7eb9(\u5373\u5750\u6807), \u4ee5\u4fbf\u8fdb\u884c\u77e9\u9635\u8fd0\u7b97\u548c\u7ebf\u6027\u65e0\u5173\u6027\u5224\u65ad.<br \/>\n$\\\\$ \u8fd9\u5c31\u5f15\u51fa\u4e86\u6838\u5fc3\u95ee\u9898: \u7528\u4ec0\u4e48\u5de5\u5177\u53bb&#8221;\u6d4b\u91cf&#8221; \u6216&#8221;\u523b\u753b&#8221; \u8fd9\u4e9b\u540c\u8c03\u7c7b.<\/p>\n<p><strong>12.1.2 \u4ee3\u6570\u89c6\u89d2: \u5bf9\u5076\u7a7a\u95f4$H^1$\u7684\u672c\u8d28(&#8220;\u6d4b\u91cf\u5de5\u5177&#8221; \u7684\u96c6\u5408)<\/strong><\/p>\n<blockquote>\n<p>\u5f53\u7cfb\u6570\u53d6\u5728\u57df\u4e0a, \u4f8b\u5982$\\mathbb{Z}_2$\u65f6, $H^1(X; \\mathbb{Z}_2)$\u662f\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4, \u5e76\u4e14\u53ef\u81ea\u7136\u7406\u89e3\u4e3a$H_1(X; \\mathbb{Z}_2)$\u7684\u5bf9\u5076\u7a7a\u95f4. \u66f4\u4e00\u822c\u5730, \u4e0a\u540c\u8c03\u7fa4\u662f\u7531\u540c\u8c03\u7fa4\u4e0a\u7684\u4ee3\u6570\u6d4b\u91cf\u6620\u5c04\u7ec4\u6210\u7684\u5bf9\u8c61; \u5f53\u7cfb\u6570\u4e0d\u662f\u57df\u65f6, \u5e94\u79f0\u4e3a\u5bf9\u5076\u7fa4\u6216\u5bf9\u5076\u6a21, \u800c\u4e0d\u5b9c\u7b80\u5355\u79f0\u4e3a\u5bf9\u5076\u7a7a\u95f4.<\/p>\n<\/blockquote>\n<p>\u5728\u7ebf\u6027\u4ee3\u6570\u4e2d, \u7ed9\u5b9a\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4$V$, \u5176\u5bf9\u5076\u7a7a\u95f4$V^*$\u662f\u7531\u6240\u6709\u4ece$V$\u6620\u5c04\u5230\u6807\u91cf\u57df(\u5982$\\mathbb{Z}_2$) \u7684\u7ebf\u6027\u6cdb\u51fd\u7ec4\u6210\u7684\u96c6\u5408. \u6620\u5c04\u5230\u4ee3\u6570\u62d3\u6251\u4e2d:<br \/>\n$\\\\$ $\\cdot$ \u540c\u8c03\u7fa4$H_1$: \u626e\u6f14\u5411\u91cf\u7a7a\u95f4$V$\u7684\u89d2\u8272, \u5176\u5143\u7d20\u662f\u540c\u8c03\u7c7b(\u53ef\u7b80\u5355\u7406\u89e3\u4e3a\u95ed\u5408\u66f2\u7ebf\u7c7b).<br \/>\n$\\\\$ $\\cdot$ \u4e0a\u540c\u8c03\u7fa4$H^1$: \u626e\u6f14\u5bf9\u5076\u7a7a\u95f4$V^*$\u7684\u89d2\u8272. $H^1$\u4e2d\u7684\u6bcf\u4e00\u4e2a\u5143\u7d20(\u4e0a\u540c\u8c03\u7c7b), \u672c\u8d28\u4e0a\u662f\u4e00\u4e2a\u7ebf\u6027\u6d4b\u91cf\u5668.<br \/>\n$\\\\$ $\\cdot$ \u523b\u753b: \u5f53\u6211\u4eec\u5c06\u4e00\u4e2a$H^1$\u4e2d\u7684\u6d4b\u91cf\u5668$f$\u4f5c\u7528\u4e8e\u4e00\u4e2a$H_1$\u4e2d\u7684\u540c\u8c03\u7c7b$[\\gamma]$\u65f6, \u4f1a\u8f93\u51fa\u4e00\u4e2a\u6807\u91cf$f([\\gamma]) \\in \\mathbb{Z}_2$. \u8fd9\u4e2a\u6807\u91cf\u5c31\u662f$\\gamma$\u5728\u63a2\u6d4b\u5668$f$\u4e0b\u7684\u4ee3\u6570\u5750\u6807.<br \/>\n$\\\\$ \u7ed3\u8bba: \u5f15\u5165\u5bf9\u5076\u7a7a\u95f4$H^1$, \u5c31\u662f\u4e3a\u4e86\u83b7\u5f97\u4e00\u7ec4&#8221;\u4ee3\u6570\u5c3a\u5b50&#8221;, \u628a\u62bd\u8c61\u7684\u66f2\u7ebf$\\gamma$\u6620\u5c04\u4e3a\u5177\u4f53\u7684\u6570\u5b57.<\/p>\n<p><strong>12.1.3 \u51e0\u4f55\u89c6\u89d2: \u4ec0\u4e48\u662f&#8221;\u51e0\u4f55\u4ee3\u8868\u5143&#8221;(\u5e9e\u52a0\u83b1\u5bf9\u5076\u7684\u5177\u8c61\u5316)?<\/strong><\/p>\n<p>\u867d\u7136$H^1$\u5728\u4ee3\u6570\u4e0a\u662f\u7ebf\u6027\u6cdb\u51fd, \u4f46\u5728\u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u548c\u51e0\u4f55\u5904\u7406\u4e2d, \u7eaf\u4ee3\u6570\u5b9a\u4e49(\u5982\u5355\u7eaf\u4e0a\u94feSimplicial Cochains\u6216\u5fb7\u62c9\u59c6\u5fae\u5206\u5f62\u5f0fDifferential Forms) \u4e0d\u591f\u76f4\u89c2, \u4e14\u96be\u4ee5\u4e0e\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u7684\u51e0\u4f55\u7f51\u683c\u76f4\u63a5\u4ea4\u4e92. \u56e0\u6b64, \u6211\u4eec\u9700\u8981\u5c06\u62bd\u8c61\u7684$H^1$\u5143\u7d20\u5177\u8c61\u5316\u4e3a\u51e0\u4f55\u5b9e\u4f53, \u8fd9\u5c31\u662f\u6240\u8c13\u7684\u51e0\u4f55\u4ee3\u8868\u5143. \u8fd9\u4e00\u8f6c\u5316\u7684\u6570\u5b66\u57fa\u77f3\u662f\u5e9e\u52a0\u83b1\u5bf9\u5076.<br \/>\n$\\\\$ $\\cdot$ \u5e9e\u52a0\u83b1\u5bf9\u5076\u5b9a\u7406: \u5728\u4e00\u4e2a$n$\u7ef4\u95ed\u53ef\u5b9a\u5411\u6d41\u5f62\u4e0a, $k$\u7ef4\u540c\u8c03\u7fa4\u4e0e$(n &#8211; k)$\u7ef4\u4e0a\u540c\u8c03\u7fa4\u540c\u6784.<br \/>\n$\\\\$ $\\cdot$ \u5728\u4e8c\u7ef4\u66f2\u9762($n = 2$) \u4e0a\u7684\u5e94\u7528: \u4e00\u7ef4\u4e0a\u540c\u8c03\u7fa4$H^1$\u4e0e\u4e00\u7ef4\u540c\u8c03\u7fa4$H_1$\u540c\u6784. \u8fd9\u610f\u5473\u7740, $H^1$\u4e2d\u7684\u62bd\u8c61\u6d4b\u91cf\u5668\u53ef\u4ee5\u901a\u8fc7\u66f2\u9762\u4e0a\u7684\u4e00\u7ef4\u5faa\u73af\u6216\u540c\u8c03\u7c7b\u4ee3\u8868\u6765\u51e0\u4f55\u5316\u7406\u89e3; \u4ee3\u8868\u5143\u4e0d\u5fc5\u603b\u662f\u4e00\u6761\u5355\u72ec\u7684\u7b80\u5355\u95ed\u5408\u66f2\u7ebf.<br \/>\n$\\\\$ $\\cdot$ \u5728\u4e09\u7ef4\u7a7a\u95f4\u5d4c\u5165\u4e2d\u7684\u63a8\u5e7f: \u5f53\u66f2\u9762\u5d4c\u5165\u5728\u4e09\u7ef4\u7a7a\u95f4$\\mathbb{R}^3$\u4e2d\u65f6, \u4e3a\u4e86\u8ba1\u7b97\u66f4\u7a33\u5b9a\u7684\u62d3\u6251\u4e0d\u53d8\u91cf(\u5982\u907f\u514d\u66f2\u7ebf\u5728\u66f2\u9762\u4e0a\u76f8\u4ea4\u7684\u5947\u70b9), \u7b97\u6cd5\u901a\u5e38\u4f1a\u5c06\u66f2\u9762\u4e0a\u7684\u63a2\u6d4b\u66f2\u7ebf\u7a0d\u5fae\u63a8\u5165\u4e09\u7ef4\u7a7a\u95f4\u7684\u5185\u90e8\u6216\u5916\u90e8.<br \/>\n$\\\\$ \u7ed3\u8bba: \u82e5\u91c7\u7528\u8fd9\u4e2a\u7c7b\u6bd4, \u8bba\u6587\u4e2d\u7684\u6270\u52a8\u7ebf\u96c6\u5408$\\{ \\alpha_j \\}$\u53ef\u4ee5\u88ab\u770b\u4f5c\u4e00\u7ec4\u51e0\u4f55&#8221;\u5c3a\u5b50&#8221;; \u4f46\u4e25\u683c\u5730\u8bf4, \u539f\u6587\u53ea\u9700\u8981\u5b83\u4eec\u4f5c\u4e3a\u6270\u52a8\u540e\u7684\u68c0\u6d4b\u66f2\u7ebf, \u7528\u6765\u548c$\\gamma_i$\u8ba1\u7b97\u73af\u7ed5\u6570.<\/p>\n<p><strong>12.1.4 \u64cd\u4f5c\u89c6\u89d2: \u5982\u4f55\u5b9e\u73b0&#8221;\u523b\u753b&#8221;(\u53cc\u7ebf\u6027\u914d\u5bf9)?<\/strong><\/p>\n<p>\u6709\u4e86\u540c\u8c03\u7c7b(\u88ab\u6d4b\u5bf9\u8c61$\\gamma_i$) \u548c\u68c0\u6d4b\u66f2\u7ebf$\\alpha_j$, \u6700\u540e\u4e00\u6b65\u662f\u5b9a\u4e49\u5b83\u4eec\u4e4b\u95f4\u7684\u4f5c\u7528\u6cd5\u5219. \u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d, \u4e24\u6761\u4e0d\u76f8\u4ea4\u7684\u95ed\u5408\u66f2\u7ebf$\\gamma$\u548c$\\alpha$\u4e4b\u95f4\u6700\u7ecf\u5178\u7684\u62d3\u6251\u4e0d\u53d8\u91cf\u662f\u73af\u7ed5\u6570($Lk$).<br \/>\n$\\\\$ $\\cdot$ \u51e0\u4f55\u8ba1\u7b97: \u8ba1\u7b97$\\gamma$\u7a7f\u8fc7\u4ee5$\\alpha$\u4e3a\u8fb9\u754c\u7684\u4efb\u610f\u66f2\u9762\u7684\u4ee3\u6570\u6b21\u6570.<br \/>\n$\\\\$ $\\cdot$ \u4ee3\u6570\u610f\u4e49: \u5728$H^1$\u7c7b\u6bd4\u4e0b, \u73af\u7ed5\u6570$Lk(\\gamma, \\alpha)$\u53ef\u4ee5\u88ab\u770b\u6210\u4e00\u79cd$H_1 \\times H^1 \\longrightarrow \\mathbb{Z}_2$\u5f0f\u7684\u53cc\u7ebf\u6027\u6d4b\u91cf. \u5728\u8bba\u6587\u7b97\u6cd5\u4e2d, \u66f4\u76f4\u63a5\u7684\u8bf4\u6cd5\u662f: \u5b83\u8bb0\u5f55\u76ee\u6807\u66f2\u7ebf$\\gamma$\u4e0e\u6270\u52a8\u68c0\u6d4b\u66f2\u7ebf$\\alpha$\u4e4b\u95f4\u7684$\\mathbb{Z}_2$\u73af\u7ed5\u5173\u7cfb.<br \/>\n$\\\\$ \u6574\u4e2a&#8221;\u523b\u753b&#8221; \u8fc7\u7a0b\u7684\u95ed\u73af:<br \/>\n$\\\\$ $\\cdot$ 1) \u63d0\u53d6\u4e00\u6761\u66f2\u9762\u5faa\u73af$\\gamma_i$(\u5c5e\u4e8e$H_1$).<br \/>\n$\\\\$ $\\cdot$ 2) \u62ff\u51fa\u4e00\u6761\u6270\u52a8\u68c0\u6d4b\u66f2\u7ebf$\\alpha_j$(\u53ef\u7c7b\u6bd4\u4e3a\u6d4b\u91cf\u5c3a\u5b50).<br \/>\n$\\\\$ $\\cdot$ 3) \u8ba1\u7b97\u5b83\u4eec\u7684\u73af\u7ed5\u6570$Lk(\\gamma_i, \\alpha_j)$, \u5f97\u5230\u4e00\u4e2a\u6574\u6570.<br \/>\n$\\\\$ $\\cdot$ 4) \u8fd9\u4e2a\u6574\u6570\u5c31\u662f$\\gamma_i$\u5728$\\alpha_j$\u8fd9\u4e2a\u7ef4\u5ea6\u4e0a\u7684\u4ee3\u6570\u5750\u6807. \u904d\u5386\u6240\u6709$2g$\u628a\u5c3a\u5b50, \u5c31\u5f97\u5230\u4e86$\\gamma_i$\u7684\u5b8c\u6574\u5750\u6807\u5411\u91cf.<\/p>\n<p><strong>12.1.5 \u5b66\u672f\u603b\u7ed3<\/strong><\/p>\n<p>\u4ece\u8f85\u52a9\u7406\u89e3\u7684\u89d2\u5ea6\u770b, $H^1$\u548c\u5e9e\u52a0\u83b1\u5bf9\u5076\u63d0\u4f9b\u4e86\u4e00\u79cd\u89e3\u91ca: \u540c\u8c03\u7fa4$H_1$\u63cf\u8ff0\u7a7a\u95f4\u4e2d\u7684&#8221;\u6d1e&#8221;, \u800c\u73af\u7ed5\u6570\u53ef\u4ee5\u50cf\u6d4b\u91cf\u5668\u4e00\u6837\u628a\u66f2\u7ebf\u76f8\u5bf9\u4e8e\u4e00\u7ec4\u68c0\u6d4b\u66f2\u7ebf\u7684\u5173\u7cfb\u7f16\u7801\u6210$\\mathbb{Z}_2$\u5750\u6807. \u4f46\u56de\u5230\u8bba\u6587\u7b97\u6cd5\u672c\u8eab, \u5b83\u5e76\u4e0d\u9700\u8981\u663e\u5f0f\u6784\u9020\u4e0a\u540c\u8c03\u7c7b; \u5b83\u771f\u6b63\u8ba1\u7b97\u7684\u662f$\\{ \\gamma_i, \\overline{\\gamma_i} \\}$\u4e0e\u5176\u6270\u52a8\u7248\u672c$\\{ \\alpha_j \\}$\u4e4b\u95f4\u7684\u73af\u7ed5\u6570\u77e9\u9635, \u518d\u5229\u7528\u8fd9\u4e2a\u77e9\u9635\u7684\u53ef\u9006\u6027\u5b8c\u6210\u628a\u624b\u57fa\u548c\u96a7\u9053\u57fa\u7684\u5206\u79bb.<\/p>\n<p><strong>12.2 \u9644\u5f55B<\/strong><\/p>\n<p><strong>12.2.1 \u66f2\u9762\u7684\u6807\u51c6 CW \u590d\u5f62\u7ed3\u6784<\/strong><\/p>\n<p>\u4e8f\u683c\u4e3a $g$ \u7684\u53ef\u5b9a\u5411\u95ed\u66f2\u9762$M = \\Sigma_g$\u5177\u6709\u7ecf\u5178\u7684\u6807\u51c6\u591a\u8fb9\u5f62\u8868\u793a: \u53d6\u4e00\u4e2a$4g$\u8fb9\u5f62, \u6309\u5982\u4e0b\u65b9\u5f0f\u6807\u6ce8\u8fb9\u7684\u5b9a\u5411\u5e76\u7c98\u5408:$$W = a_1 b_1 a_1^{-1} b_1^{-1} \\; a_2 b_2 a_2^{-1} b_2^{-1} \\;\\cdots\\; a_g b_g a_g^{-1} b_g^{-1},$$\u7531\u6b64\u5f97\u5230\u4e00\u4e2a\u6807\u51c6\u7684CW\u590d\u5f62\u7ed3\u6784:<br \/>\n$\\\\$ $\\cdot$ 0-\u80de\u8154: 1\u4e2a, \u8bb0\u4e3a$e^0$(\u6240\u6709\u9876\u70b9\u5728\u7c98\u5408\u540e\u5f52\u4e3a\u4e00\u70b9);<br \/>\n$\\\\$ $\\cdot$ 1-\u80de\u8154: $2g$\u4e2a, \u8bb0\u4e3a$a_1$, $b_1$, $a_2$, $b_2$, $\\cdots$, $a_g$, $b_g$;<br \/>\n$\\\\$ $\\cdot$ 2-\u80de\u8154: 1\u4e2a, \u8bb0\u4e3a$e^2$(\u591a\u8fb9\u5f62\u5185\u90e8).<\/p>\n<p><strong>12.2.2 \u6574\u7cfb\u6570\u540c\u8c03\u7fa4$H_1(M;\\mathbb{Z})$\u7684\u8ba1\u7b97<\/strong><\/p>\n<p><strong>12.2.2.1 \u80de\u8154\u94fe\u590d\u5f62<\/strong><br \/>\n$$0 \\xrightarrow{\\;\\partial_3\\;} C_2 \\xrightarrow{\\;\\partial_2\\;} C_1 \\xrightarrow{\\;\\partial_1\\;} C_0 \\xrightarrow{\\;\\partial_0\\;} 0,$$\u5176\u4e2d\u5404\u94fe\u7fa4\u4e3a:$$C_0 = \\mathbb{Z}\\langle e^0 \\rangle \\cong \\mathbb{Z}, \\\\ C_1 = \\mathbb{Z}\\langle a_1, b_1, \\ldots, a_g, b_g \\rangle \\cong \\mathbb{Z}^{2g}, \\\\ C_2 = \\mathbb{Z}\\langle e^2 \\rangle \\cong \\mathbb{Z}.$$<br \/>\n<strong>12.2.2.2 \u8ba1\u7b97\u8fb9\u754c\u540c\u6001$\\partial_1$<\/strong><\/p>\n<p>\u6bcf\u4e2a1-\u80de\u8154$a_i$(\u6216$b_i$) \u7684\u8d77\u70b9\u548c\u7ec8\u70b9\u5728\u7c98\u5408\u540e\u5747\u4e3a\u540c\u4e00\u4e2a0-\u80de\u8154$e^0$, \u56e0\u6b64:$$\\partial_1(a_i) = e^0 &#8211; e^0 = 0, \\\\ \\partial_1(b_i) = e^0 &#8211; e^0 = 0 \\quad (1 \\leqslant i \\leqslant g).$$\u7ed3\u8bba: $\\partial_1 C_1 = 0$, \u5373$\\ker \\partial_1 = C_1 \\cong \\mathbb{Z}^{2g}$.<\/p>\n<p><strong>12.2.2.3 \u8ba1\u7b97\u8fb9\u754c\u540c\u6001$\\partial_2$<\/strong><\/p>\n<p>\u6839\u636e\u80de\u8154\u8fb9\u754c\u516c\u5f0f, 2-\u80de\u8154$e^2$\u7684\u8fb9\u754c\u7531\u7c98\u5408\u8bcd$W$\u4e2d\u54041-\u80de\u8154\u51fa\u73b0\u6b21\u6570\u7684\u4ee3\u6570\u548c(\u6307\u6570\u4e4b\u548c) \u51b3\u5b9a:$$\\partial_2(e^2) = \\sum_{j=1}^{g} \\left( \\deg_{a_j}(W) \\cdot a_j + \\deg_{b_j}(W) \\cdot b_j \\right),$$\u5176\u4e2d, $\\deg_x(W)$\u8868\u793a\u5b57\u6bcd$x$\u5728\u7c98\u5408\u8bcd$W$\u4e2d\u7684\u6307\u6570\u4e4b\u548c.<br \/>\n$\\\\$ \u5728$W$\u4e2d, \u5bf9\u6bcf\u4e2a$i = 1$, $\\ldots$, $g$:<br \/>\n$\\\\$ $\\cdot$ $a_i$\u4ee5\u6307\u6570+1\u51fa\u73b0\u4e00\u6b21, \u4ee5\u6307\u6570-1\u51fa\u73b0\u4e00\u6b21$\\implies \\deg_{a_i}(W) = (+1) + $$ (-1) = $$ 0$;<br \/>\n$\\\\$ $\\cdot$ $b_i$\u4ee5\u6307\u6570+1\u51fa\u73b0\u4e00\u6b21, \u4ee5\u6307\u6570-1\u51fa\u73b0\u4e00\u6b21$\\implies \\deg_{b_i}(W) = (+1) + $$ (-1) = $$ 0$.<br \/>\n$\\\\$ \u56e0\u6b64:$$\\partial_2(e^2) = \\sum_{i=1}^{g} (0 \\cdot a_i + 0 \\cdot b_i) = 0.$$\u7ed3\u8bba: $\\partial_2 = 0$, \u5373$\\operatorname{im} \\partial_2 = 0$.<\/p>\n<p><strong>12.2.2.4 \u540c\u8c03\u7fa4<\/strong><\/p>\n<p>$$\\boxed{H_1(M;\\mathbb{Z}) = \\frac{\\ker \\partial_1}{\\operatorname{im} \\partial_2} = \\frac{\\mathbb{Z}^{2g}}{0} \\cong \\mathbb{Z}^{2g}},$$\u8fd9\u662f\u4e00\u4e2a\u79e9\u4e3a$2g$\u7684\u81ea\u7531\u963f\u8d1d\u5c14\u7fa4, \u5176\u4e00\u7ec4\u6807\u51c6\u57fa\u4e3a$\\{[a_1], [b_1], \\ldots, [a_g], [b_g]\\}$.<\/p>\n<div style=\"text-align:right;\">$\\blacksquare$<\/div>\n<p><strong>12.2.3 $\\mathbb{Z}_2$ \u7cfb\u6570\u540c\u8c03\u7fa4$H_1(M;\\mathbb{Z}_2)$\u7684\u8ba1\u7b97<\/strong><\/p>\n<p>\u6709\u4e24\u79cd\u4e25\u683c\u7684\u65b9\u6cd5\u53ef\u4ee5\u5f97\u5230\u8fd9\u4e2a\u7ed3\u679c.<\/p>\n<p><strong>12.2.3.1 \u65b9\u6cd5\u4e00: \u76f4\u63a5\u8ba1\u7b97$\\mathbb{Z}_2$\u80de\u8154\u540c\u8c03<\/strong><\/p>\n<p>\u5c06\u94fe\u7fa4\u7684\u7cfb\u6570\u6362\u4e3a$\\mathbb{Z}_2$:$$C_k(M;\\mathbb{Z}_2) = C_k(M;\\mathbb{Z}) \\otimes_{\\mathbb{Z}} \\mathbb{Z}_2,$$\u56e0\u6b64:$$C_0(M;\\mathbb{Z}_2) \\cong \\mathbb{Z}_2, \\\\ C_1(M;\\mathbb{Z}_2) \\cong \\mathbb{Z}_2^{2g}, \\\\ C_2(M;\\mathbb{Z}_2) \\cong \\mathbb{Z}_2.$$\u8fb9\u754c\u540c\u6001\u4e3a\u539f\u8fb9\u754c\u540c\u6001\u6a212\u7ea6\u5316:$$\\overline{\\partial}_k = \\partial_k \\otimes \\operatorname{id}_{\\mathbb{Z}_2}.$$\u7531\u4e8e$\\partial_1 = 0$\u4e14$\\partial_2 = 0$, \u7ea6\u5316\u540e\u4ecd\u6709:$$\\overline{\\partial}_1 = 0, \\\\ \\overline{\\partial}_2 = 0.$$\u4e8e\u662f:$$\\boxed{H_1(M;\\mathbb{Z}_2) = \\frac{\\ker \\overline{\\partial}_1}{\\operatorname{im} \\overline{\\partial}_2} = \\frac{\\mathbb{Z}_2^{2g}}{0} \\cong (\\mathbb{Z}_2)^{2g}}.$$$\\mathbb{Z}_2$\u662f\u4e00\u4e2a\u57df, \u56e0\u6b64$(\\mathbb{Z}_2)^{2g}$\u81ea\u7136\u6210\u4e3a$\\mathbb{Z}_2$\u4e0a\u7684$2g$\u7ef4\u5411\u91cf\u7a7a\u95f4.<\/p>\n<div style=\"text-align:right;\">$\\blacksquare$<\/div>\n<p><strong>12.2.3.2 \u5229\u7528\u6cdb\u7cfb\u6570\u5b9a\u7406(UCT)<\/strong><\/p>\n<p>\u6cdb\u7cfb\u6570\u5b9a\u7406\u7ed9\u51fa\u77ed\u6b63\u5408\u5217:$$0 \\longrightarrow H_1(M;\\mathbb{Z}) \\otimes_{\\mathbb{Z}} \\mathbb{Z}_2 \\longrightarrow H_1(M;\\mathbb{Z}_2) \\longrightarrow \\operatorname{Tor}_1^{\\mathbb{Z}}(H_0(M;\\mathbb{Z}),\\, \\mathbb{Z}_2) \\longrightarrow 0.$$\u9010\u9879\u8ba1\u7b97:<br \/>\n$\\\\$ $\\cdot$ \u5de6\u9879: $H_1(M;\\mathbb{Z}) \\otimes \\mathbb{Z}_2 \\cong \\mathbb{Z}^{2g} \\otimes \\mathbb{Z}_2 \\cong (\\mathbb{Z} \\otimes \\mathbb{Z}_2)^{2g} \\cong \\mathbb{Z}_2^{2g}$;<br \/>\n$\\\\$ $\\cdot$ \u53f3\u9879: $H_0(M;\\mathbb{Z}) \\cong \\mathbb{Z}$(\u56e0\u4e3a$M$\u662f\u8fde\u901a\u7684\u95ed\u66f2\u9762). \u7531\u4e8e$\\mathbb{Z}$\u662f\u81ea\u7531\u963f\u8d1d\u5c14\u7fa4(\u6295\u5c04\u6a21), \u6709:$$\\operatorname{Tor}_1^{\\mathbb{Z}}(\\mathbb{Z},\\, \\mathbb{Z}_2) = 0.$$\u56e0\u6b64\u77ed\u6b63\u5408\u5217\u5316\u4e3a:$$0 \\longrightarrow \\mathbb{Z}_2^{2g} \\longrightarrow H_1(M;\\mathbb{Z}_2) \\longrightarrow 0.$$\u8fd9\u7ed9\u51fa\u5178\u8303\u540c\u6784:$$\\boxed{H_1(M;\\mathbb{Z}_2) \\cong \\mathbb{Z}_2^{2g} = (\\mathbb{Z}_2)^{2g}}.$$<\/p>\n<div style=\"text-align:right;\">$\\blacksquare$<\/div>\n<p><strong>12.2.4 \u603b\u7ed3\u4e0e\u9644\u6ce8<\/strong><\/p>\n<p>$\\\\$ $\\cdot$ $H_1(M;\\mathbb{Z})$: \u7ed3\u679c\u4e3a$\\mathbb{Z}^{2g}$, \u4ee3\u6570\u7ed3\u6784\u4e3a\u79e9$2g$\u7684\u81ea\u7531\u963f\u8d1d\u5c14\u7fa4;<br \/>\n$\\\\$ $\\cdot$ $H_1(M;\\mathbb{Z}_2)$: \u7ed3\u679c\u4e3a$(\\mathbb{Z}_2)^{2g}$, \u4ee3\u6570\u7ed3\u6784\u4e3a$\\mathbb{Z}_2$\u4e0a$2g$\u7ef4\u5411\u91cf\u7a7a\u95f4.<br \/>\n$\\\\$ \u5173\u952e\u6d1e\u5bdf: \u6574\u4e2a\u8bc1\u660e\u7684\u6838\u5fc3\u5728\u4e8e$\\partial_2 = 0$, \u800c\u8fd9\u6070\u597d\u6765\u6e90\u4e8e\u7c98\u5408\u8bcd$W$\u7684\u4ea4\u6362\u5b50\u7ed3\u6784$[a_i, b_i] = a_i b_i a_i^{-1} b_i^{-1}$\u2014\u2014 \u6bcf\u4e2a\u751f\u6210\u5143\u5728$W$\u4e2d\u4ee5+1\u548c-1\u5404\u51fa\u73b0\u4e00\u6b21, \u4f7f\u5f97\u6307\u6570\u4e4b\u548c(\u5373\u540c\u8c03\u610f\u4e49\u4e0b\u7684\u8fb9\u754c) \u6070\u597d\u4e3a\u96f6. \u8fd9\u6b63\u662f\u53ef\u5b9a\u5411\u66f2\u9762\u62d3\u6251\u7684\u4ee3\u6570\u53cd\u6620: \u540c\u8c03&#8221;\u770b\u4e0d\u5230&#8221; \u4ea4\u6362\u5b50, \u56e0\u6b64$2g$\u4e2a\u751f\u6210\u5143\u5728\u540c\u8c03\u4e2d\u5b8c\u5168\u72ec\u7acb\u5730\u5b58\u6d3b\u4e0b\u6765.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u57285\u670823\u65e5\u7ec8\u4e8e\u7ed3\u675f\u4e862026\u5e74\u4e0a\u534a\u5e74\u7cfb\u7edf\u67b6\u6784\u8bbe\u8ba1\u5e08\u8f6f\u8003, \u8003\u5b8c\u4ee5\u540e\u5fc3\u90fd\u51c9\u4e86, \u4eca\u5e74\u7684\u6cc4\u9898\u4e8b\u4ef6\u6682\u4e14\u4e0d\u63d0, \u81ea\u5df1 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2026\/06\/23\/interpretation_paper_efficient_computation_handle_tunnel_loops_via_reeb_graphs\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u300aAn Efficient Computation of Handle and Tunnel Loops via Reeb Graphs\u300b\u8bba\u6587\u89e3\u8bfb<\/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":[15,24,5],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/5258"}],"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=5258"}],"version-history":[{"count":111,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/5258\/revisions"}],"predecessor-version":[{"id":5350,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/5258\/revisions\/5350"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=5258"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=5258"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=5258"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}