{"id":1027,"date":"2025-10-24T10:22:39","date_gmt":"2025-10-24T08:22:39","guid":{"rendered":"http:\/\/blog.mathyuan.com\/?p=1027"},"modified":"2025-10-24T10:24:03","modified_gmt":"2025-10-24T08:24:03","slug":"heegaard-splitting","status":"publish","type":"post","link":"https:\/\/blog.mathyuan.com\/?p=1027","title":{"rendered":"\u4e09\u7ef4\u6d41\u5f62\u7684Heegaard\u5206\u89e3"},"content":{"rendered":"\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u672c\u6587\u4e2d\u6240\u79f0\u300c\u6d41\u5f62\u300d\u5747\u4e3a\u5e26\u8fb9\u62d3\u6251\u6d41\u5f62\u3002<\/p>\n<\/blockquote>\n\n\n\n<p>\u4e00\u79cd\u6700\u5e38\u7528\u7684\u5f97\u5230\u65b0\u4e09\u7ef4\u6d41\u5f62\u7684\u529e\u6cd5\u662f\uff1a\u53d6\u51fa\u4e24\u4e2a\u5b9a\u5411\u5e26\u8fb9\u4e09\u7ef4\u6d41\u5f62$M_1$\u548c$M_2$\uff0c\u5177\u6709\u540c\u80da\u7684\u8fb9\u754c$\\partial M_1\\approx\\partial M_2=\\Sigma_g$\uff0c\u5c06\u8fd9\u4e24\u4e2a\u6d41\u5f62\u6cbf\u7740\u8fb9\u754c\uff0c\u7528\u67d0\u4e2a\u540c\u80da$f:\\Sigma_g\\rightarrow \\Sigma_g$\u7c98\u63a5\u8d77\u6765\uff0c\u5f97\u5230\u95ed\u4e09\u7ef4\u6d41\u5f62<br>$$<br>M:=M_1\\cup_f M_2.<br>$$<br>\u5e76\u4e14\u9886\u7ed3\u5f15\u7406\u544a\u8bc9\u6211\u4eec\uff1a\u7c98\u63a5\u6620\u5c04\u7684\u540c\u75d5\u7c7b$[f]\\in\\mathrm{Mod}(\\Sigma_g)$\u51b3\u5b9a\u4e86\u4ea7\u51fa\u6d41\u5f62$M$\u7684\u540c\u80da\u578b\u3002\u56de\u8fc7\u5934\u6765\u95ee\uff1a\u4ec0\u4e48\u6837\u7684\u4e09\u7ef4\u6d41\u5f62$M_1$\u4ee5$\\Sigma_g$\u4e3a\u8fb9\u754c\uff1f\u81ea\u7136\u4f1a\u60f3\u5230\u5b9e\u5fc3\u73af\u9762\u53ca\u5176\u8fb9\u754c\u548c\uff0c\u8fd9\u5c31\u628a\u6240\u6709\u4fe1\u606f\u786e\u5b9a\u4e86\uff1a\u5982\u679c\u4e00\u4e2a\u4e09\u7ef4\u6d41\u5f62$M$\u53ef\u4ee5\u901a\u8fc7\u4e24\u4e2a\u76f8\u540c\u7684\u5b9e\u5fc3\u73af\u9762\uff0c\u7531$f\\in\\mathrm{Mod}(\\Sigma_g)$\u7c98\u63a5\u8fb9\u754c\u5f97\u5230\uff0c\u90a3\u4e48\u8fd9\u6837\u7684\u62c6\u5206\u65b9\u6cd5\u5c31\u79f0\u4e3a\u8fd9\u4e2a\u4e09\u7ef4\u6d41\u5f62$M$\u7684Heegaard\u5206\u89e3\u3002<\/p>\n\n\n\n<p>\u7136\u800c\uff0c\u5b9a\u4e49\u5e76\u4e0d\u4fdd\u8bc1\u6240\u6709\u7684\u4e09\u7ef4\u6d41\u5f62\u90fd\u80fd\u591f\u8fd9\u6837\u62c6\u5206\uff0c\u6b32\u884cHeegaard\u5206\u89e3\u4e8b\uff0c\u8fd8\u9700\u4e0b\u9762\u7684\u5b9a\u7406\u3002<\/p>\n\n\n\n<p><strong>Theorem.<\/strong> \u4efb\u610f\u4e09\u7ef4\u6d41\u5f62\u90fd\u6709Heegaard\u5206\u89e3\u3002<br><strong>Proof.<\/strong> \u6709\u7ed3\u679c\u8bf4\u300c\u4efb\u610f\u4e09\u7ef4\u6d41\u5f62\u90fd\u6709\u4e09\u89d2\u5256\u5206\u300d\uff0c\u6211\u4eec\u57fa\u4e8e\u8be5\u7ed3\u679c\u7ed9\u51fa\u4e00\u4e2a\u6784\u9020\u6027\u8bc1\u660e\u3002\u53d6\u5b9a\u4e09\u7ef4\u6d41\u5f62$M$\u7684\u4e00\u4e2a\u4e09\u89d2\u5256\u5206T\uff0c\u5c06T\u7684\u6240\u6709\u4fe1\u606f\u8fdb\u884c\u5982\u4e0b\u6539\u9020\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u9876\u70b9\uff1a\u66ff\u6362\u4e3a$D^3$;<\/li>\n\n\n\n<li>\u8fb9\uff1a\u66ff\u6362\u4e3a$D^2\\times I$\uff0c\u60f3\u8c61\u4e00\u4e2a\u957f\u800c\u7ec6\u7684\u5706\u67f1\uff0c\u4e24\u5934\u51f9\u9677\uff1b<\/li>\n\n\n\n<li>\u4e09\u89d2\u5f62\uff1a\u66ff\u6362\u4e3a$\\Delta\\times I$\uff0c\u60f3\u8c61\u4e00\u4e2a\u54ac\u4e86\u4e09\u53e3\u7684\u53d1\u7cd5\uff1b<\/li>\n\n\n\n<li>\u4e09\u68f1\u9525\uff1a\u66ff\u6362\u4e3a$D^3$\u3002<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"651\" src=\"http:\/\/blog.mathyuan.com\/wp-content\/uploads\/2025\/10\/1761294215263-1024x651.png\" alt=\"\" class=\"wp-image-1039\" srcset=\"https:\/\/blog.mathyuan.com\/wp-content\/uploads\/2025\/10\/1761294215263-1024x651.png 1024w, https:\/\/blog.mathyuan.com\/wp-content\/uploads\/2025\/10\/1761294215263-300x191.png 300w, https:\/\/blog.mathyuan.com\/wp-content\/uploads\/2025\/10\/1761294215263-768x488.png 768w, https:\/\/blog.mathyuan.com\/wp-content\/uploads\/2025\/10\/1761294215263.png 1245w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>\u5982\u6b64\uff0c\u53ef\u4ee4$H(T)$\u4e3a\u6240\u6709\u9876\u70b9\u548c\u8fb9\u7684\u66ff\u6362\u6784\u6210\u76842-handlebody\uff0c$H'(T)$\u4e3a\u4e09\u89d2\u5f62\u548c\u4e09\u68f1\u9525\u66ff\u6362\u6784\u6210\u7684handlebody\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u4e09\u89d2\u5256\u5206\u7ed9\u51fa\u4e86\u8fb9\u754c\u7c98\u63a5\u7684\u65b9\u6cd5\uff0c\u5982\u6b64\u5f97\u5230Heegaard\u5206\u89e3\u3002$\\square$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>\u4e00\u4e2a\u6d41\u5f62\u4e0a\u53ef\u80fd\u5b58\u5728\u5f88\u591a\u4e0d\u540c\u7684Heegaard\u5206\u89e3\u3002\u6bd4\u5982\uff0c\u7ed9\u5b9a\u4e00\u4e2aHeegaard\u5206\u89e3$M=H_g\\cup_f H_g$\uff0c\u8003\u8651$\\mathrm{Mod}(\\Sigma_g)\\rightarrow\\mathrm{Mod}(\\Sigma_{g+1})$\u81ea\u7136\u5d4c\u5165\uff0c\u4e8e\u662f\u53ef\u4ee5\u901a\u8fc7\u5e73\u51e1\u7c98\u63a5\u73af\u9762 (add unknotted 1-handle)\uff0c\u6765\u4f7f\u5f97Heegaard\u5206\u89e3\u7684\u4e8f\u683c\u589e\u957f<br>$$<br>M=H_{g+1}\\cup_{f&#8217;} H_{g+1}<br>$$<br>\u8fd9\u6837\u7684\u8fc7\u7a0b\u79f0\u4e3a\u7a33\u5b9a\u5316\u3002\u5982\u679c\u4e24\u4e2aHeegaard\u5206\u89e3\u80fd\u901a\u8fc7\u7a33\u5b9a\u5316\u5f97\u5230\u540c\u4e00\u4e2aHeegaard\u5206\u89e3\uff0c\u90a3\u4e48\u8fd9\u4e24\u4e2aHeegaard\u5206\u89e3\u79f0\u4f5c\u662f\u7a33\u5b9a\u7b49\u4ef7\u7684\u3002\u4e8b\u5b9e\u4e0a\uff1a<\/p>\n\n\n\n<p><strong>Theorem.<\/strong> \u540c\u80da\u7684\u4e09\u7ef4\u6d41\u5f62\u7684\u4e0d\u540cHeegaard\u5206\u89e3\u4e4b\u95f4\u662f\u7a33\u5b9a\u7b49\u4ef7\u7684\u3002<br><strong>Proof.<\/strong> \u8bc1\u660e\u5206\u4e3a\u4e24\u6b65<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u7531\u4e09\u89d2\u5256\u5206$T$\u548c$T&#8217;$\u5f97\u6765\u7684Heegaard\u5206\u89e3\u662f\u7a33\u5b9a\u7b49\u4ef7\u7684\uff1b<\/li>\n\n\n\n<li>\u5bf9\u4e8e\u4efb\u610fHeegaard\u5206\u89e3\uff0c\u5b58\u5728\u67d0\u4e2a\u4e09\u89d2\u5256\u5206$T$\uff0c\u4f7f\u5f97\u8be5Heegaard\u5206\u89e3\u4e0e$T$\u7ed9\u51fa\u7684Heegaard\u5206\u89e3\u65f6\u7a33\u5b9a\u7b49\u4ef7\u7684\u3002<\/li>\n<\/ol>\n\n\n\n<p>\u7b2c\u4e00\u6b65\u7684\u8bc1\u660e\u4f9d\u9760\u300c\u4efb\u610f\u4e24\u4e2a\u4e09\u89d2\u5256\u5206\u53ef\u4ee5\u901a\u8fc7\u91cd\u5fc3\u5256\u5206 (barycentric subdivision)\uff0c\u627e\u5230\u5171\u540c\u7684\u5b50\u4e09\u89d2\u5256\u5206\u300d\uff0c\u5e76\u4e14\u91cd\u5fc3\u5256\u5206\u53ef\u4ee5\u7528\u300c\u7c98\u63a5unknotted 1-handle\u300d\u5b9e\u73b0\u3002<\/p>\n\n\n\n<p>\u7b2c\u4e8c\u6b65\u7684\u8bc1\u660e\uff1a\u5148\u53d6\u5b9a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Heegaard\u5206\u89e3$M=H_g\\cup H&#8217;_g$\uff1b<\/li>\n\n\n\n<li>$\\Gamma$\uff1a1-\u67c4\u4f53$H_g$\u7684\u4e00\u4e2a\u300c\u8f74\u5fc3\u56fe\u300d\uff0c\u90a3\u4e48$\\Gamma$\u5c31\u662f\u4e00\u4e2a\u53ea\u6709\u4e00\u4e2a\u9876\u70b9\u548c$g$\u6761\u95ed\u66f2\u7ebf\u7684\u56fe\uff1b<\/li>\n\n\n\n<li>$T$\uff1a$M$\u7684\u4e09\u89d2\u5256\u5206\uff0c\u4f7f\u5f97$H_g$\u548c$H_g&#8217;$\u90fd\u662f\u5b50\u590d\u5f62\uff1b<\/li>\n<\/ul>\n\n\n\n<p>\u4ee4$\\tau=T|_{H_g}$\u548c$\\tau&#8217;=T|_{H&#8217;_g}$\uff0c\u90a3\u4e48\u901a\u8fc7\u9002\u5f53\u53d6\u5b50\u5256\u5206\uff0c\u53ef\u4ee5\u4f7f\u5f97$\\tau$\u6ee1\u8db3\u4e0b\u9762\u6761\u4ef6(a)\uff0c$\\tau&#8217;$\u6ee1\u8db3\u4e0b\u9762\u6761\u4ef6(b)\u3002<\/p>\n\n\n\n<p>\u200b (a) $\\tau^{(1)}$\u7684\u4e09\u7ef4\u52a0\u539a\uff0c\u53ef\u4ee5\u7531$\\Gamma$\u7684\u4e09\u7ef4\u52a0\u539a\uff0c\u4ec5\u901a\u8fc7\u7c98\u63a5unknotted handles\u5f97\u5230\uff1b<br>\u200b (b) $\\tau&#8217;^{(1)}$\u7684\u4e09\u7ef4\u52a0\u539a\uff0c\u53ef\u4ee5\u7531$\\tau&#8217;^{(1)}|_{\\partial H&#8217;_g}$\u7684\u4e09\u7ef4\u52a0\u539a\uff0c\u4ec5\u901a\u8fc7\u7c98\u63a5unknotted handles\u5f97\u5230\uff1b<\/p>\n\n\n\n<p>\u8bb0\u300c\u52a0\u539a\u300d\u4e3a$U(\\cdot)$\uff0c\u56e0\u4e3a$\\tau&#8217;$\u6ee1\u8db3\u6761\u4ef6(b)\uff0c\u5e76\u4e14$\\tau&#8217;$\u4e0e$\\tau$\u5728\u8fb9\u754c\u4e0a\u76f8\u7b49\uff0c\u6240\u4ee5<br>$$<br>\\begin{aligned}<br>U(\\tau&#8217;^{(1)})&amp;=U(\\tau&#8217;^{(1)}|_{\\partial H_g})\\cup\\text{unknotted 1-handles}\\\\&amp;=U(\\tau^{(1)}|_{\\partial H_g})\\cup\\text{unknotted 1-handles}\\\\<br>\\end{aligned}<br>$$<br>\u53e6\u5916\uff0c$\\tau$\u53c8\u6ee1\u8db3\u6761\u4ef6(a)\uff0c\u4e14$\\tau$\u548c$\\tau&#8217;$\u5408\u8d77\u6765\u5f97\u5230\u6574\u4e2a$T$\uff0c\u6240\u4ee5<br>$$<br>\\begin{aligned}<br>H(T)=U(T^{(1)})&amp;=U(\\tau^{(1)})\\cup\\text{{unknotted 1-handles}}\\\\<br>&amp;=U(\\Gamma)\\cup\\text{\u66f4\u591aunknotted 1-handles}\\\\<br>&amp;=H\\cup\\text{\u66f4\u591aunknotted 1-handles}<br>\\end{aligned}<br>$$<br>\u5373\u662f\u8bf4\u7ed9\u5b9a\u7684Heegaard\u5206\u89e3\uff0c\u548c$T$\u7ed9\u51fa\u7684Heegaard\u5206\u89e3\u7a33\u5b9a\u7b49\u4ef7\uff0c\u8fd9\u5c31\u5b8c\u6210\u4e86\u8bc1\u660e\u3002$\\square$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n","protected":false},"excerpt":{"rendered":"<p>Heegaard\u5206\u89e3\u5b58\u5728\u4e4b\u7406\u7531 <a class=\"more-link\" href=\"https:\/\/blog.mathyuan.com\/?p=1027\">\u7ee7\u7eed\u9605\u8bfb <span class=\"screen-reader-text\">  \u4e09\u7ef4\u6d41\u5f62\u7684Heegaard\u5206\u89e3<\/span><span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[17,8],"class_list":["post-1027","post","type-post","status-publish","format-standard","hentry","category-notes","tag-17","tag-8"],"_links":{"self":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts\/1027","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1027"}],"version-history":[{"count":12,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts\/1027\/revisions"}],"predecessor-version":[{"id":1041,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts\/1027\/revisions\/1041"}],"wp:attachment":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1027"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1027"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1027"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}