{"id":2374,"date":"2022-10-14T13:55:29","date_gmt":"2022-10-14T05:55:29","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=2374"},"modified":"2025-02-26T11:14:45","modified_gmt":"2025-02-26T03:14:45","slug":"deck_transformation_mark","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/10\/14\/deck_transformation_mark\/","title":{"rendered":"\u590d\u8fed\u53d8\u6362\u6ce8\u8bb0"},"content":{"rendered":"<p><script type=\"text\/javascript\" async src=\"https:\/\/www.caiqinyi.cn\/wp-content\/MathJax\/MathJax.js?config=TeX-AMS_CHTML\">\n<\/script><br \/>\n<script type=\"text\/x-mathjax-config\">\n    MathJax.Hub.Config({\n        tex2jax: {inlineMath: [['$','$']]},\n        TeX: {equationNumbers: {autoNumber: [\"AMS\"], useLabelIds: true}},\n        \"HTML-CSS\": {linebreaks: {automatic: true}},\n        SVG: {linebreaks: {automatic: true}}\n    });\n<\/script><\/p>\n<p>\u7ec8\u4e8e\u57fa\u672c\u5b8c\u6210\u4e86\u590d\u8fed\u53d8\u6362\u4e00\u8282\u5185\u5bb9\u7684\u5b66\u4e60\u5566, \u540c\u65f6\u672c\u6587\u5e94\u8be5\u4e5f\u662f\u6211\u5b66\u4e60\u5305\u5fd7\u5f3a\u8001\u5e08\u7684\u300a\u70b9\u96c6\u62d3\u6251\u4e0e\u4ee3\u6570\u62d3\u6251\u5f15\u8bba\u300b\u4e00\u4e66\u7684\u6700\u540e\u4e00\u7bc7\u7b14\u8bb0~ \u4eca\u5929\u8bf7\u4e86\u4e2a\u5047, \u4e0b\u5348\u8981\u53bb\u5b81\u6ce2, \u56e0\u6b64\u5148\u628a\u6587\u7ae0\u53d1\u5e03\u51fa\u6765, \u8def\u4e0a\u518d\u68c0\u67e5\u4e00\u4e0b\u6587\u7ae0\u6709\u6ca1\u6709\u4ec0\u4e48\u95ee\u9898~<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/math.stackexchange.com\/questions\/3015280\/why-is-a-space-homeomorphic-to-the-orbit-space-of-its-universal-cover-by-its-dec\">Why is a space homeomorphic to the orbit space of its universal cover by its deck transformations?<\/a><br \/>\n2. <a href=\"https:\/\/math.stackexchange.com\/questions\/183086\/the-double-cover-of-klein-bottle?rq=1\">The double cover of Klein bottle<\/a><br \/>\n3. <a href=\"https:\/\/www2.math.upenn.edu\/~qze\/math601s20\/hw6.pdf\">MATH 601 ALGEBRAIC TOPOLOGY HW 6 SELECTED SOLUTIONS SKETCH\/HINT<\/a><br \/>\n4. <a href=\"https:\/\/math.stackexchange.com\/questions\/904762\/how-to-show-the-fundamental-group-of-torus-is-abelian-in-a-homotopic-way\">How to show the fundamental group of torus is abelian in a homotopic way?<\/a><br \/>\n5. <a href=\"https:\/\/www.youtube.com\/watch?v=nLcr-DWVEto\">The fundamental Group of the Torus is abelian<\/a><\/p>\n<p>1. \u590d\u8fed\u53d8\u6362\u662f\u4e00\u79cd\u95ed\u9053\u8def\u7c7b\u5bf9\u5e94\u7684\u53d8\u6362(\u53ef\u53c2\u8003<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/08\/21\/winding_number_non-closed_curve\/\">\u975e\u95ed\u66f2\u7ebf\u4e0a\u7684\u5708\u6570<\/a>) \u7684\u63a8\u5e7f. \u4efb\u53d6\u4e00\u4e2a\u5747\u5300\u590d\u8fed\u90bb\u57df$U$, \u6211\u4eec\u77e5\u9053$p^{-1}(U)$\u662f\u5168\u7a7a\u95f4\u4e2d\u7684\u4e00\u5806\u540c\u80da\u4e8e$U$\u5e76\u4e14\u4e24\u4e24\u4e0d\u76f8\u4ea4\u7684\u5b50\u96c6\u7684\u5e76\u96c6, \u5982\u679c\u628a\u8fd9\u4e9b\u5b50\u96c6\u60f3\u8c61\u6210\u4e00\u53e0\u6251\u514b\u724c\u7684\u8bdd, \u590d\u8fed\u53d8\u6362\u5c31\u50cf\u662f\u4e00\u4e2a\u6d17\u724c\u7684\u64cd\u4f5c. \u5173\u4e8e\u590d\u8fed\u53d8\u6362\u7684\u6700\u5b9e\u7528\u7684\u7ed3\u8bba\u5c31\u662f: \u6cdb\u590d\u8fed\u7a7a\u95f4\u7684\u590d\u8fed\u53d8\u6362\u7fa4\u4e0e\u5e95\u7a7a\u95f4\u7684\u57fa\u672c\u7fa4\u540c\u6784.<br \/>\n$\\\\$ \u590d\u8fed\u53d8\u6362\u53ef\u4ee5\u89e3\u51b3\u628a\u57fa\u672c\u7fa4\u548c\u4e00\u6761\u7ea4\u7ef4\u4e2d\u7684\u70b9\u5bf9\u5e94\u8d77\u6765\u540e\u4e0d\u80fd\u8868\u73b0\u51fa\u7fa4\u7684\u7ed3\u6784(\u5373\u9053\u8def\u7684\u4e58\u6cd5) \u7684\u95ee\u9898, \u5373\u628a\u57fa\u672c\u7fa4\u548c\u590d\u8fed\u53d8\u6362\u7fa4\u4e2d\u7684\u5143\u7d20\u5bf9\u5e94\u8d77\u6765\u540e, \u9053\u8def\u7684\u4e58\u6cd5\u53ef\u4ee5\u5bf9\u5e94\u5230\u6620\u5c04\u7684\u590d\u5408. \u5982\u6b64\u4e00\u6765, \u57fa\u672c\u7fa4\u7684\u7ed3\u6784\u4e5f\u5c31\u80fd\u591f\u66f4\u4e3a\u6e05\u6670\u5730\u8868\u73b0\u51fa\u6765. \u6b64\u5916, \u4f7f\u7528\u590d\u8fed\u53d8\u6362\u8868\u73b0\u57fa\u672c\u7fa4\u7684\u7ed3\u6784, \u4e5f\u4e0d\u53d7\u5168\u7a7a\u95f4\u4e2d\u57fa\u70b9\u7684\u9009\u62e9\u65b9\u5f0f\u7684\u5f71\u54cd.<br \/>\n$\\\\$ \u5bf9\u4e8e\u6b63\u5219\u590d\u8fed\u6620\u5c04\u6765\u8bf4, \u590d\u8fed\u53d8\u6362\u7fa4\u53ef\u4ee5\u548c\u4e00\u6839\u7ea4\u7ef4\u4e2d\u7684\u70b9\u5efa\u7acb\u4e00\u4e00\u5bf9\u5e94(\u628a\u6bcf\u4e2a$h$\u5bf9\u5e94\u5230$h(e)$). \u524d\u9762\u6211\u4eec\u8bb2\u8fc7\u6cdb\u590d\u8fed\u7a7a\u95f4\u7684\u5e95\u7a7a\u95f4\u7684\u57fa\u672c\u7fa4\u548c\u4e00\u6839\u7ea4\u7ef4\u4e2d\u7684\u70b9\u4e00\u4e00\u5bf9\u5e94, \u73b0\u5728\u5c31\u53ef\u4ee5\u7528\u590d\u8fed\u53d8\u6362\u7fa4\u6765\u4ee3\u66ff\u90a3\u6839\u7ea4\u7ef4\u53bb\u5efa\u7acb\u4e00\u4e00\u5bf9\u5e94, \u800c\u4e14\u8fd9\u4e2a\u5bf9\u5e94\u662f\u7fa4\u7684\u540c\u6784. \u6ce8\u610f, \u4e0d\u6b63\u5219\u7684\u590d\u8fed\u7a7a\u95f4\u5176\u5b9e\u8fd8\u662f\u5f88\u5e38\u89c1\u7684. \u6b63\u5219\u590d\u8fed\u53ea\u662f\u770b\u4e0a\u53bb\u6700\u6574\u9f50\u3001\u6700\u597d\u7528\u7684\u4e00\u7c7b\u590d\u8fed\u800c\u5df2.<\/p>\n<p>2. \u7c7b\u6bd4\u5706\u5468\u4e0a\u95ed\u9053\u8def\u7c7b\u7684\u5708\u6570\u5b9a\u4e49, \u53ef\u4ee5\u5b9a\u4e49\u5e7f\u4e49\u7684&#8221;\u4f9d\u5708\u6570\u5e73\u79fb&#8221; \u89c4\u5219\u4e3a\u5982\u4e0b\u6620\u5c04:$$q: \\pi_1(B, b) \\to \\mathscr{D}(E, p).$$\u5982\u679c\u95ed\u9053\u8def\u7c7b$\\alpha \\in \\pi_1(B, b)$\u7684\u4ece$e$\u51fa\u53d1\u7684\u63d0\u5347\u4ee5$e&#8217;$\u4e3a\u7ec8\u70b9, \u5219\u53d6$q(\\alpha) $$ = h_{e&#8217;}$(\u628a$e$\u9001\u5230$e&#8217;$). \u8bbe$p: E \\to B$\u662f\u6b63\u5219\u590d\u8fed\u6620\u5c04, \u53d6\u5b9a$e \\in E$\u53ca$b $$ = p(e)$. \u5219\u6620\u5c04$q$\u662f\u4e00\u4e2a\u6ee1\u540c\u6001.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u9996\u5148\u8bc1\u660e$q$\u662f\u6ee1\u5c04. \u4efb\u53d6\u590d\u8fed\u53d8\u6362$h \\in \\mathscr{D}(E, p)$, \u4e0d\u59a8\u8bbe$h(e) = $$ e&#8217;$, \u53d6$E$\u4e2d\u4e00\u6761\u4ece$e$\u5230$e&#8217;$\u7684\u9053\u8def\u7c7b$\\alpha^\\uparrow$, \u518d\u53d6$\\alpha = p(\\alpha^\\uparrow)$, \u5219\u663e\u7136$\\alpha \\in $$ \\pi_1(B, b)$, \u5e76\u4e14$q(\\alpha) = h$.<br \/>\n$\\\\$ \u518d\u6765\u8bc1\u660e$q$\u662f\u540c\u6001. \u4efb\u53d6$B$\u4e2d\u7684\u4e24\u6761\u4ee5$b$\u4e3a\u57fa\u70b9\u7684\u95ed\u9053\u8def\u7c7b$\\alpha, \\beta$, \u8bbe\u5b83\u4eec\u7684\u4ee5$e$\u4e3a\u8d77\u70b9\u7684\u63d0\u5347\u5206\u522b\u4e3a$\\alpha^\\uparrow$\u548c$\\beta^\\uparrow$. \u8003\u8651\u590d\u8fed\u53d8\u6362$h = q(\\alpha)$, \u5219\u9053\u8def\u7c7b$h(\\beta^\\uparrow)$\u4ee5$\\alpha^\\uparrow$\u7684\u7ec8\u70b9$h(e)$\u4e3a\u8d77\u70b9, \u5e76\u4e14\u4e5f\u662f$\\beta$\u7684\u63d0\u5347.<br \/>\n$\\\\$ \u4e8e\u662f$\\alpha^\\uparrow h(\\beta^\\uparrow)$\u5c31\u662f$\\alpha \\beta$\u7684\u4ee5$e$\u4e3a\u8d77\u70b9\u7684\u63d0\u5347\u9053\u8def\u7c7b. \u590d\u8fed\u53d8\u6362$q(\\alpha \\beta)$\u5c31\u662f\u628a$e$\u53d8\u5230\u8be5\u9053\u8def\u7c7b\u7ec8\u70b9\u7684\u590d\u8fed\u53d8\u6362. \u6ce8\u610f, $q(\\beta)$\u628a$e$\u53d8\u5230$\\beta^\\uparrow$\u7684\u7ec8\u70b9, \u56e0\u6b64$q(\\alpha) \\circ q(\\beta)$\u628a$e$\u53d8\u5230$h(\\beta^\\uparrow)$\u7684\u7ec8\u70b9, \u8fd9\u548c$e$\u5728$q(\\alpha \\beta)$\u4e0b\u7684\u50cf\u662f\u4e00\u6837\u7684. \u56e0\u6b64$q(\\alpha \\beta) = q(\\alpha) q(\\beta)$, \u8bf4\u660e$q$\u662f\u540c\u6001.<\/p>\n<p>3. \u5982\u679c$p: E \\to B$\u662f\u4e2a\u6b63\u5219\u590d\u8fed, $\\mathscr{D}(E, p)$\u662f\u76f8\u5e94\u7684\u590d\u8fed\u53d8\u6362\u7fa4, \u5219$B$\u5c31\u53ef\u4ee5\u770b\u6210$E$\u5173\u4e8e$\\mathscr{D}(E, p)$\u8fd9\u4e2a\u7fa4\u4f5c\u7528\u7684\u8f68\u9053\u7a7a\u95f4.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/10\/orbit_space_and_deck_transformations.png\" alt=\"\" width=\"668\" height=\"263\" class=\"aligncenter size-full wp-image-2391\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/10\/orbit_space_and_deck_transformations.png 668w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/10\/orbit_space_and_deck_transformations-300x118.png 300w\" sizes=\"(max-width: 668px) 100vw, 668px\" \/><\/p>\n<p>4. (1) \u5199\u51fa\u4e00\u4e2a\u4ece$E^2 \\backslash \\{ 0 \\}$\u5230\u81ea\u8eab\u76844\u91cd\u6b63\u5219\u590d\u8fed.<br \/>\n$\\\\$ (2) \u8bc1\u660e$E^2 \\backslash \\{ 0 \\}$\u4e0a\u7684\u4efb\u4f55\u590d\u8fed\u6620\u5c04\u90fd\u662f\u6b63\u5219\u7684.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> (1) \u91c7\u7528\u590d\u5750\u6807, \u7136\u540e\u8003\u8651\u6620\u5c04$f(z) = z^4$. \u9700\u8981\u6ce8\u610f\u7684\u662f, \u6b64\u5904\u91c7\u7528\u590d\u5750\u6807\u7684\u505a\u6cd5\u5e76\u4e0d\u610f\u5473\u7740$E^2 \\backslash \\{ 0 \\}$\u4e0e$S^1$\u540c\u80da. \u4e8b\u5b9e\u4e0a, \u5f53\u4ece$S^1$\u4e0a\u6316\u53bb\u4e24\u70b9\u540e, \u5f97\u5230\u7684\u62d3\u6251\u7a7a\u95f4\u662f\u4e0d\u8fde\u901a\u7684; \u800c\u4ece$E^2 \\backslash \\{ 0 \\}$\u4e0a\u6316\u53bb\u4e24\u70b9\u540e, \u5f97\u5230\u7684\u62d3\u6251\u7a7a\u95f4\u4f9d\u7136\u662f\u8fde\u901a\u7684, \u6545$E^2 \\backslash \\{ 0 \\}$\u4e0e$S^1$\u5e76\u4e0d\u540c\u80da. $S^1$\u4e3a$E^2 \\backslash \\{ 0 \\}$\u7684\u5f3a\u5f62\u53d8\u6536\u7f29\u6838, \u4e8c\u8005\u4ec5\u4ec5\u662f\u540c\u4f26\u7b49\u4ef7\u7684(\u5373\u57fa\u672c\u7fa4\u662f\u540c\u80da\u7684).<br \/>\n$\\\\$ (2) $\\pi_1(E^2 \\backslash \\{ 0 \\}) \\cong Z$\u662f\u4ea4\u6362\u7fa4, \u5b83\u7684\u4efb\u4f55\u5b50\u7fa4\u90fd\u662f\u6b63\u89c4\u5b50\u7fa4. \u6545\u53d6\u4efb\u610f\u590d\u8fed\u6620\u5c04$p: E \\to E^2 \\backslash \\{ 0 \\}$, $p_\\pi(\\pi_1(E, e))$\u5747\u4e3a$\\pi_1(E^2 \\backslash \\{ 0 \\})$\u7684\u6b63\u89c4\u5b50\u7fa4. \u7531\u6b63\u5219\u7684\u590d\u8fed\u6620\u5c04\u7684\u5b9a\u4e49\u53ef\u77e5, \u547d\u9898\u5f97\u8bc1.<\/p>\n<p>5. (1) \u63cf\u8ff0\u4e00\u4e0bKlein\u74f6\u7684\u6cdb\u590d\u8fed\u6620\u5c04.<br \/>\n$\\\\$ (2) \u8bc1\u660e\u5728Klein\u74f6\u4e0a\u4efb\u53d6\u4e00\u6761\u95ed\u9053\u8def$a$, \u5982\u679c$a^2$\u96f6\u4f26, \u5219$a$\u4e00\u5b9a\u4e5f\u96f6\u4f26.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> (1) \u5bf9\u4e8e$\\forall c > 0$, \u5728$E^2$\u4e0a\u5b9a\u4e49\u4e00\u4e2a\u7b49\u4ef7\u5173\u7cfb$\\sim$, \u4f7f\u5f97$$(x, y) \\sim (x, y + kc), k \\in Z, \\\\ (x, y) \\sim (x + \\ell, (-1)^\\ell y), \\ell \\in Z.$$\u8be5\u7b49\u4ef7\u5173\u7cfb$\\sim$\u4ee5\u77e9\u5f62$[0, 1] \\times [0, c]$\u4e3a\u57fa\u672c\u57df, \u5546\u7a7a\u95f4$E^2 \/ \\sim$\u540c\u80da\u4e8e\u4e00\u4e2a&#8221;\u957f\u5ea6&#8221; \u4e3a1, &#8220;\u5bbd\u5ea6&#8221; \u4e3a$c$\u7684Klein\u74f6$K$, \u5e76\u4e14\u7c98\u5408\u6620\u5c04$p: E^2 \\to E^2 \/ \\sim$\u662f\u6cdb\u590d\u8fed\u6620\u5c04.<br \/>\n$\\\\$ \u6b64\u5916, \u6211\u4eec\u77e5\u9053$\\pi_1(K) = \\left \\langle a, b | aba^{-1}b \\right \\rangle$. \u5bf9\u4e8e$\\forall a, b \\in \\pi_1(K)$, \u5206\u522b\u5bf9\u5e94\u7684\u590d\u8fed\u53d8\u6362\u4e3a$$h_a : (x, y) \\mapsto (x + \\ell, (-1)^\\ell y), \\ell \\in Z, \\\\ h_b : (x, y) \\mapsto (x, y + kc), k \\in Z.$$(2) \u5bf9\u4e8e\u7b2c(1)\u5c0f\u95ee\u4e2d\u5b9a\u4e49\u7684\u6cdb\u590d\u8fed\u6620\u5c04$p$, \u590d\u8fed\u53d8\u6362\u5747\u5177\u6709$$h_{mn} : E^2 \\to E^2, (x, y) \\mapsto (x + m, (-1)^m y + n)$$\u7684\u5f62\u5f0f($m, n \\in Z$). \u5e94\u7528\u4e66\u4e0aP258\u7684\u5b9a\u74065.6.1, \u53ef\u77e5$\\mathscr{D}(E, p) \\cong $$ \\pi_1(K)$, \u6545$\\left \\langle a \\right \\rangle \\in \\pi_1(K)$\u5bf9\u5e94\u4e8e$h_{mn} \\in \\mathscr{D}(E, p)$, $\\left \\langle a^2 \\right \\rangle = \\left \\langle a \\right \\rangle \\left \\langle a \\right \\rangle$\u5bf9\u5e94\u4e8e$h_{mn}^2$, $\\pi_1(K)$\u4e2d\u96f6\u4f26\u7684\u9053\u8def\u7c7b\u5bf9\u5e94\u4e8e$id_\\mathscr{D}$, \u4ece\u800c$a^2$\u96f6\u4f26\u610f\u5473\u7740$h_{mn}^2 = id_\\mathscr{D}$. \u5bf9\u4e8e$\\forall (x, y) \\in E^2$,$$h_{mn}^2((x, y)) = h_{mn}((x + m, (-1)^m y + n)) \\\\ = (x + 2m, (-1)^m ((-1)^m y + n) + n) \\\\ = (x + 2m, (-1)^{2m} y + (-1)^m n + n) = (x, y),$$\u6545$2m = 0 \\Rightarrow m = 0$, $(-1)^{2m} y + (-1)^m n + n = y \\Rightarrow $$ n = 0$, \u4ece\u800c\u6211\u4eec\u6709$h_{mn} = id_\\mathscr{D}$. \u7efc\u4e0a\u6240\u8ff0, \u547d\u9898\u5f97\u8bc1.<\/p>\n<p>6. \u8bbe$p: E \\to B$\u662f\u6b63\u5219\u590d\u8fed\u6620\u5c04. \u5982\u679c$f_1 : X \\to E$, $f_2 : X \\to E$\u90fd\u662f$g: X $$ \\to B$\u7684\u63d0\u5347, \u8bc1\u660e\u5b58\u5728\u590d\u8fed\u53d8\u6362$h: E \\to E$, \u4f7f\u5f97$f_2 = $$ h \\circ f_1$.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u53d6\u5b9a$x \\in X$, \u8bbe$f_1(x) = e_1$, $f_2(x) = e_2$, \u5219\u5b58\u5728\u590d\u8fed\u53d8\u6362$h$\u4f7f\u5f97$h(e_1) = $$ e_2$. \u56e0\u4e3a$f_1$\u662f$g: X \\to B$\u7684\u63d0\u5347, \u6545$p \\circ f_1 = g$, \u4ece\u800c\u6211\u4eec\u6709$$p \\circ (h \\circ f_1) = (p \\circ h) \\circ f_1 = p \\circ f_1 = g,$$\u5373$h \\circ f_1$\u4ea6\u4e3a$g: X \\to B$\u7684\u63d0\u5347. \u53c8$f_2$\u4e0e$h \\circ f_1$\u5728$x$\u4e0a\u53d6\u503c\u76f8\u540c, \u7531\u590d\u8fed\u7a7a\u95f4\u4e2d\u7684\u63d0\u5347\u7684\u552f\u4e00\u6027\u53ef\u77e5, $f_2 = h \\circ f_1$, \u547d\u9898\u5f97\u8bc1.<\/p>\n<p>7. \u5199\u51fa\u73af\u9762$T^2$\u4e0a\u7684\u4e24\u4e2a\u4e0d\u540c\u4f26\u7684\u81ea\u540c\u80da, \u5e76\u8bc1\u660e\u5b83\u4eec\u786e\u5b9e\u76f8\u4e92\u4e0d\u540c\u4f26.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u91c7\u7528\u590d\u5750\u6807, \u5373\u8bbe$T^2 = \\{ (z_1, z_2) \\in C^2 | |z_1| = |z_2| = 1 \\}$. \u8003\u8651\u81ea\u540c\u80da$f: $$ (z_1, z_2) \\mapsto (z_2, z_1)$\u4ee5\u53ca$id_{T^2}$. \u5982\u679c\u5b83\u4eec\u540c\u4f26, \u5219\u8bf1\u5bfc\u7684$\\pi_1(T^2, (1, 1))$\u4e0a\u7684\u81ea\u540c\u6784$f_\\pi$\u548c$id_\\pi$\u53ea\u5dee\u4e00\u4e2a\u5171\u8f6d. \u7531\u4e8e$\\pi_1(T^2)$\u662f\u4ea4\u6362\u7fa4(\u540c\u80da\u4e8e$Z \\times Z$), \u56e0\u6b64$f_\\pi = $$ id_\\pi$. \u800c\u5f53\u4ee5$((1, 0), (0, 1))$\u4e3a\u57fa\u70b9\u7684\u95ed\u9053\u8def\u7c7b\u4e2d\u5b58\u5728$z_1, z_2$\u5206\u91cf\u4e0d\u540c\u7684\u70b9\u65f6, \u5176\u5728$f_\\pi$\u4e0e$id_\\pi$\u4e0b\u7684\u50cf\u662f\u4e0d\u540c\u7684(\u6b64\u65f6, \u7ecf$f_\\pi$\u4f5c\u7528\u540e\u7684\u95ed\u9053\u8def\u7c7b\u65b9\u5411\u4e0e\u539f\u5148\u7684\u95ed\u9053\u8def\u7c7b\u65b9\u5411\u4e00\u822c\u662f\u76f8\u53cd\u7684), \u77db\u76fe.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u7ec8\u4e8e\u57fa\u672c\u5b8c\u6210\u4e86\u590d\u8fed\u53d8\u6362\u4e00\u8282\u5185\u5bb9\u7684\u5b66\u4e60\u5566, \u540c\u65f6\u672c\u6587\u5e94\u8be5\u4e5f\u662f\u6211\u5b66\u4e60\u5305\u5fd7\u5f3a\u8001\u5e08\u7684\u300a\u70b9\u96c6\u62d3\u6251\u4e0e\u4ee3\u6570\u62d3\u6251\u5f15\u8bba\u300b\u4e00\u4e66\u7684\u6700 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/10\/14\/deck_transformation_mark\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u590d\u8fed\u53d8\u6362\u6ce8\u8bb0<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/2374"}],"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=2374"}],"version-history":[{"count":72,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/2374\/revisions"}],"predecessor-version":[{"id":3621,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/2374\/revisions\/3621"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2374"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2374"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2374"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}